Embed Size (px)
Citation preview
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
1
GEF1100ndashAutumn 2014 16092014
Prof Dr Kirstin Kruumlger (MetOs UiO since October 2013)
Research interests
bull Stratospheric ozone bull Dynamics of the stratosphere bull Influence of volcanic eruptions on climate earth system bull Stratosphere-ocean interactions
Who am I
Prof Dr Kirstin Kruumlger
kkruegergeouiono
climate modelling
observations
ship expeditions
Volcanic excursion
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
3 Addition not in book
Motivation 1 Motivation
Marshall and Plumb (2008)
4
- Derivation of the general equation of motion - Different horizontal wind forms - General circulation of atmosphere ocean
Equations of fluid motions
bull Hydrodynamics focuses on moving liquids and gases
bull Hydrodynamics and thermodynamics are the foundations of meteorology and oceanography
bull Equations of motion for the atmosphere and ocean are constituted by macroscopic conservation laws for substances impulses and internal energy (hydrodynamics and thermodynamics)
The characteristics of very small volume elements are looked at (eg overall mass average speed)
rarr Phenomenological derivation of hydrodynamics based on the conservation of momentum mass and energy
5 hydro (greek) water
2 Basics
Differentiation following the motion - Lagrange and Euler perspectives
T(Xt) T(Xt+1)
χ=Xt Lagrange
Euler
T(x2t2) u(x2t2) T(x1t1) u(x1t1)
Parcel of air trajectory
Fixed box in space
Lagrange perspective The control volume is an air or seawater parcel that always consists of the same particles and which moves along with the wind or the current Euler perspective The control volume is anchored stationary within the coordinate system eg a cube through which the air or the seawater flows through
6
2 Basics
Differentiation following the motion - Lagrange ndash Euler derivatives
7
2 Basics
Example wind blows over hill cloud forms at the ridge of the lee wave steady state assumption eq cloud does not change in time Lagrangian derivative (after Lagrange 1736-1813)
120597120597119862119862120597120597119905119905 fixed particle ne 0
Eulerian derivative (after Euler 1707-1783)
120597120597119862119862120597120597119905119905 fixed point in space = 0
C = C(xyzt) quantity ie cloud amount 120597120597120597120597119905119905
partial derivatives other variables are kept fixed during the differentiation
Leonhard Euler (CH)
Joseph-Louis Lagrange (I)
Differentiation following the motion mathematically
8
2 Basics
For arbitrary small variations
120575120575119862119862 = 120597120597119862119862120597120597119905119905
120575120575119905119905 + 120597120597119862119862120597120597119909119909
120575120575119909119909 + 120597120597119862119862120597120597119910119910
120575120575119910119910 + 120597120597119862119862120597120597119911119911
120575120575119911119911
120575120575119862119862 119891119891119891119891119909119909119891119891119891119891 119901119901119901119901119901119901119905119905119891119891119901119901119901119901119891119891 =
120597120597119862119862120597120597119905119905
+ 119906119906120597120597119862119862120597120597119909119909
+ 119907119907 120597120597119862119862120597120597119910119910
+ w 120597120597119862119862120597120597119911119911
120575120575119905119905
In limit of small variations
120597120597119862119862120597120597119905119905 119891119891119891119891119909119909119891119891119891119891 119901119901119901119901119901119901119905119905119891119891119901119901119901119901119891119891 =
120597120597119862119862120597120597119905119905
+ 119906119906 120597120597119862119862120597120597119909119909
+ 119907119907 120597120597119862119862120597120597119910119910
+ w 120597120597119862119862120597120597119911119911
= 119863119863119862119862119863119863119905119905
119863119863
119863119863119905119905 119863119863
119863119863119905119905= 120597120597
120597120597119905119905+ 119906119906 120597120597
120597120597119909119909+ 119907119907 120597120597
120597120597119910119910 + w 120597120597
120597120597119911119911 equiv 120597120597
120597120597119905119905 + 119958119958 ∙ 120513120513
δ Greek small delta infinitesimal small size
Insert 120575120575119909119909 = u120575120575119905119905 120575120575119910119910 = v120575120575119905119905 120575120575119911119911 = w120575120575119905119905
Differentiation following the motion mathematically
9
2 Basics
Lagrangian or total derivative (after Lagrange 1736-1813)
119863119863119863119863119905119905 fixed particle equiv 120597120597
120597120597119905119905+ 119906119906 120597120597
120597120597119909119909+ 119907119907 120597120597
120597120597119910119910 + w 120597120597
120597120597119911119911
equiv 120597120597120597120597119905119905
+ 119958119958 ∙ 120513120513 (6-1)
ldquoTime rate of change of some characteristic of a particular element of fluid which in general is changing its positionrdquo Eulerian derivative (after Euler 1707-1783)
120597120597120597120597119905119905 fixed point
ldquoTime rate of change of some characteristic at a fixed point in space but with constantly changing fluid element because the fluid is movingrdquo
119958119958 = (uvw) velocity vector 120513120513 equiv 120597120597
120597120597119909119909 120597120597
120597120597119910119910 120597120597
120597120597119911119911 gradient operator
called ldquonablardquo
Nomenclature summary
bull δ Greek small delta infinitesimal small size eg δV= small air volume bull partial derivative of a quantity with respect to a coordinate eg x bull total differential derivative of a quantity with respect to all dependent coordinates bull Lagrangian (or total) derivative
10
xpartpart
dxd
2 Basics
Trajectories started 850 hPa - top between 2604-29041986 - bottom between 3004-05051986
Example nuclear reactor accident Chernobyl on 26041986 0130 am Trajectories (=flight path) are used for the prediction of air movement
11
2 Basics
Kraus (2004)
Differentiation following the motion - Lagrangian perspective
Think about examples in your everyday life for Eulerrsquos and Lagrangersquos differentiation Give three examples
Lagrange and Euler differentiation
12
2 Basics
22 Basics - Mathematical add on
- Vector operations - Nabla operator
13
22 Basics - Mathematical add on September 16 2014
Modified slides from Clemens Simmer University Bonn
Vector operations - Multiplication with a scalar a -
=
====
z
y
x
z
y
x
ii
avavav
vvv
aavavavva
bull With a scalar multiplication the vector stays a vector
bull Each element of a vector is multiplied individually with a scalar a
bull The vector extends (or shortens) itself by the factor a bull Convention With the multiplication scalar ndash vector we
donrsquot use a point (like with scalar ndash scalar)
22 Mathematical add on
14
Vector operations - Scalar product -
iii
iiiizzyyxx
z
y
x
z
y
x
vfvfvffvfvfvfff
vvv
vffv sum =equiv=++=
sdot
=sdot=sdot
bull The scalar product of two vectors is a scalar bull It is multiplied component-wise then added bull Convention The scalar product is marked by a multiplication sign (middot) bull It is at a maximum with parallel vectors and disappears when the
vectors are perpendicular to each other
f
α
v
αcosfvfv
=sdot
22 Mathematical add on
15
Vector operations ndash vector product
bcacbabac
babababababa
babakbabajbabaibbbaaakji
axbbac
xyyx
zxxz
yzzy
xyyxzxxzyzzy
zyx
zyx
perpandperp=times=
minusminusminus
=
minus+minusminusminus==minus=times=
sin
)()()(
α
a b
bull The vector product of two vectors is again a vector bull Convention The vector product or cross product is marked by an ldquoxrdquo bull If is turned to on the fastest route possible the vector (from the vector product) points in the direction in which a right-hand screw would move (right hand rule) bull With parallel vectors it disappears and it reaches its maximum when the
two vectors are at right angles to each other
16
22 Mathematical add on
Graphically - Pooling of the spatial gradients towards the spatial coordinate axes rarr vector
- Gradient points towards the largest increase of the quantity
- Amount is the magnitude of the derivative pointing towards the largest increase
Nabla operator ndash spatial gradient
with
Operator -Nabla
equivpart=nabla=nabla
nabla
partpartpartpartpartpart
z
y
x
i
17
bull Nabla is a (vector) operator that works to the right
bull It only results in a value if it stands left to an arithmetic expression bull If it stands to the right of an arithmetic expression it keeps its operator function (and ldquowaitsrdquo for application)
22 Mathematical add on
TTgradT
z
y
x
zTyTxT
=
=equivnabla
partpartpartpartpartpart
partpartpartpartpartpart
zw
yv
xu
wvu
udivu
z
y
x
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpart
minuspartpart
partpart
minuspartpart
partpart
minuspartpart
==
times
equivequivtimesnabla part
partpartpart
partpart
partpartpartpartpartpart
yu
xv
xw
zu
zv
yw
wvu
kji
wvu
urotu zyx
z
y
x
Vector product ldquoxrdquo vector
Scalar product ldquordquo scalar
Product with a Scalar vector
Nabla operator 22 Mathematical add on
18
Nabla operator ndash algorithms
=nablanenabla=
=
partpartpartpartpartpart
partpartpartpartpartpart
partpartpartpartpartpart
z
y
x
z
y
x
zTyTxT
TTT
TTT
zw
yv
xu
wvu
u
zw
yv
xu
wvu
udivu
z
y
x
z
y
x
partpart
+partpart
+partpart
=
sdot
=nablasdot
ne
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpartpartpartpartpart
22 Mathematical add on
Note Nabla is a (vector) operator ie the sequence must not be changed here
19
State variables for atmosphere and ocean
u Wind = (uvw) in ms T Temperature in degC or K p Pressure in Nm2 = 1 Pa (1 hPa = 102 Pa = 1 mbar) Specifically for the atmosphere q Specific humidity in kg water vapourkg moist air Specifically for the ocean S Salinity in kg saltkg seawater (often the salt content is expressed in practical salinity unit psu = kg salt1000 kg seawater) 20
2 Basics
I Laws of motion for a fluid parcel in x-y-z- directions (applying Newtons 1 and 2 law)
II Conservation of mass
III Law of thermodynamics (including motion) rarr5 equations for the evolution of the fluid (5 unknowns)
21
3 Equations of motion - nonrotating fluid 3 Equations of motion - nonrotating fluid
Momentum conservation The conservation of momentum is represented by
Newtonrsquos first law (ldquoLex primardquo) (momentum = mass middot velocity)
momentum = const at absence of forces
Momentum sentence (ldquoLex secundardquo) temporal change in momentum = Force (mass middot acceleration = Force)
22
Fu=
dtdM t time M mass u velocity vector F Force vector
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Forces on a fluid parcel
23
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119958119958119863119863119905119905
= F (Eq 6-2)
120588120588 density (120575120575119909119909 120575120575119910119910 120575120575119911119911) fluid parcel with infinitesimal dimensions 120575120575119872119872 mass of the parcel 120575120575119872119872 = 120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119958119958 parcel velocity F net force
119863119863119958119958119863119863119905119905
= 120597120597119958119958120597120597119905119905
+ u 120597120597119958119958120597120597119909119909
+v 120597120597119958119958120597120597119910119910
+w 120597120597119958119958120597120597119911119911
= 120597120597119958119958120597120597119905119905
+ (119958119958 ∙ 120571120571) 119958119958
Gravitational force
24
Fgravity= g 120575120575119872119872
Fgravity= minusg 120588120588119963119963 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-3)
Fgravity Gravitational force M mass g gravity acceleration g = const 120588120588 density 119963119963 unit vector in upward direction
Marshall and Plumb (2008)
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Pressure gradient force
25
Fpressure= 119865119865119909119909 119865119865119910119910 119865119865119911119911
=minus 120597120597119901119901120597120597119909119909
120597120597119901119901120597120597119910119910
120597120597119901119901120597120597119911119911
120575120575119909119909 120575120575119910119910 120575120575119911119911
=minus 120571120571119901119901 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-4)
F(A) = p ∙ 119860119860 = 119901119901 (x minus 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911
F(B) = p ∙ 119861119861 = minus119901119901 (x + 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911 Fx = F(A) + F(B)
Apply Taylor expansion (A21) at midpoint of parcel and neglect small terms (see book)
119901119901 pressure
Friction force
26
3 Equations of motion - nonrotating fluid
- Friction force operates on the earthrsquos surface
- the greater the surface roughness the higher the friction force
- the greater the wind velocity the higher the friction force
- friction force takes effect in vertical distances of ~ 100 m up to 1000 m (rarr atmospheric boundary layer Chapter 7)
Ffriction = 120588120588 ℱ 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-5)
ℱ frictional force per unit mass 120588120588 density
Equations of motion bull All three forces together in Eq 6-2 gives
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119854119854119863119863119905119905
= F119901119901119901119901119891119891119901119901119901119901119906119906119901119901119891119891 + F119892119892119901119901119901119901119907119907119891119891119905119905119910119910 + F119891119891119901119901119891119891119901119901119905119905119891119891119891119891119891119891
bull Re-arranging leads to equation of motion for fluid parcels
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = ℱ (Equation 6-6)
27
3 Equations of motion - nonrotating fluid
120588120588 density u velocity vector F Force vector ℱ frictional force per unit mass p pressure g gravity acceleration 119963119963 unit vector in z-direction
Equations of motion ndash component form
(Eq 6-6) in Cartesian coordinates
bull 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
bull 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
bull 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
28
3 Equations of motion - nonrotating fluid
119906119906 zonal velocity 119907119907meridional velocity w vertical velocity 120588120588 density ℱ frictional force per unit mass p pressure g gravity acceleration
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Research interests
bull Stratospheric ozone bull Dynamics of the stratosphere bull Influence of volcanic eruptions on climate earth system bull Stratosphere-ocean interactions
Who am I
Prof Dr Kirstin Kruumlger
kkruegergeouiono
climate modelling
observations
ship expeditions
Volcanic excursion
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
3 Addition not in book
Motivation 1 Motivation
Marshall and Plumb (2008)
4
- Derivation of the general equation of motion - Different horizontal wind forms - General circulation of atmosphere ocean
Equations of fluid motions
bull Hydrodynamics focuses on moving liquids and gases
bull Hydrodynamics and thermodynamics are the foundations of meteorology and oceanography
bull Equations of motion for the atmosphere and ocean are constituted by macroscopic conservation laws for substances impulses and internal energy (hydrodynamics and thermodynamics)
The characteristics of very small volume elements are looked at (eg overall mass average speed)
rarr Phenomenological derivation of hydrodynamics based on the conservation of momentum mass and energy
5 hydro (greek) water
2 Basics
Differentiation following the motion - Lagrange and Euler perspectives
T(Xt) T(Xt+1)
χ=Xt Lagrange
Euler
T(x2t2) u(x2t2) T(x1t1) u(x1t1)
Parcel of air trajectory
Fixed box in space
Lagrange perspective The control volume is an air or seawater parcel that always consists of the same particles and which moves along with the wind or the current Euler perspective The control volume is anchored stationary within the coordinate system eg a cube through which the air or the seawater flows through
6
2 Basics
Differentiation following the motion - Lagrange ndash Euler derivatives
7
2 Basics
Example wind blows over hill cloud forms at the ridge of the lee wave steady state assumption eq cloud does not change in time Lagrangian derivative (after Lagrange 1736-1813)
120597120597119862119862120597120597119905119905 fixed particle ne 0
Eulerian derivative (after Euler 1707-1783)
120597120597119862119862120597120597119905119905 fixed point in space = 0
C = C(xyzt) quantity ie cloud amount 120597120597120597120597119905119905
partial derivatives other variables are kept fixed during the differentiation
Leonhard Euler (CH)
Joseph-Louis Lagrange (I)
Differentiation following the motion mathematically
8
2 Basics
For arbitrary small variations
120575120575119862119862 = 120597120597119862119862120597120597119905119905
120575120575119905119905 + 120597120597119862119862120597120597119909119909
120575120575119909119909 + 120597120597119862119862120597120597119910119910
120575120575119910119910 + 120597120597119862119862120597120597119911119911
120575120575119911119911
120575120575119862119862 119891119891119891119891119909119909119891119891119891119891 119901119901119901119901119901119901119905119905119891119891119901119901119901119901119891119891 =
120597120597119862119862120597120597119905119905
+ 119906119906120597120597119862119862120597120597119909119909
+ 119907119907 120597120597119862119862120597120597119910119910
+ w 120597120597119862119862120597120597119911119911
120575120575119905119905
In limit of small variations
120597120597119862119862120597120597119905119905 119891119891119891119891119909119909119891119891119891119891 119901119901119901119901119901119901119905119905119891119891119901119901119901119901119891119891 =
120597120597119862119862120597120597119905119905
+ 119906119906 120597120597119862119862120597120597119909119909
+ 119907119907 120597120597119862119862120597120597119910119910
+ w 120597120597119862119862120597120597119911119911
= 119863119863119862119862119863119863119905119905
119863119863
119863119863119905119905 119863119863
119863119863119905119905= 120597120597
120597120597119905119905+ 119906119906 120597120597
120597120597119909119909+ 119907119907 120597120597
120597120597119910119910 + w 120597120597
120597120597119911119911 equiv 120597120597
120597120597119905119905 + 119958119958 ∙ 120513120513
δ Greek small delta infinitesimal small size
Insert 120575120575119909119909 = u120575120575119905119905 120575120575119910119910 = v120575120575119905119905 120575120575119911119911 = w120575120575119905119905
Differentiation following the motion mathematically
9
2 Basics
Lagrangian or total derivative (after Lagrange 1736-1813)
119863119863119863119863119905119905 fixed particle equiv 120597120597
120597120597119905119905+ 119906119906 120597120597
120597120597119909119909+ 119907119907 120597120597
120597120597119910119910 + w 120597120597
120597120597119911119911
equiv 120597120597120597120597119905119905
+ 119958119958 ∙ 120513120513 (6-1)
ldquoTime rate of change of some characteristic of a particular element of fluid which in general is changing its positionrdquo Eulerian derivative (after Euler 1707-1783)
120597120597120597120597119905119905 fixed point
ldquoTime rate of change of some characteristic at a fixed point in space but with constantly changing fluid element because the fluid is movingrdquo
119958119958 = (uvw) velocity vector 120513120513 equiv 120597120597
120597120597119909119909 120597120597
120597120597119910119910 120597120597
120597120597119911119911 gradient operator
called ldquonablardquo
Nomenclature summary
bull δ Greek small delta infinitesimal small size eg δV= small air volume bull partial derivative of a quantity with respect to a coordinate eg x bull total differential derivative of a quantity with respect to all dependent coordinates bull Lagrangian (or total) derivative
10
xpartpart
dxd
2 Basics
Trajectories started 850 hPa - top between 2604-29041986 - bottom between 3004-05051986
Example nuclear reactor accident Chernobyl on 26041986 0130 am Trajectories (=flight path) are used for the prediction of air movement
11
2 Basics
Kraus (2004)
Differentiation following the motion - Lagrangian perspective
Think about examples in your everyday life for Eulerrsquos and Lagrangersquos differentiation Give three examples
Lagrange and Euler differentiation
12
2 Basics
22 Basics - Mathematical add on
- Vector operations - Nabla operator
13
22 Basics - Mathematical add on September 16 2014
Modified slides from Clemens Simmer University Bonn
Vector operations - Multiplication with a scalar a -
=
====
z
y
x
z
y
x
ii
avavav
vvv
aavavavva
bull With a scalar multiplication the vector stays a vector
bull Each element of a vector is multiplied individually with a scalar a
bull The vector extends (or shortens) itself by the factor a bull Convention With the multiplication scalar ndash vector we
donrsquot use a point (like with scalar ndash scalar)
22 Mathematical add on
14
Vector operations - Scalar product -
iii
iiiizzyyxx
z
y
x
z
y
x
vfvfvffvfvfvfff
vvv
vffv sum =equiv=++=
sdot
=sdot=sdot
bull The scalar product of two vectors is a scalar bull It is multiplied component-wise then added bull Convention The scalar product is marked by a multiplication sign (middot) bull It is at a maximum with parallel vectors and disappears when the
vectors are perpendicular to each other
f
α
v
αcosfvfv
=sdot
22 Mathematical add on
15
Vector operations ndash vector product
bcacbabac
babababababa
babakbabajbabaibbbaaakji
axbbac
xyyx
zxxz
yzzy
xyyxzxxzyzzy
zyx
zyx
perpandperp=times=
minusminusminus
=
minus+minusminusminus==minus=times=
sin
)()()(
α
a b
bull The vector product of two vectors is again a vector bull Convention The vector product or cross product is marked by an ldquoxrdquo bull If is turned to on the fastest route possible the vector (from the vector product) points in the direction in which a right-hand screw would move (right hand rule) bull With parallel vectors it disappears and it reaches its maximum when the
two vectors are at right angles to each other
16
22 Mathematical add on
Graphically - Pooling of the spatial gradients towards the spatial coordinate axes rarr vector
- Gradient points towards the largest increase of the quantity
- Amount is the magnitude of the derivative pointing towards the largest increase
Nabla operator ndash spatial gradient
with
Operator -Nabla
equivpart=nabla=nabla
nabla
partpartpartpartpartpart
z
y
x
i
17
bull Nabla is a (vector) operator that works to the right
bull It only results in a value if it stands left to an arithmetic expression bull If it stands to the right of an arithmetic expression it keeps its operator function (and ldquowaitsrdquo for application)
22 Mathematical add on
TTgradT
z
y
x
zTyTxT
=
=equivnabla
partpartpartpartpartpart
partpartpartpartpartpart
zw
yv
xu
wvu
udivu
z
y
x
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpart
minuspartpart
partpart
minuspartpart
partpart
minuspartpart
==
times
equivequivtimesnabla part
partpartpart
partpart
partpartpartpartpartpart
yu
xv
xw
zu
zv
yw
wvu
kji
wvu
urotu zyx
z
y
x
Vector product ldquoxrdquo vector
Scalar product ldquordquo scalar
Product with a Scalar vector
Nabla operator 22 Mathematical add on
18
Nabla operator ndash algorithms
=nablanenabla=
=
partpartpartpartpartpart
partpartpartpartpartpart
partpartpartpartpartpart
z
y
x
z
y
x
zTyTxT
TTT
TTT
zw
yv
xu
wvu
u
zw
yv
xu
wvu
udivu
z
y
x
z
y
x
partpart
+partpart
+partpart
=
sdot
=nablasdot
ne
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpartpartpartpartpart
22 Mathematical add on
Note Nabla is a (vector) operator ie the sequence must not be changed here
19
State variables for atmosphere and ocean
u Wind = (uvw) in ms T Temperature in degC or K p Pressure in Nm2 = 1 Pa (1 hPa = 102 Pa = 1 mbar) Specifically for the atmosphere q Specific humidity in kg water vapourkg moist air Specifically for the ocean S Salinity in kg saltkg seawater (often the salt content is expressed in practical salinity unit psu = kg salt1000 kg seawater) 20
2 Basics
I Laws of motion for a fluid parcel in x-y-z- directions (applying Newtons 1 and 2 law)
II Conservation of mass
III Law of thermodynamics (including motion) rarr5 equations for the evolution of the fluid (5 unknowns)
21
3 Equations of motion - nonrotating fluid 3 Equations of motion - nonrotating fluid
Momentum conservation The conservation of momentum is represented by
Newtonrsquos first law (ldquoLex primardquo) (momentum = mass middot velocity)
momentum = const at absence of forces
Momentum sentence (ldquoLex secundardquo) temporal change in momentum = Force (mass middot acceleration = Force)
22
Fu=
dtdM t time M mass u velocity vector F Force vector
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Forces on a fluid parcel
23
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119958119958119863119863119905119905
= F (Eq 6-2)
120588120588 density (120575120575119909119909 120575120575119910119910 120575120575119911119911) fluid parcel with infinitesimal dimensions 120575120575119872119872 mass of the parcel 120575120575119872119872 = 120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119958119958 parcel velocity F net force
119863119863119958119958119863119863119905119905
= 120597120597119958119958120597120597119905119905
+ u 120597120597119958119958120597120597119909119909
+v 120597120597119958119958120597120597119910119910
+w 120597120597119958119958120597120597119911119911
= 120597120597119958119958120597120597119905119905
+ (119958119958 ∙ 120571120571) 119958119958
Gravitational force
24
Fgravity= g 120575120575119872119872
Fgravity= minusg 120588120588119963119963 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-3)
Fgravity Gravitational force M mass g gravity acceleration g = const 120588120588 density 119963119963 unit vector in upward direction
Marshall and Plumb (2008)
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Pressure gradient force
25
Fpressure= 119865119865119909119909 119865119865119910119910 119865119865119911119911
=minus 120597120597119901119901120597120597119909119909
120597120597119901119901120597120597119910119910
120597120597119901119901120597120597119911119911
120575120575119909119909 120575120575119910119910 120575120575119911119911
=minus 120571120571119901119901 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-4)
F(A) = p ∙ 119860119860 = 119901119901 (x minus 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911
F(B) = p ∙ 119861119861 = minus119901119901 (x + 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911 Fx = F(A) + F(B)
Apply Taylor expansion (A21) at midpoint of parcel and neglect small terms (see book)
119901119901 pressure
Friction force
26
3 Equations of motion - nonrotating fluid
- Friction force operates on the earthrsquos surface
- the greater the surface roughness the higher the friction force
- the greater the wind velocity the higher the friction force
- friction force takes effect in vertical distances of ~ 100 m up to 1000 m (rarr atmospheric boundary layer Chapter 7)
Ffriction = 120588120588 ℱ 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-5)
ℱ frictional force per unit mass 120588120588 density
Equations of motion bull All three forces together in Eq 6-2 gives
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119854119854119863119863119905119905
= F119901119901119901119901119891119891119901119901119901119901119906119906119901119901119891119891 + F119892119892119901119901119901119901119907119907119891119891119905119905119910119910 + F119891119891119901119901119891119891119901119901119905119905119891119891119891119891119891119891
bull Re-arranging leads to equation of motion for fluid parcels
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = ℱ (Equation 6-6)
27
3 Equations of motion - nonrotating fluid
120588120588 density u velocity vector F Force vector ℱ frictional force per unit mass p pressure g gravity acceleration 119963119963 unit vector in z-direction
Equations of motion ndash component form
(Eq 6-6) in Cartesian coordinates
bull 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
bull 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
bull 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
28
3 Equations of motion - nonrotating fluid
119906119906 zonal velocity 119907119907meridional velocity w vertical velocity 120588120588 density ℱ frictional force per unit mass p pressure g gravity acceleration
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
3 Addition not in book
Motivation 1 Motivation
Marshall and Plumb (2008)
4
- Derivation of the general equation of motion - Different horizontal wind forms - General circulation of atmosphere ocean
Equations of fluid motions
bull Hydrodynamics focuses on moving liquids and gases
bull Hydrodynamics and thermodynamics are the foundations of meteorology and oceanography
bull Equations of motion for the atmosphere and ocean are constituted by macroscopic conservation laws for substances impulses and internal energy (hydrodynamics and thermodynamics)
The characteristics of very small volume elements are looked at (eg overall mass average speed)
rarr Phenomenological derivation of hydrodynamics based on the conservation of momentum mass and energy
5 hydro (greek) water
2 Basics
Differentiation following the motion - Lagrange and Euler perspectives
T(Xt) T(Xt+1)
χ=Xt Lagrange
Euler
T(x2t2) u(x2t2) T(x1t1) u(x1t1)
Parcel of air trajectory
Fixed box in space
Lagrange perspective The control volume is an air or seawater parcel that always consists of the same particles and which moves along with the wind or the current Euler perspective The control volume is anchored stationary within the coordinate system eg a cube through which the air or the seawater flows through
6
2 Basics
Differentiation following the motion - Lagrange ndash Euler derivatives
7
2 Basics
Example wind blows over hill cloud forms at the ridge of the lee wave steady state assumption eq cloud does not change in time Lagrangian derivative (after Lagrange 1736-1813)
120597120597119862119862120597120597119905119905 fixed particle ne 0
Eulerian derivative (after Euler 1707-1783)
120597120597119862119862120597120597119905119905 fixed point in space = 0
C = C(xyzt) quantity ie cloud amount 120597120597120597120597119905119905
partial derivatives other variables are kept fixed during the differentiation
Leonhard Euler (CH)
Joseph-Louis Lagrange (I)
Differentiation following the motion mathematically
8
2 Basics
For arbitrary small variations
120575120575119862119862 = 120597120597119862119862120597120597119905119905
120575120575119905119905 + 120597120597119862119862120597120597119909119909
120575120575119909119909 + 120597120597119862119862120597120597119910119910
120575120575119910119910 + 120597120597119862119862120597120597119911119911
120575120575119911119911
120575120575119862119862 119891119891119891119891119909119909119891119891119891119891 119901119901119901119901119901119901119905119905119891119891119901119901119901119901119891119891 =
120597120597119862119862120597120597119905119905
+ 119906119906120597120597119862119862120597120597119909119909
+ 119907119907 120597120597119862119862120597120597119910119910
+ w 120597120597119862119862120597120597119911119911
120575120575119905119905
In limit of small variations
120597120597119862119862120597120597119905119905 119891119891119891119891119909119909119891119891119891119891 119901119901119901119901119901119901119905119905119891119891119901119901119901119901119891119891 =
120597120597119862119862120597120597119905119905
+ 119906119906 120597120597119862119862120597120597119909119909
+ 119907119907 120597120597119862119862120597120597119910119910
+ w 120597120597119862119862120597120597119911119911
= 119863119863119862119862119863119863119905119905
119863119863
119863119863119905119905 119863119863
119863119863119905119905= 120597120597
120597120597119905119905+ 119906119906 120597120597
120597120597119909119909+ 119907119907 120597120597
120597120597119910119910 + w 120597120597
120597120597119911119911 equiv 120597120597
120597120597119905119905 + 119958119958 ∙ 120513120513
δ Greek small delta infinitesimal small size
Insert 120575120575119909119909 = u120575120575119905119905 120575120575119910119910 = v120575120575119905119905 120575120575119911119911 = w120575120575119905119905
Differentiation following the motion mathematically
9
2 Basics
Lagrangian or total derivative (after Lagrange 1736-1813)
119863119863119863119863119905119905 fixed particle equiv 120597120597
120597120597119905119905+ 119906119906 120597120597
120597120597119909119909+ 119907119907 120597120597
120597120597119910119910 + w 120597120597
120597120597119911119911
equiv 120597120597120597120597119905119905
+ 119958119958 ∙ 120513120513 (6-1)
ldquoTime rate of change of some characteristic of a particular element of fluid which in general is changing its positionrdquo Eulerian derivative (after Euler 1707-1783)
120597120597120597120597119905119905 fixed point
ldquoTime rate of change of some characteristic at a fixed point in space but with constantly changing fluid element because the fluid is movingrdquo
119958119958 = (uvw) velocity vector 120513120513 equiv 120597120597
120597120597119909119909 120597120597
120597120597119910119910 120597120597
120597120597119911119911 gradient operator
called ldquonablardquo
Nomenclature summary
bull δ Greek small delta infinitesimal small size eg δV= small air volume bull partial derivative of a quantity with respect to a coordinate eg x bull total differential derivative of a quantity with respect to all dependent coordinates bull Lagrangian (or total) derivative
10
xpartpart
dxd
2 Basics
Trajectories started 850 hPa - top between 2604-29041986 - bottom between 3004-05051986
Example nuclear reactor accident Chernobyl on 26041986 0130 am Trajectories (=flight path) are used for the prediction of air movement
11
2 Basics
Kraus (2004)
Differentiation following the motion - Lagrangian perspective
Think about examples in your everyday life for Eulerrsquos and Lagrangersquos differentiation Give three examples
Lagrange and Euler differentiation
12
2 Basics
22 Basics - Mathematical add on
- Vector operations - Nabla operator
13
22 Basics - Mathematical add on September 16 2014
Modified slides from Clemens Simmer University Bonn
Vector operations - Multiplication with a scalar a -
=
====
z
y
x
z
y
x
ii
avavav
vvv
aavavavva
bull With a scalar multiplication the vector stays a vector
bull Each element of a vector is multiplied individually with a scalar a
bull The vector extends (or shortens) itself by the factor a bull Convention With the multiplication scalar ndash vector we
donrsquot use a point (like with scalar ndash scalar)
22 Mathematical add on
14
Vector operations - Scalar product -
iii
iiiizzyyxx
z
y
x
z
y
x
vfvfvffvfvfvfff
vvv
vffv sum =equiv=++=
sdot
=sdot=sdot
bull The scalar product of two vectors is a scalar bull It is multiplied component-wise then added bull Convention The scalar product is marked by a multiplication sign (middot) bull It is at a maximum with parallel vectors and disappears when the
vectors are perpendicular to each other
f
α
v
αcosfvfv
=sdot
22 Mathematical add on
15
Vector operations ndash vector product
bcacbabac
babababababa
babakbabajbabaibbbaaakji
axbbac
xyyx
zxxz
yzzy
xyyxzxxzyzzy
zyx
zyx
perpandperp=times=
minusminusminus
=
minus+minusminusminus==minus=times=
sin
)()()(
α
a b
bull The vector product of two vectors is again a vector bull Convention The vector product or cross product is marked by an ldquoxrdquo bull If is turned to on the fastest route possible the vector (from the vector product) points in the direction in which a right-hand screw would move (right hand rule) bull With parallel vectors it disappears and it reaches its maximum when the
two vectors are at right angles to each other
16
22 Mathematical add on
Graphically - Pooling of the spatial gradients towards the spatial coordinate axes rarr vector
- Gradient points towards the largest increase of the quantity
- Amount is the magnitude of the derivative pointing towards the largest increase
Nabla operator ndash spatial gradient
with
Operator -Nabla
equivpart=nabla=nabla
nabla
partpartpartpartpartpart
z
y
x
i
17
bull Nabla is a (vector) operator that works to the right
bull It only results in a value if it stands left to an arithmetic expression bull If it stands to the right of an arithmetic expression it keeps its operator function (and ldquowaitsrdquo for application)
22 Mathematical add on
TTgradT
z
y
x
zTyTxT
=
=equivnabla
partpartpartpartpartpart
partpartpartpartpartpart
zw
yv
xu
wvu
udivu
z
y
x
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpart
minuspartpart
partpart
minuspartpart
partpart
minuspartpart
==
times
equivequivtimesnabla part
partpartpart
partpart
partpartpartpartpartpart
yu
xv
xw
zu
zv
yw
wvu
kji
wvu
urotu zyx
z
y
x
Vector product ldquoxrdquo vector
Scalar product ldquordquo scalar
Product with a Scalar vector
Nabla operator 22 Mathematical add on
18
Nabla operator ndash algorithms
=nablanenabla=
=
partpartpartpartpartpart
partpartpartpartpartpart
partpartpartpartpartpart
z
y
x
z
y
x
zTyTxT
TTT
TTT
zw
yv
xu
wvu
u
zw
yv
xu
wvu
udivu
z
y
x
z
y
x
partpart
+partpart
+partpart
=
sdot
=nablasdot
ne
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpartpartpartpartpart
22 Mathematical add on
Note Nabla is a (vector) operator ie the sequence must not be changed here
19
State variables for atmosphere and ocean
u Wind = (uvw) in ms T Temperature in degC or K p Pressure in Nm2 = 1 Pa (1 hPa = 102 Pa = 1 mbar) Specifically for the atmosphere q Specific humidity in kg water vapourkg moist air Specifically for the ocean S Salinity in kg saltkg seawater (often the salt content is expressed in practical salinity unit psu = kg salt1000 kg seawater) 20
2 Basics
I Laws of motion for a fluid parcel in x-y-z- directions (applying Newtons 1 and 2 law)
II Conservation of mass
III Law of thermodynamics (including motion) rarr5 equations for the evolution of the fluid (5 unknowns)
21
3 Equations of motion - nonrotating fluid 3 Equations of motion - nonrotating fluid
Momentum conservation The conservation of momentum is represented by
Newtonrsquos first law (ldquoLex primardquo) (momentum = mass middot velocity)
momentum = const at absence of forces
Momentum sentence (ldquoLex secundardquo) temporal change in momentum = Force (mass middot acceleration = Force)
22
Fu=
dtdM t time M mass u velocity vector F Force vector
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Forces on a fluid parcel
23
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119958119958119863119863119905119905
= F (Eq 6-2)
120588120588 density (120575120575119909119909 120575120575119910119910 120575120575119911119911) fluid parcel with infinitesimal dimensions 120575120575119872119872 mass of the parcel 120575120575119872119872 = 120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119958119958 parcel velocity F net force
119863119863119958119958119863119863119905119905
= 120597120597119958119958120597120597119905119905
+ u 120597120597119958119958120597120597119909119909
+v 120597120597119958119958120597120597119910119910
+w 120597120597119958119958120597120597119911119911
= 120597120597119958119958120597120597119905119905
+ (119958119958 ∙ 120571120571) 119958119958
Gravitational force
24
Fgravity= g 120575120575119872119872
Fgravity= minusg 120588120588119963119963 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-3)
Fgravity Gravitational force M mass g gravity acceleration g = const 120588120588 density 119963119963 unit vector in upward direction
Marshall and Plumb (2008)
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Pressure gradient force
25
Fpressure= 119865119865119909119909 119865119865119910119910 119865119865119911119911
=minus 120597120597119901119901120597120597119909119909
120597120597119901119901120597120597119910119910
120597120597119901119901120597120597119911119911
120575120575119909119909 120575120575119910119910 120575120575119911119911
=minus 120571120571119901119901 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-4)
F(A) = p ∙ 119860119860 = 119901119901 (x minus 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911
F(B) = p ∙ 119861119861 = minus119901119901 (x + 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911 Fx = F(A) + F(B)
Apply Taylor expansion (A21) at midpoint of parcel and neglect small terms (see book)
119901119901 pressure
Friction force
26
3 Equations of motion - nonrotating fluid
- Friction force operates on the earthrsquos surface
- the greater the surface roughness the higher the friction force
- the greater the wind velocity the higher the friction force
- friction force takes effect in vertical distances of ~ 100 m up to 1000 m (rarr atmospheric boundary layer Chapter 7)
Ffriction = 120588120588 ℱ 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-5)
ℱ frictional force per unit mass 120588120588 density
Equations of motion bull All three forces together in Eq 6-2 gives
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119854119854119863119863119905119905
= F119901119901119901119901119891119891119901119901119901119901119906119906119901119901119891119891 + F119892119892119901119901119901119901119907119907119891119891119905119905119910119910 + F119891119891119901119901119891119891119901119901119905119905119891119891119891119891119891119891
bull Re-arranging leads to equation of motion for fluid parcels
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = ℱ (Equation 6-6)
27
3 Equations of motion - nonrotating fluid
120588120588 density u velocity vector F Force vector ℱ frictional force per unit mass p pressure g gravity acceleration 119963119963 unit vector in z-direction
Equations of motion ndash component form
(Eq 6-6) in Cartesian coordinates
bull 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
bull 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
bull 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
28
3 Equations of motion - nonrotating fluid
119906119906 zonal velocity 119907119907meridional velocity w vertical velocity 120588120588 density ℱ frictional force per unit mass p pressure g gravity acceleration
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Motivation 1 Motivation
Marshall and Plumb (2008)
4
- Derivation of the general equation of motion - Different horizontal wind forms - General circulation of atmosphere ocean
Equations of fluid motions
bull Hydrodynamics focuses on moving liquids and gases
bull Hydrodynamics and thermodynamics are the foundations of meteorology and oceanography
bull Equations of motion for the atmosphere and ocean are constituted by macroscopic conservation laws for substances impulses and internal energy (hydrodynamics and thermodynamics)
The characteristics of very small volume elements are looked at (eg overall mass average speed)
rarr Phenomenological derivation of hydrodynamics based on the conservation of momentum mass and energy
5 hydro (greek) water
2 Basics
Differentiation following the motion - Lagrange and Euler perspectives
T(Xt) T(Xt+1)
χ=Xt Lagrange
Euler
T(x2t2) u(x2t2) T(x1t1) u(x1t1)
Parcel of air trajectory
Fixed box in space
Lagrange perspective The control volume is an air or seawater parcel that always consists of the same particles and which moves along with the wind or the current Euler perspective The control volume is anchored stationary within the coordinate system eg a cube through which the air or the seawater flows through
6
2 Basics
Differentiation following the motion - Lagrange ndash Euler derivatives
7
2 Basics
Example wind blows over hill cloud forms at the ridge of the lee wave steady state assumption eq cloud does not change in time Lagrangian derivative (after Lagrange 1736-1813)
120597120597119862119862120597120597119905119905 fixed particle ne 0
Eulerian derivative (after Euler 1707-1783)
120597120597119862119862120597120597119905119905 fixed point in space = 0
C = C(xyzt) quantity ie cloud amount 120597120597120597120597119905119905
partial derivatives other variables are kept fixed during the differentiation
Leonhard Euler (CH)
Joseph-Louis Lagrange (I)
Differentiation following the motion mathematically
8
2 Basics
For arbitrary small variations
120575120575119862119862 = 120597120597119862119862120597120597119905119905
120575120575119905119905 + 120597120597119862119862120597120597119909119909
120575120575119909119909 + 120597120597119862119862120597120597119910119910
120575120575119910119910 + 120597120597119862119862120597120597119911119911
120575120575119911119911
120575120575119862119862 119891119891119891119891119909119909119891119891119891119891 119901119901119901119901119901119901119905119905119891119891119901119901119901119901119891119891 =
120597120597119862119862120597120597119905119905
+ 119906119906120597120597119862119862120597120597119909119909
+ 119907119907 120597120597119862119862120597120597119910119910
+ w 120597120597119862119862120597120597119911119911
120575120575119905119905
In limit of small variations
120597120597119862119862120597120597119905119905 119891119891119891119891119909119909119891119891119891119891 119901119901119901119901119901119901119905119905119891119891119901119901119901119901119891119891 =
120597120597119862119862120597120597119905119905
+ 119906119906 120597120597119862119862120597120597119909119909
+ 119907119907 120597120597119862119862120597120597119910119910
+ w 120597120597119862119862120597120597119911119911
= 119863119863119862119862119863119863119905119905
119863119863
119863119863119905119905 119863119863
119863119863119905119905= 120597120597
120597120597119905119905+ 119906119906 120597120597
120597120597119909119909+ 119907119907 120597120597
120597120597119910119910 + w 120597120597
120597120597119911119911 equiv 120597120597
120597120597119905119905 + 119958119958 ∙ 120513120513
δ Greek small delta infinitesimal small size
Insert 120575120575119909119909 = u120575120575119905119905 120575120575119910119910 = v120575120575119905119905 120575120575119911119911 = w120575120575119905119905
Differentiation following the motion mathematically
9
2 Basics
Lagrangian or total derivative (after Lagrange 1736-1813)
119863119863119863119863119905119905 fixed particle equiv 120597120597
120597120597119905119905+ 119906119906 120597120597
120597120597119909119909+ 119907119907 120597120597
120597120597119910119910 + w 120597120597
120597120597119911119911
equiv 120597120597120597120597119905119905
+ 119958119958 ∙ 120513120513 (6-1)
ldquoTime rate of change of some characteristic of a particular element of fluid which in general is changing its positionrdquo Eulerian derivative (after Euler 1707-1783)
120597120597120597120597119905119905 fixed point
ldquoTime rate of change of some characteristic at a fixed point in space but with constantly changing fluid element because the fluid is movingrdquo
119958119958 = (uvw) velocity vector 120513120513 equiv 120597120597
120597120597119909119909 120597120597
120597120597119910119910 120597120597
120597120597119911119911 gradient operator
called ldquonablardquo
Nomenclature summary
bull δ Greek small delta infinitesimal small size eg δV= small air volume bull partial derivative of a quantity with respect to a coordinate eg x bull total differential derivative of a quantity with respect to all dependent coordinates bull Lagrangian (or total) derivative
10
xpartpart
dxd
2 Basics
Trajectories started 850 hPa - top between 2604-29041986 - bottom between 3004-05051986
Example nuclear reactor accident Chernobyl on 26041986 0130 am Trajectories (=flight path) are used for the prediction of air movement
11
2 Basics
Kraus (2004)
Differentiation following the motion - Lagrangian perspective
Think about examples in your everyday life for Eulerrsquos and Lagrangersquos differentiation Give three examples
Lagrange and Euler differentiation
12
2 Basics
22 Basics - Mathematical add on
- Vector operations - Nabla operator
13
22 Basics - Mathematical add on September 16 2014
Modified slides from Clemens Simmer University Bonn
Vector operations - Multiplication with a scalar a -
=
====
z
y
x
z
y
x
ii
avavav
vvv
aavavavva
bull With a scalar multiplication the vector stays a vector
bull Each element of a vector is multiplied individually with a scalar a
bull The vector extends (or shortens) itself by the factor a bull Convention With the multiplication scalar ndash vector we
donrsquot use a point (like with scalar ndash scalar)
22 Mathematical add on
14
Vector operations - Scalar product -
iii
iiiizzyyxx
z
y
x
z
y
x
vfvfvffvfvfvfff
vvv
vffv sum =equiv=++=
sdot
=sdot=sdot
bull The scalar product of two vectors is a scalar bull It is multiplied component-wise then added bull Convention The scalar product is marked by a multiplication sign (middot) bull It is at a maximum with parallel vectors and disappears when the
vectors are perpendicular to each other
f
α
v
αcosfvfv
=sdot
22 Mathematical add on
15
Vector operations ndash vector product
bcacbabac
babababababa
babakbabajbabaibbbaaakji
axbbac
xyyx
zxxz
yzzy
xyyxzxxzyzzy
zyx
zyx
perpandperp=times=
minusminusminus
=
minus+minusminusminus==minus=times=
sin
)()()(
α
a b
bull The vector product of two vectors is again a vector bull Convention The vector product or cross product is marked by an ldquoxrdquo bull If is turned to on the fastest route possible the vector (from the vector product) points in the direction in which a right-hand screw would move (right hand rule) bull With parallel vectors it disappears and it reaches its maximum when the
two vectors are at right angles to each other
16
22 Mathematical add on
Graphically - Pooling of the spatial gradients towards the spatial coordinate axes rarr vector
- Gradient points towards the largest increase of the quantity
- Amount is the magnitude of the derivative pointing towards the largest increase
Nabla operator ndash spatial gradient
with
Operator -Nabla
equivpart=nabla=nabla
nabla
partpartpartpartpartpart
z
y
x
i
17
bull Nabla is a (vector) operator that works to the right
bull It only results in a value if it stands left to an arithmetic expression bull If it stands to the right of an arithmetic expression it keeps its operator function (and ldquowaitsrdquo for application)
22 Mathematical add on
TTgradT
z
y
x
zTyTxT
=
=equivnabla
partpartpartpartpartpart
partpartpartpartpartpart
zw
yv
xu
wvu
udivu
z
y
x
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpart
minuspartpart
partpart
minuspartpart
partpart
minuspartpart
==
times
equivequivtimesnabla part
partpartpart
partpart
partpartpartpartpartpart
yu
xv
xw
zu
zv
yw
wvu
kji
wvu
urotu zyx
z
y
x
Vector product ldquoxrdquo vector
Scalar product ldquordquo scalar
Product with a Scalar vector
Nabla operator 22 Mathematical add on
18
Nabla operator ndash algorithms
=nablanenabla=
=
partpartpartpartpartpart
partpartpartpartpartpart
partpartpartpartpartpart
z
y
x
z
y
x
zTyTxT
TTT
TTT
zw
yv
xu
wvu
u
zw
yv
xu
wvu
udivu
z
y
x
z
y
x
partpart
+partpart
+partpart
=
sdot
=nablasdot
ne
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpartpartpartpartpart
22 Mathematical add on
Note Nabla is a (vector) operator ie the sequence must not be changed here
19
State variables for atmosphere and ocean
u Wind = (uvw) in ms T Temperature in degC or K p Pressure in Nm2 = 1 Pa (1 hPa = 102 Pa = 1 mbar) Specifically for the atmosphere q Specific humidity in kg water vapourkg moist air Specifically for the ocean S Salinity in kg saltkg seawater (often the salt content is expressed in practical salinity unit psu = kg salt1000 kg seawater) 20
2 Basics
I Laws of motion for a fluid parcel in x-y-z- directions (applying Newtons 1 and 2 law)
II Conservation of mass
III Law of thermodynamics (including motion) rarr5 equations for the evolution of the fluid (5 unknowns)
21
3 Equations of motion - nonrotating fluid 3 Equations of motion - nonrotating fluid
Momentum conservation The conservation of momentum is represented by
Newtonrsquos first law (ldquoLex primardquo) (momentum = mass middot velocity)
momentum = const at absence of forces
Momentum sentence (ldquoLex secundardquo) temporal change in momentum = Force (mass middot acceleration = Force)
22
Fu=
dtdM t time M mass u velocity vector F Force vector
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Forces on a fluid parcel
23
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119958119958119863119863119905119905
= F (Eq 6-2)
120588120588 density (120575120575119909119909 120575120575119910119910 120575120575119911119911) fluid parcel with infinitesimal dimensions 120575120575119872119872 mass of the parcel 120575120575119872119872 = 120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119958119958 parcel velocity F net force
119863119863119958119958119863119863119905119905
= 120597120597119958119958120597120597119905119905
+ u 120597120597119958119958120597120597119909119909
+v 120597120597119958119958120597120597119910119910
+w 120597120597119958119958120597120597119911119911
= 120597120597119958119958120597120597119905119905
+ (119958119958 ∙ 120571120571) 119958119958
Gravitational force
24
Fgravity= g 120575120575119872119872
Fgravity= minusg 120588120588119963119963 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-3)
Fgravity Gravitational force M mass g gravity acceleration g = const 120588120588 density 119963119963 unit vector in upward direction
Marshall and Plumb (2008)
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Pressure gradient force
25
Fpressure= 119865119865119909119909 119865119865119910119910 119865119865119911119911
=minus 120597120597119901119901120597120597119909119909
120597120597119901119901120597120597119910119910
120597120597119901119901120597120597119911119911
120575120575119909119909 120575120575119910119910 120575120575119911119911
=minus 120571120571119901119901 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-4)
F(A) = p ∙ 119860119860 = 119901119901 (x minus 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911
F(B) = p ∙ 119861119861 = minus119901119901 (x + 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911 Fx = F(A) + F(B)
Apply Taylor expansion (A21) at midpoint of parcel and neglect small terms (see book)
119901119901 pressure
Friction force
26
3 Equations of motion - nonrotating fluid
- Friction force operates on the earthrsquos surface
- the greater the surface roughness the higher the friction force
- the greater the wind velocity the higher the friction force
- friction force takes effect in vertical distances of ~ 100 m up to 1000 m (rarr atmospheric boundary layer Chapter 7)
Ffriction = 120588120588 ℱ 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-5)
ℱ frictional force per unit mass 120588120588 density
Equations of motion bull All three forces together in Eq 6-2 gives
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119854119854119863119863119905119905
= F119901119901119901119901119891119891119901119901119901119901119906119906119901119901119891119891 + F119892119892119901119901119901119901119907119907119891119891119905119905119910119910 + F119891119891119901119901119891119891119901119901119905119905119891119891119891119891119891119891
bull Re-arranging leads to equation of motion for fluid parcels
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = ℱ (Equation 6-6)
27
3 Equations of motion - nonrotating fluid
120588120588 density u velocity vector F Force vector ℱ frictional force per unit mass p pressure g gravity acceleration 119963119963 unit vector in z-direction
Equations of motion ndash component form
(Eq 6-6) in Cartesian coordinates
bull 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
bull 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
bull 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
28
3 Equations of motion - nonrotating fluid
119906119906 zonal velocity 119907119907meridional velocity w vertical velocity 120588120588 density ℱ frictional force per unit mass p pressure g gravity acceleration
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Equations of fluid motions
bull Hydrodynamics focuses on moving liquids and gases
bull Hydrodynamics and thermodynamics are the foundations of meteorology and oceanography
bull Equations of motion for the atmosphere and ocean are constituted by macroscopic conservation laws for substances impulses and internal energy (hydrodynamics and thermodynamics)
The characteristics of very small volume elements are looked at (eg overall mass average speed)
rarr Phenomenological derivation of hydrodynamics based on the conservation of momentum mass and energy
5 hydro (greek) water
2 Basics
Differentiation following the motion - Lagrange and Euler perspectives
T(Xt) T(Xt+1)
χ=Xt Lagrange
Euler
T(x2t2) u(x2t2) T(x1t1) u(x1t1)
Parcel of air trajectory
Fixed box in space
Lagrange perspective The control volume is an air or seawater parcel that always consists of the same particles and which moves along with the wind or the current Euler perspective The control volume is anchored stationary within the coordinate system eg a cube through which the air or the seawater flows through
6
2 Basics
Differentiation following the motion - Lagrange ndash Euler derivatives
7
2 Basics
Example wind blows over hill cloud forms at the ridge of the lee wave steady state assumption eq cloud does not change in time Lagrangian derivative (after Lagrange 1736-1813)
120597120597119862119862120597120597119905119905 fixed particle ne 0
Eulerian derivative (after Euler 1707-1783)
120597120597119862119862120597120597119905119905 fixed point in space = 0
C = C(xyzt) quantity ie cloud amount 120597120597120597120597119905119905
partial derivatives other variables are kept fixed during the differentiation
Leonhard Euler (CH)
Joseph-Louis Lagrange (I)
Differentiation following the motion mathematically
8
2 Basics
For arbitrary small variations
120575120575119862119862 = 120597120597119862119862120597120597119905119905
120575120575119905119905 + 120597120597119862119862120597120597119909119909
120575120575119909119909 + 120597120597119862119862120597120597119910119910
120575120575119910119910 + 120597120597119862119862120597120597119911119911
120575120575119911119911
120575120575119862119862 119891119891119891119891119909119909119891119891119891119891 119901119901119901119901119901119901119905119905119891119891119901119901119901119901119891119891 =
120597120597119862119862120597120597119905119905
+ 119906119906120597120597119862119862120597120597119909119909
+ 119907119907 120597120597119862119862120597120597119910119910
+ w 120597120597119862119862120597120597119911119911
120575120575119905119905
In limit of small variations
120597120597119862119862120597120597119905119905 119891119891119891119891119909119909119891119891119891119891 119901119901119901119901119901119901119905119905119891119891119901119901119901119901119891119891 =
120597120597119862119862120597120597119905119905
+ 119906119906 120597120597119862119862120597120597119909119909
+ 119907119907 120597120597119862119862120597120597119910119910
+ w 120597120597119862119862120597120597119911119911
= 119863119863119862119862119863119863119905119905
119863119863
119863119863119905119905 119863119863
119863119863119905119905= 120597120597
120597120597119905119905+ 119906119906 120597120597
120597120597119909119909+ 119907119907 120597120597
120597120597119910119910 + w 120597120597
120597120597119911119911 equiv 120597120597
120597120597119905119905 + 119958119958 ∙ 120513120513
δ Greek small delta infinitesimal small size
Insert 120575120575119909119909 = u120575120575119905119905 120575120575119910119910 = v120575120575119905119905 120575120575119911119911 = w120575120575119905119905
Differentiation following the motion mathematically
9
2 Basics
Lagrangian or total derivative (after Lagrange 1736-1813)
119863119863119863119863119905119905 fixed particle equiv 120597120597
120597120597119905119905+ 119906119906 120597120597
120597120597119909119909+ 119907119907 120597120597
120597120597119910119910 + w 120597120597
120597120597119911119911
equiv 120597120597120597120597119905119905
+ 119958119958 ∙ 120513120513 (6-1)
ldquoTime rate of change of some characteristic of a particular element of fluid which in general is changing its positionrdquo Eulerian derivative (after Euler 1707-1783)
120597120597120597120597119905119905 fixed point
ldquoTime rate of change of some characteristic at a fixed point in space but with constantly changing fluid element because the fluid is movingrdquo
119958119958 = (uvw) velocity vector 120513120513 equiv 120597120597
120597120597119909119909 120597120597
120597120597119910119910 120597120597
120597120597119911119911 gradient operator
called ldquonablardquo
Nomenclature summary
bull δ Greek small delta infinitesimal small size eg δV= small air volume bull partial derivative of a quantity with respect to a coordinate eg x bull total differential derivative of a quantity with respect to all dependent coordinates bull Lagrangian (or total) derivative
10
xpartpart
dxd
2 Basics
Trajectories started 850 hPa - top between 2604-29041986 - bottom between 3004-05051986
Example nuclear reactor accident Chernobyl on 26041986 0130 am Trajectories (=flight path) are used for the prediction of air movement
11
2 Basics
Kraus (2004)
Differentiation following the motion - Lagrangian perspective
Think about examples in your everyday life for Eulerrsquos and Lagrangersquos differentiation Give three examples
Lagrange and Euler differentiation
12
2 Basics
22 Basics - Mathematical add on
- Vector operations - Nabla operator
13
22 Basics - Mathematical add on September 16 2014
Modified slides from Clemens Simmer University Bonn
Vector operations - Multiplication with a scalar a -
=
====
z
y
x
z
y
x
ii
avavav
vvv
aavavavva
bull With a scalar multiplication the vector stays a vector
bull Each element of a vector is multiplied individually with a scalar a
bull The vector extends (or shortens) itself by the factor a bull Convention With the multiplication scalar ndash vector we
donrsquot use a point (like with scalar ndash scalar)
22 Mathematical add on
14
Vector operations - Scalar product -
iii
iiiizzyyxx
z
y
x
z
y
x
vfvfvffvfvfvfff
vvv
vffv sum =equiv=++=
sdot
=sdot=sdot
bull The scalar product of two vectors is a scalar bull It is multiplied component-wise then added bull Convention The scalar product is marked by a multiplication sign (middot) bull It is at a maximum with parallel vectors and disappears when the
vectors are perpendicular to each other
f
α
v
αcosfvfv
=sdot
22 Mathematical add on
15
Vector operations ndash vector product
bcacbabac
babababababa
babakbabajbabaibbbaaakji
axbbac
xyyx
zxxz
yzzy
xyyxzxxzyzzy
zyx
zyx
perpandperp=times=
minusminusminus
=
minus+minusminusminus==minus=times=
sin
)()()(
α
a b
bull The vector product of two vectors is again a vector bull Convention The vector product or cross product is marked by an ldquoxrdquo bull If is turned to on the fastest route possible the vector (from the vector product) points in the direction in which a right-hand screw would move (right hand rule) bull With parallel vectors it disappears and it reaches its maximum when the
two vectors are at right angles to each other
16
22 Mathematical add on
Graphically - Pooling of the spatial gradients towards the spatial coordinate axes rarr vector
- Gradient points towards the largest increase of the quantity
- Amount is the magnitude of the derivative pointing towards the largest increase
Nabla operator ndash spatial gradient
with
Operator -Nabla
equivpart=nabla=nabla
nabla
partpartpartpartpartpart
z
y
x
i
17
bull Nabla is a (vector) operator that works to the right
bull It only results in a value if it stands left to an arithmetic expression bull If it stands to the right of an arithmetic expression it keeps its operator function (and ldquowaitsrdquo for application)
22 Mathematical add on
TTgradT
z
y
x
zTyTxT
=
=equivnabla
partpartpartpartpartpart
partpartpartpartpartpart
zw
yv
xu
wvu
udivu
z
y
x
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpart
minuspartpart
partpart
minuspartpart
partpart
minuspartpart
==
times
equivequivtimesnabla part
partpartpart
partpart
partpartpartpartpartpart
yu
xv
xw
zu
zv
yw
wvu
kji
wvu
urotu zyx
z
y
x
Vector product ldquoxrdquo vector
Scalar product ldquordquo scalar
Product with a Scalar vector
Nabla operator 22 Mathematical add on
18
Nabla operator ndash algorithms
=nablanenabla=
=
partpartpartpartpartpart
partpartpartpartpartpart
partpartpartpartpartpart
z
y
x
z
y
x
zTyTxT
TTT
TTT
zw
yv
xu
wvu
u
zw
yv
xu
wvu
udivu
z
y
x
z
y
x
partpart
+partpart
+partpart
=
sdot
=nablasdot
ne
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpartpartpartpartpart
22 Mathematical add on
Note Nabla is a (vector) operator ie the sequence must not be changed here
19
State variables for atmosphere and ocean
u Wind = (uvw) in ms T Temperature in degC or K p Pressure in Nm2 = 1 Pa (1 hPa = 102 Pa = 1 mbar) Specifically for the atmosphere q Specific humidity in kg water vapourkg moist air Specifically for the ocean S Salinity in kg saltkg seawater (often the salt content is expressed in practical salinity unit psu = kg salt1000 kg seawater) 20
2 Basics
I Laws of motion for a fluid parcel in x-y-z- directions (applying Newtons 1 and 2 law)
II Conservation of mass
III Law of thermodynamics (including motion) rarr5 equations for the evolution of the fluid (5 unknowns)
21
3 Equations of motion - nonrotating fluid 3 Equations of motion - nonrotating fluid
Momentum conservation The conservation of momentum is represented by
Newtonrsquos first law (ldquoLex primardquo) (momentum = mass middot velocity)
momentum = const at absence of forces
Momentum sentence (ldquoLex secundardquo) temporal change in momentum = Force (mass middot acceleration = Force)
22
Fu=
dtdM t time M mass u velocity vector F Force vector
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Forces on a fluid parcel
23
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119958119958119863119863119905119905
= F (Eq 6-2)
120588120588 density (120575120575119909119909 120575120575119910119910 120575120575119911119911) fluid parcel with infinitesimal dimensions 120575120575119872119872 mass of the parcel 120575120575119872119872 = 120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119958119958 parcel velocity F net force
119863119863119958119958119863119863119905119905
= 120597120597119958119958120597120597119905119905
+ u 120597120597119958119958120597120597119909119909
+v 120597120597119958119958120597120597119910119910
+w 120597120597119958119958120597120597119911119911
= 120597120597119958119958120597120597119905119905
+ (119958119958 ∙ 120571120571) 119958119958
Gravitational force
24
Fgravity= g 120575120575119872119872
Fgravity= minusg 120588120588119963119963 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-3)
Fgravity Gravitational force M mass g gravity acceleration g = const 120588120588 density 119963119963 unit vector in upward direction
Marshall and Plumb (2008)
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Pressure gradient force
25
Fpressure= 119865119865119909119909 119865119865119910119910 119865119865119911119911
=minus 120597120597119901119901120597120597119909119909
120597120597119901119901120597120597119910119910
120597120597119901119901120597120597119911119911
120575120575119909119909 120575120575119910119910 120575120575119911119911
=minus 120571120571119901119901 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-4)
F(A) = p ∙ 119860119860 = 119901119901 (x minus 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911
F(B) = p ∙ 119861119861 = minus119901119901 (x + 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911 Fx = F(A) + F(B)
Apply Taylor expansion (A21) at midpoint of parcel and neglect small terms (see book)
119901119901 pressure
Friction force
26
3 Equations of motion - nonrotating fluid
- Friction force operates on the earthrsquos surface
- the greater the surface roughness the higher the friction force
- the greater the wind velocity the higher the friction force
- friction force takes effect in vertical distances of ~ 100 m up to 1000 m (rarr atmospheric boundary layer Chapter 7)
Ffriction = 120588120588 ℱ 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-5)
ℱ frictional force per unit mass 120588120588 density
Equations of motion bull All three forces together in Eq 6-2 gives
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119854119854119863119863119905119905
= F119901119901119901119901119891119891119901119901119901119901119906119906119901119901119891119891 + F119892119892119901119901119901119901119907119907119891119891119905119905119910119910 + F119891119891119901119901119891119891119901119901119905119905119891119891119891119891119891119891
bull Re-arranging leads to equation of motion for fluid parcels
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = ℱ (Equation 6-6)
27
3 Equations of motion - nonrotating fluid
120588120588 density u velocity vector F Force vector ℱ frictional force per unit mass p pressure g gravity acceleration 119963119963 unit vector in z-direction
Equations of motion ndash component form
(Eq 6-6) in Cartesian coordinates
bull 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
bull 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
bull 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
28
3 Equations of motion - nonrotating fluid
119906119906 zonal velocity 119907119907meridional velocity w vertical velocity 120588120588 density ℱ frictional force per unit mass p pressure g gravity acceleration
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Differentiation following the motion - Lagrange and Euler perspectives
T(Xt) T(Xt+1)
χ=Xt Lagrange
Euler
T(x2t2) u(x2t2) T(x1t1) u(x1t1)
Parcel of air trajectory
Fixed box in space
Lagrange perspective The control volume is an air or seawater parcel that always consists of the same particles and which moves along with the wind or the current Euler perspective The control volume is anchored stationary within the coordinate system eg a cube through which the air or the seawater flows through
6
2 Basics
Differentiation following the motion - Lagrange ndash Euler derivatives
7
2 Basics
Example wind blows over hill cloud forms at the ridge of the lee wave steady state assumption eq cloud does not change in time Lagrangian derivative (after Lagrange 1736-1813)
120597120597119862119862120597120597119905119905 fixed particle ne 0
Eulerian derivative (after Euler 1707-1783)
120597120597119862119862120597120597119905119905 fixed point in space = 0
C = C(xyzt) quantity ie cloud amount 120597120597120597120597119905119905
partial derivatives other variables are kept fixed during the differentiation
Leonhard Euler (CH)
Joseph-Louis Lagrange (I)
Differentiation following the motion mathematically
8
2 Basics
For arbitrary small variations
120575120575119862119862 = 120597120597119862119862120597120597119905119905
120575120575119905119905 + 120597120597119862119862120597120597119909119909
120575120575119909119909 + 120597120597119862119862120597120597119910119910
120575120575119910119910 + 120597120597119862119862120597120597119911119911
120575120575119911119911
120575120575119862119862 119891119891119891119891119909119909119891119891119891119891 119901119901119901119901119901119901119905119905119891119891119901119901119901119901119891119891 =
120597120597119862119862120597120597119905119905
+ 119906119906120597120597119862119862120597120597119909119909
+ 119907119907 120597120597119862119862120597120597119910119910
+ w 120597120597119862119862120597120597119911119911
120575120575119905119905
In limit of small variations
120597120597119862119862120597120597119905119905 119891119891119891119891119909119909119891119891119891119891 119901119901119901119901119901119901119905119905119891119891119901119901119901119901119891119891 =
120597120597119862119862120597120597119905119905
+ 119906119906 120597120597119862119862120597120597119909119909
+ 119907119907 120597120597119862119862120597120597119910119910
+ w 120597120597119862119862120597120597119911119911
= 119863119863119862119862119863119863119905119905
119863119863
119863119863119905119905 119863119863
119863119863119905119905= 120597120597
120597120597119905119905+ 119906119906 120597120597
120597120597119909119909+ 119907119907 120597120597
120597120597119910119910 + w 120597120597
120597120597119911119911 equiv 120597120597
120597120597119905119905 + 119958119958 ∙ 120513120513
δ Greek small delta infinitesimal small size
Insert 120575120575119909119909 = u120575120575119905119905 120575120575119910119910 = v120575120575119905119905 120575120575119911119911 = w120575120575119905119905
Differentiation following the motion mathematically
9
2 Basics
Lagrangian or total derivative (after Lagrange 1736-1813)
119863119863119863119863119905119905 fixed particle equiv 120597120597
120597120597119905119905+ 119906119906 120597120597
120597120597119909119909+ 119907119907 120597120597
120597120597119910119910 + w 120597120597
120597120597119911119911
equiv 120597120597120597120597119905119905
+ 119958119958 ∙ 120513120513 (6-1)
ldquoTime rate of change of some characteristic of a particular element of fluid which in general is changing its positionrdquo Eulerian derivative (after Euler 1707-1783)
120597120597120597120597119905119905 fixed point
ldquoTime rate of change of some characteristic at a fixed point in space but with constantly changing fluid element because the fluid is movingrdquo
119958119958 = (uvw) velocity vector 120513120513 equiv 120597120597
120597120597119909119909 120597120597
120597120597119910119910 120597120597
120597120597119911119911 gradient operator
called ldquonablardquo
Nomenclature summary
bull δ Greek small delta infinitesimal small size eg δV= small air volume bull partial derivative of a quantity with respect to a coordinate eg x bull total differential derivative of a quantity with respect to all dependent coordinates bull Lagrangian (or total) derivative
10
xpartpart
dxd
2 Basics
Trajectories started 850 hPa - top between 2604-29041986 - bottom between 3004-05051986
Example nuclear reactor accident Chernobyl on 26041986 0130 am Trajectories (=flight path) are used for the prediction of air movement
11
2 Basics
Kraus (2004)
Differentiation following the motion - Lagrangian perspective
Think about examples in your everyday life for Eulerrsquos and Lagrangersquos differentiation Give three examples
Lagrange and Euler differentiation
12
2 Basics
22 Basics - Mathematical add on
- Vector operations - Nabla operator
13
22 Basics - Mathematical add on September 16 2014
Modified slides from Clemens Simmer University Bonn
Vector operations - Multiplication with a scalar a -
=
====
z
y
x
z
y
x
ii
avavav
vvv
aavavavva
bull With a scalar multiplication the vector stays a vector
bull Each element of a vector is multiplied individually with a scalar a
bull The vector extends (or shortens) itself by the factor a bull Convention With the multiplication scalar ndash vector we
donrsquot use a point (like with scalar ndash scalar)
22 Mathematical add on
14
Vector operations - Scalar product -
iii
iiiizzyyxx
z
y
x
z
y
x
vfvfvffvfvfvfff
vvv
vffv sum =equiv=++=
sdot
=sdot=sdot
bull The scalar product of two vectors is a scalar bull It is multiplied component-wise then added bull Convention The scalar product is marked by a multiplication sign (middot) bull It is at a maximum with parallel vectors and disappears when the
vectors are perpendicular to each other
f
α
v
αcosfvfv
=sdot
22 Mathematical add on
15
Vector operations ndash vector product
bcacbabac
babababababa
babakbabajbabaibbbaaakji
axbbac
xyyx
zxxz
yzzy
xyyxzxxzyzzy
zyx
zyx
perpandperp=times=
minusminusminus
=
minus+minusminusminus==minus=times=
sin
)()()(
α
a b
bull The vector product of two vectors is again a vector bull Convention The vector product or cross product is marked by an ldquoxrdquo bull If is turned to on the fastest route possible the vector (from the vector product) points in the direction in which a right-hand screw would move (right hand rule) bull With parallel vectors it disappears and it reaches its maximum when the
two vectors are at right angles to each other
16
22 Mathematical add on
Graphically - Pooling of the spatial gradients towards the spatial coordinate axes rarr vector
- Gradient points towards the largest increase of the quantity
- Amount is the magnitude of the derivative pointing towards the largest increase
Nabla operator ndash spatial gradient
with
Operator -Nabla
equivpart=nabla=nabla
nabla
partpartpartpartpartpart
z
y
x
i
17
bull Nabla is a (vector) operator that works to the right
bull It only results in a value if it stands left to an arithmetic expression bull If it stands to the right of an arithmetic expression it keeps its operator function (and ldquowaitsrdquo for application)
22 Mathematical add on
TTgradT
z
y
x
zTyTxT
=
=equivnabla
partpartpartpartpartpart
partpartpartpartpartpart
zw
yv
xu
wvu
udivu
z
y
x
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpart
minuspartpart
partpart
minuspartpart
partpart
minuspartpart
==
times
equivequivtimesnabla part
partpartpart
partpart
partpartpartpartpartpart
yu
xv
xw
zu
zv
yw
wvu
kji
wvu
urotu zyx
z
y
x
Vector product ldquoxrdquo vector
Scalar product ldquordquo scalar
Product with a Scalar vector
Nabla operator 22 Mathematical add on
18
Nabla operator ndash algorithms
=nablanenabla=
=
partpartpartpartpartpart
partpartpartpartpartpart
partpartpartpartpartpart
z
y
x
z
y
x
zTyTxT
TTT
TTT
zw
yv
xu
wvu
u
zw
yv
xu
wvu
udivu
z
y
x
z
y
x
partpart
+partpart
+partpart
=
sdot
=nablasdot
ne
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpartpartpartpartpart
22 Mathematical add on
Note Nabla is a (vector) operator ie the sequence must not be changed here
19
State variables for atmosphere and ocean
u Wind = (uvw) in ms T Temperature in degC or K p Pressure in Nm2 = 1 Pa (1 hPa = 102 Pa = 1 mbar) Specifically for the atmosphere q Specific humidity in kg water vapourkg moist air Specifically for the ocean S Salinity in kg saltkg seawater (often the salt content is expressed in practical salinity unit psu = kg salt1000 kg seawater) 20
2 Basics
I Laws of motion for a fluid parcel in x-y-z- directions (applying Newtons 1 and 2 law)
II Conservation of mass
III Law of thermodynamics (including motion) rarr5 equations for the evolution of the fluid (5 unknowns)
21
3 Equations of motion - nonrotating fluid 3 Equations of motion - nonrotating fluid
Momentum conservation The conservation of momentum is represented by
Newtonrsquos first law (ldquoLex primardquo) (momentum = mass middot velocity)
momentum = const at absence of forces
Momentum sentence (ldquoLex secundardquo) temporal change in momentum = Force (mass middot acceleration = Force)
22
Fu=
dtdM t time M mass u velocity vector F Force vector
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Forces on a fluid parcel
23
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119958119958119863119863119905119905
= F (Eq 6-2)
120588120588 density (120575120575119909119909 120575120575119910119910 120575120575119911119911) fluid parcel with infinitesimal dimensions 120575120575119872119872 mass of the parcel 120575120575119872119872 = 120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119958119958 parcel velocity F net force
119863119863119958119958119863119863119905119905
= 120597120597119958119958120597120597119905119905
+ u 120597120597119958119958120597120597119909119909
+v 120597120597119958119958120597120597119910119910
+w 120597120597119958119958120597120597119911119911
= 120597120597119958119958120597120597119905119905
+ (119958119958 ∙ 120571120571) 119958119958
Gravitational force
24
Fgravity= g 120575120575119872119872
Fgravity= minusg 120588120588119963119963 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-3)
Fgravity Gravitational force M mass g gravity acceleration g = const 120588120588 density 119963119963 unit vector in upward direction
Marshall and Plumb (2008)
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Pressure gradient force
25
Fpressure= 119865119865119909119909 119865119865119910119910 119865119865119911119911
=minus 120597120597119901119901120597120597119909119909
120597120597119901119901120597120597119910119910
120597120597119901119901120597120597119911119911
120575120575119909119909 120575120575119910119910 120575120575119911119911
=minus 120571120571119901119901 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-4)
F(A) = p ∙ 119860119860 = 119901119901 (x minus 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911
F(B) = p ∙ 119861119861 = minus119901119901 (x + 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911 Fx = F(A) + F(B)
Apply Taylor expansion (A21) at midpoint of parcel and neglect small terms (see book)
119901119901 pressure
Friction force
26
3 Equations of motion - nonrotating fluid
- Friction force operates on the earthrsquos surface
- the greater the surface roughness the higher the friction force
- the greater the wind velocity the higher the friction force
- friction force takes effect in vertical distances of ~ 100 m up to 1000 m (rarr atmospheric boundary layer Chapter 7)
Ffriction = 120588120588 ℱ 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-5)
ℱ frictional force per unit mass 120588120588 density
Equations of motion bull All three forces together in Eq 6-2 gives
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119854119854119863119863119905119905
= F119901119901119901119901119891119891119901119901119901119901119906119906119901119901119891119891 + F119892119892119901119901119901119901119907119907119891119891119905119905119910119910 + F119891119891119901119901119891119891119901119901119905119905119891119891119891119891119891119891
bull Re-arranging leads to equation of motion for fluid parcels
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = ℱ (Equation 6-6)
27
3 Equations of motion - nonrotating fluid
120588120588 density u velocity vector F Force vector ℱ frictional force per unit mass p pressure g gravity acceleration 119963119963 unit vector in z-direction
Equations of motion ndash component form
(Eq 6-6) in Cartesian coordinates
bull 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
bull 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
bull 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
28
3 Equations of motion - nonrotating fluid
119906119906 zonal velocity 119907119907meridional velocity w vertical velocity 120588120588 density ℱ frictional force per unit mass p pressure g gravity acceleration
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Differentiation following the motion - Lagrange ndash Euler derivatives
7
2 Basics
Example wind blows over hill cloud forms at the ridge of the lee wave steady state assumption eq cloud does not change in time Lagrangian derivative (after Lagrange 1736-1813)
120597120597119862119862120597120597119905119905 fixed particle ne 0
Eulerian derivative (after Euler 1707-1783)
120597120597119862119862120597120597119905119905 fixed point in space = 0
C = C(xyzt) quantity ie cloud amount 120597120597120597120597119905119905
partial derivatives other variables are kept fixed during the differentiation
Leonhard Euler (CH)
Joseph-Louis Lagrange (I)
Differentiation following the motion mathematically
8
2 Basics
For arbitrary small variations
120575120575119862119862 = 120597120597119862119862120597120597119905119905
120575120575119905119905 + 120597120597119862119862120597120597119909119909
120575120575119909119909 + 120597120597119862119862120597120597119910119910
120575120575119910119910 + 120597120597119862119862120597120597119911119911
120575120575119911119911
120575120575119862119862 119891119891119891119891119909119909119891119891119891119891 119901119901119901119901119901119901119905119905119891119891119901119901119901119901119891119891 =
120597120597119862119862120597120597119905119905
+ 119906119906120597120597119862119862120597120597119909119909
+ 119907119907 120597120597119862119862120597120597119910119910
+ w 120597120597119862119862120597120597119911119911
120575120575119905119905
In limit of small variations
120597120597119862119862120597120597119905119905 119891119891119891119891119909119909119891119891119891119891 119901119901119901119901119901119901119905119905119891119891119901119901119901119901119891119891 =
120597120597119862119862120597120597119905119905
+ 119906119906 120597120597119862119862120597120597119909119909
+ 119907119907 120597120597119862119862120597120597119910119910
+ w 120597120597119862119862120597120597119911119911
= 119863119863119862119862119863119863119905119905
119863119863
119863119863119905119905 119863119863
119863119863119905119905= 120597120597
120597120597119905119905+ 119906119906 120597120597
120597120597119909119909+ 119907119907 120597120597
120597120597119910119910 + w 120597120597
120597120597119911119911 equiv 120597120597
120597120597119905119905 + 119958119958 ∙ 120513120513
δ Greek small delta infinitesimal small size
Insert 120575120575119909119909 = u120575120575119905119905 120575120575119910119910 = v120575120575119905119905 120575120575119911119911 = w120575120575119905119905
Differentiation following the motion mathematically
9
2 Basics
Lagrangian or total derivative (after Lagrange 1736-1813)
119863119863119863119863119905119905 fixed particle equiv 120597120597
120597120597119905119905+ 119906119906 120597120597
120597120597119909119909+ 119907119907 120597120597
120597120597119910119910 + w 120597120597
120597120597119911119911
equiv 120597120597120597120597119905119905
+ 119958119958 ∙ 120513120513 (6-1)
ldquoTime rate of change of some characteristic of a particular element of fluid which in general is changing its positionrdquo Eulerian derivative (after Euler 1707-1783)
120597120597120597120597119905119905 fixed point
ldquoTime rate of change of some characteristic at a fixed point in space but with constantly changing fluid element because the fluid is movingrdquo
119958119958 = (uvw) velocity vector 120513120513 equiv 120597120597
120597120597119909119909 120597120597
120597120597119910119910 120597120597
120597120597119911119911 gradient operator
called ldquonablardquo
Nomenclature summary
bull δ Greek small delta infinitesimal small size eg δV= small air volume bull partial derivative of a quantity with respect to a coordinate eg x bull total differential derivative of a quantity with respect to all dependent coordinates bull Lagrangian (or total) derivative
10
xpartpart
dxd
2 Basics
Trajectories started 850 hPa - top between 2604-29041986 - bottom between 3004-05051986
Example nuclear reactor accident Chernobyl on 26041986 0130 am Trajectories (=flight path) are used for the prediction of air movement
11
2 Basics
Kraus (2004)
Differentiation following the motion - Lagrangian perspective
Think about examples in your everyday life for Eulerrsquos and Lagrangersquos differentiation Give three examples
Lagrange and Euler differentiation
12
2 Basics
22 Basics - Mathematical add on
- Vector operations - Nabla operator
13
22 Basics - Mathematical add on September 16 2014
Modified slides from Clemens Simmer University Bonn
Vector operations - Multiplication with a scalar a -
=
====
z
y
x
z
y
x
ii
avavav
vvv
aavavavva
bull With a scalar multiplication the vector stays a vector
bull Each element of a vector is multiplied individually with a scalar a
bull The vector extends (or shortens) itself by the factor a bull Convention With the multiplication scalar ndash vector we
donrsquot use a point (like with scalar ndash scalar)
22 Mathematical add on
14
Vector operations - Scalar product -
iii
iiiizzyyxx
z
y
x
z
y
x
vfvfvffvfvfvfff
vvv
vffv sum =equiv=++=
sdot
=sdot=sdot
bull The scalar product of two vectors is a scalar bull It is multiplied component-wise then added bull Convention The scalar product is marked by a multiplication sign (middot) bull It is at a maximum with parallel vectors and disappears when the
vectors are perpendicular to each other
f
α
v
αcosfvfv
=sdot
22 Mathematical add on
15
Vector operations ndash vector product
bcacbabac
babababababa
babakbabajbabaibbbaaakji
axbbac
xyyx
zxxz
yzzy
xyyxzxxzyzzy
zyx
zyx
perpandperp=times=
minusminusminus
=
minus+minusminusminus==minus=times=
sin
)()()(
α
a b
bull The vector product of two vectors is again a vector bull Convention The vector product or cross product is marked by an ldquoxrdquo bull If is turned to on the fastest route possible the vector (from the vector product) points in the direction in which a right-hand screw would move (right hand rule) bull With parallel vectors it disappears and it reaches its maximum when the
two vectors are at right angles to each other
16
22 Mathematical add on
Graphically - Pooling of the spatial gradients towards the spatial coordinate axes rarr vector
- Gradient points towards the largest increase of the quantity
- Amount is the magnitude of the derivative pointing towards the largest increase
Nabla operator ndash spatial gradient
with
Operator -Nabla
equivpart=nabla=nabla
nabla
partpartpartpartpartpart
z
y
x
i
17
bull Nabla is a (vector) operator that works to the right
bull It only results in a value if it stands left to an arithmetic expression bull If it stands to the right of an arithmetic expression it keeps its operator function (and ldquowaitsrdquo for application)
22 Mathematical add on
TTgradT
z
y
x
zTyTxT
=
=equivnabla
partpartpartpartpartpart
partpartpartpartpartpart
zw
yv
xu
wvu
udivu
z
y
x
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpart
minuspartpart
partpart
minuspartpart
partpart
minuspartpart
==
times
equivequivtimesnabla part
partpartpart
partpart
partpartpartpartpartpart
yu
xv
xw
zu
zv
yw
wvu
kji
wvu
urotu zyx
z
y
x
Vector product ldquoxrdquo vector
Scalar product ldquordquo scalar
Product with a Scalar vector
Nabla operator 22 Mathematical add on
18
Nabla operator ndash algorithms
=nablanenabla=
=
partpartpartpartpartpart
partpartpartpartpartpart
partpartpartpartpartpart
z
y
x
z
y
x
zTyTxT
TTT
TTT
zw
yv
xu
wvu
u
zw
yv
xu
wvu
udivu
z
y
x
z
y
x
partpart
+partpart
+partpart
=
sdot
=nablasdot
ne
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpartpartpartpartpart
22 Mathematical add on
Note Nabla is a (vector) operator ie the sequence must not be changed here
19
State variables for atmosphere and ocean
u Wind = (uvw) in ms T Temperature in degC or K p Pressure in Nm2 = 1 Pa (1 hPa = 102 Pa = 1 mbar) Specifically for the atmosphere q Specific humidity in kg water vapourkg moist air Specifically for the ocean S Salinity in kg saltkg seawater (often the salt content is expressed in practical salinity unit psu = kg salt1000 kg seawater) 20
2 Basics
I Laws of motion for a fluid parcel in x-y-z- directions (applying Newtons 1 and 2 law)
II Conservation of mass
III Law of thermodynamics (including motion) rarr5 equations for the evolution of the fluid (5 unknowns)
21
3 Equations of motion - nonrotating fluid 3 Equations of motion - nonrotating fluid
Momentum conservation The conservation of momentum is represented by
Newtonrsquos first law (ldquoLex primardquo) (momentum = mass middot velocity)
momentum = const at absence of forces
Momentum sentence (ldquoLex secundardquo) temporal change in momentum = Force (mass middot acceleration = Force)
22
Fu=
dtdM t time M mass u velocity vector F Force vector
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Forces on a fluid parcel
23
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119958119958119863119863119905119905
= F (Eq 6-2)
120588120588 density (120575120575119909119909 120575120575119910119910 120575120575119911119911) fluid parcel with infinitesimal dimensions 120575120575119872119872 mass of the parcel 120575120575119872119872 = 120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119958119958 parcel velocity F net force
119863119863119958119958119863119863119905119905
= 120597120597119958119958120597120597119905119905
+ u 120597120597119958119958120597120597119909119909
+v 120597120597119958119958120597120597119910119910
+w 120597120597119958119958120597120597119911119911
= 120597120597119958119958120597120597119905119905
+ (119958119958 ∙ 120571120571) 119958119958
Gravitational force
24
Fgravity= g 120575120575119872119872
Fgravity= minusg 120588120588119963119963 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-3)
Fgravity Gravitational force M mass g gravity acceleration g = const 120588120588 density 119963119963 unit vector in upward direction
Marshall and Plumb (2008)
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Pressure gradient force
25
Fpressure= 119865119865119909119909 119865119865119910119910 119865119865119911119911
=minus 120597120597119901119901120597120597119909119909
120597120597119901119901120597120597119910119910
120597120597119901119901120597120597119911119911
120575120575119909119909 120575120575119910119910 120575120575119911119911
=minus 120571120571119901119901 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-4)
F(A) = p ∙ 119860119860 = 119901119901 (x minus 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911
F(B) = p ∙ 119861119861 = minus119901119901 (x + 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911 Fx = F(A) + F(B)
Apply Taylor expansion (A21) at midpoint of parcel and neglect small terms (see book)
119901119901 pressure
Friction force
26
3 Equations of motion - nonrotating fluid
- Friction force operates on the earthrsquos surface
- the greater the surface roughness the higher the friction force
- the greater the wind velocity the higher the friction force
- friction force takes effect in vertical distances of ~ 100 m up to 1000 m (rarr atmospheric boundary layer Chapter 7)
Ffriction = 120588120588 ℱ 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-5)
ℱ frictional force per unit mass 120588120588 density
Equations of motion bull All three forces together in Eq 6-2 gives
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119854119854119863119863119905119905
= F119901119901119901119901119891119891119901119901119901119901119906119906119901119901119891119891 + F119892119892119901119901119901119901119907119907119891119891119905119905119910119910 + F119891119891119901119901119891119891119901119901119905119905119891119891119891119891119891119891
bull Re-arranging leads to equation of motion for fluid parcels
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = ℱ (Equation 6-6)
27
3 Equations of motion - nonrotating fluid
120588120588 density u velocity vector F Force vector ℱ frictional force per unit mass p pressure g gravity acceleration 119963119963 unit vector in z-direction
Equations of motion ndash component form
(Eq 6-6) in Cartesian coordinates
bull 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
bull 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
bull 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
28
3 Equations of motion - nonrotating fluid
119906119906 zonal velocity 119907119907meridional velocity w vertical velocity 120588120588 density ℱ frictional force per unit mass p pressure g gravity acceleration
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Differentiation following the motion mathematically
8
2 Basics
For arbitrary small variations
120575120575119862119862 = 120597120597119862119862120597120597119905119905
120575120575119905119905 + 120597120597119862119862120597120597119909119909
120575120575119909119909 + 120597120597119862119862120597120597119910119910
120575120575119910119910 + 120597120597119862119862120597120597119911119911
120575120575119911119911
120575120575119862119862 119891119891119891119891119909119909119891119891119891119891 119901119901119901119901119901119901119905119905119891119891119901119901119901119901119891119891 =
120597120597119862119862120597120597119905119905
+ 119906119906120597120597119862119862120597120597119909119909
+ 119907119907 120597120597119862119862120597120597119910119910
+ w 120597120597119862119862120597120597119911119911
120575120575119905119905
In limit of small variations
120597120597119862119862120597120597119905119905 119891119891119891119891119909119909119891119891119891119891 119901119901119901119901119901119901119905119905119891119891119901119901119901119901119891119891 =
120597120597119862119862120597120597119905119905
+ 119906119906 120597120597119862119862120597120597119909119909
+ 119907119907 120597120597119862119862120597120597119910119910
+ w 120597120597119862119862120597120597119911119911
= 119863119863119862119862119863119863119905119905
119863119863
119863119863119905119905 119863119863
119863119863119905119905= 120597120597
120597120597119905119905+ 119906119906 120597120597
120597120597119909119909+ 119907119907 120597120597
120597120597119910119910 + w 120597120597
120597120597119911119911 equiv 120597120597
120597120597119905119905 + 119958119958 ∙ 120513120513
δ Greek small delta infinitesimal small size
Insert 120575120575119909119909 = u120575120575119905119905 120575120575119910119910 = v120575120575119905119905 120575120575119911119911 = w120575120575119905119905
Differentiation following the motion mathematically
9
2 Basics
Lagrangian or total derivative (after Lagrange 1736-1813)
119863119863119863119863119905119905 fixed particle equiv 120597120597
120597120597119905119905+ 119906119906 120597120597
120597120597119909119909+ 119907119907 120597120597
120597120597119910119910 + w 120597120597
120597120597119911119911
equiv 120597120597120597120597119905119905
+ 119958119958 ∙ 120513120513 (6-1)
ldquoTime rate of change of some characteristic of a particular element of fluid which in general is changing its positionrdquo Eulerian derivative (after Euler 1707-1783)
120597120597120597120597119905119905 fixed point
ldquoTime rate of change of some characteristic at a fixed point in space but with constantly changing fluid element because the fluid is movingrdquo
119958119958 = (uvw) velocity vector 120513120513 equiv 120597120597
120597120597119909119909 120597120597
120597120597119910119910 120597120597
120597120597119911119911 gradient operator
called ldquonablardquo
Nomenclature summary
bull δ Greek small delta infinitesimal small size eg δV= small air volume bull partial derivative of a quantity with respect to a coordinate eg x bull total differential derivative of a quantity with respect to all dependent coordinates bull Lagrangian (or total) derivative
10
xpartpart
dxd
2 Basics
Trajectories started 850 hPa - top between 2604-29041986 - bottom between 3004-05051986
Example nuclear reactor accident Chernobyl on 26041986 0130 am Trajectories (=flight path) are used for the prediction of air movement
11
2 Basics
Kraus (2004)
Differentiation following the motion - Lagrangian perspective
Think about examples in your everyday life for Eulerrsquos and Lagrangersquos differentiation Give three examples
Lagrange and Euler differentiation
12
2 Basics
22 Basics - Mathematical add on
- Vector operations - Nabla operator
13
22 Basics - Mathematical add on September 16 2014
Modified slides from Clemens Simmer University Bonn
Vector operations - Multiplication with a scalar a -
=
====
z
y
x
z
y
x
ii
avavav
vvv
aavavavva
bull With a scalar multiplication the vector stays a vector
bull Each element of a vector is multiplied individually with a scalar a
bull The vector extends (or shortens) itself by the factor a bull Convention With the multiplication scalar ndash vector we
donrsquot use a point (like with scalar ndash scalar)
22 Mathematical add on
14
Vector operations - Scalar product -
iii
iiiizzyyxx
z
y
x
z
y
x
vfvfvffvfvfvfff
vvv
vffv sum =equiv=++=
sdot
=sdot=sdot
bull The scalar product of two vectors is a scalar bull It is multiplied component-wise then added bull Convention The scalar product is marked by a multiplication sign (middot) bull It is at a maximum with parallel vectors and disappears when the
vectors are perpendicular to each other
f
α
v
αcosfvfv
=sdot
22 Mathematical add on
15
Vector operations ndash vector product
bcacbabac
babababababa
babakbabajbabaibbbaaakji
axbbac
xyyx
zxxz
yzzy
xyyxzxxzyzzy
zyx
zyx
perpandperp=times=
minusminusminus
=
minus+minusminusminus==minus=times=
sin
)()()(
α
a b
bull The vector product of two vectors is again a vector bull Convention The vector product or cross product is marked by an ldquoxrdquo bull If is turned to on the fastest route possible the vector (from the vector product) points in the direction in which a right-hand screw would move (right hand rule) bull With parallel vectors it disappears and it reaches its maximum when the
two vectors are at right angles to each other
16
22 Mathematical add on
Graphically - Pooling of the spatial gradients towards the spatial coordinate axes rarr vector
- Gradient points towards the largest increase of the quantity
- Amount is the magnitude of the derivative pointing towards the largest increase
Nabla operator ndash spatial gradient
with
Operator -Nabla
equivpart=nabla=nabla
nabla
partpartpartpartpartpart
z
y
x
i
17
bull Nabla is a (vector) operator that works to the right
bull It only results in a value if it stands left to an arithmetic expression bull If it stands to the right of an arithmetic expression it keeps its operator function (and ldquowaitsrdquo for application)
22 Mathematical add on
TTgradT
z
y
x
zTyTxT
=
=equivnabla
partpartpartpartpartpart
partpartpartpartpartpart
zw
yv
xu
wvu
udivu
z
y
x
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpart
minuspartpart
partpart
minuspartpart
partpart
minuspartpart
==
times
equivequivtimesnabla part
partpartpart
partpart
partpartpartpartpartpart
yu
xv
xw
zu
zv
yw
wvu
kji
wvu
urotu zyx
z
y
x
Vector product ldquoxrdquo vector
Scalar product ldquordquo scalar
Product with a Scalar vector
Nabla operator 22 Mathematical add on
18
Nabla operator ndash algorithms
=nablanenabla=
=
partpartpartpartpartpart
partpartpartpartpartpart
partpartpartpartpartpart
z
y
x
z
y
x
zTyTxT
TTT
TTT
zw
yv
xu
wvu
u
zw
yv
xu
wvu
udivu
z
y
x
z
y
x
partpart
+partpart
+partpart
=
sdot
=nablasdot
ne
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpartpartpartpartpart
22 Mathematical add on
Note Nabla is a (vector) operator ie the sequence must not be changed here
19
State variables for atmosphere and ocean
u Wind = (uvw) in ms T Temperature in degC or K p Pressure in Nm2 = 1 Pa (1 hPa = 102 Pa = 1 mbar) Specifically for the atmosphere q Specific humidity in kg water vapourkg moist air Specifically for the ocean S Salinity in kg saltkg seawater (often the salt content is expressed in practical salinity unit psu = kg salt1000 kg seawater) 20
2 Basics
I Laws of motion for a fluid parcel in x-y-z- directions (applying Newtons 1 and 2 law)
II Conservation of mass
III Law of thermodynamics (including motion) rarr5 equations for the evolution of the fluid (5 unknowns)
21
3 Equations of motion - nonrotating fluid 3 Equations of motion - nonrotating fluid
Momentum conservation The conservation of momentum is represented by
Newtonrsquos first law (ldquoLex primardquo) (momentum = mass middot velocity)
momentum = const at absence of forces
Momentum sentence (ldquoLex secundardquo) temporal change in momentum = Force (mass middot acceleration = Force)
22
Fu=
dtdM t time M mass u velocity vector F Force vector
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Forces on a fluid parcel
23
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119958119958119863119863119905119905
= F (Eq 6-2)
120588120588 density (120575120575119909119909 120575120575119910119910 120575120575119911119911) fluid parcel with infinitesimal dimensions 120575120575119872119872 mass of the parcel 120575120575119872119872 = 120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119958119958 parcel velocity F net force
119863119863119958119958119863119863119905119905
= 120597120597119958119958120597120597119905119905
+ u 120597120597119958119958120597120597119909119909
+v 120597120597119958119958120597120597119910119910
+w 120597120597119958119958120597120597119911119911
= 120597120597119958119958120597120597119905119905
+ (119958119958 ∙ 120571120571) 119958119958
Gravitational force
24
Fgravity= g 120575120575119872119872
Fgravity= minusg 120588120588119963119963 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-3)
Fgravity Gravitational force M mass g gravity acceleration g = const 120588120588 density 119963119963 unit vector in upward direction
Marshall and Plumb (2008)
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Pressure gradient force
25
Fpressure= 119865119865119909119909 119865119865119910119910 119865119865119911119911
=minus 120597120597119901119901120597120597119909119909
120597120597119901119901120597120597119910119910
120597120597119901119901120597120597119911119911
120575120575119909119909 120575120575119910119910 120575120575119911119911
=minus 120571120571119901119901 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-4)
F(A) = p ∙ 119860119860 = 119901119901 (x minus 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911
F(B) = p ∙ 119861119861 = minus119901119901 (x + 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911 Fx = F(A) + F(B)
Apply Taylor expansion (A21) at midpoint of parcel and neglect small terms (see book)
119901119901 pressure
Friction force
26
3 Equations of motion - nonrotating fluid
- Friction force operates on the earthrsquos surface
- the greater the surface roughness the higher the friction force
- the greater the wind velocity the higher the friction force
- friction force takes effect in vertical distances of ~ 100 m up to 1000 m (rarr atmospheric boundary layer Chapter 7)
Ffriction = 120588120588 ℱ 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-5)
ℱ frictional force per unit mass 120588120588 density
Equations of motion bull All three forces together in Eq 6-2 gives
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119854119854119863119863119905119905
= F119901119901119901119901119891119891119901119901119901119901119906119906119901119901119891119891 + F119892119892119901119901119901119901119907119907119891119891119905119905119910119910 + F119891119891119901119901119891119891119901119901119905119905119891119891119891119891119891119891
bull Re-arranging leads to equation of motion for fluid parcels
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = ℱ (Equation 6-6)
27
3 Equations of motion - nonrotating fluid
120588120588 density u velocity vector F Force vector ℱ frictional force per unit mass p pressure g gravity acceleration 119963119963 unit vector in z-direction
Equations of motion ndash component form
(Eq 6-6) in Cartesian coordinates
bull 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
bull 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
bull 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
28
3 Equations of motion - nonrotating fluid
119906119906 zonal velocity 119907119907meridional velocity w vertical velocity 120588120588 density ℱ frictional force per unit mass p pressure g gravity acceleration
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Differentiation following the motion mathematically
9
2 Basics
Lagrangian or total derivative (after Lagrange 1736-1813)
119863119863119863119863119905119905 fixed particle equiv 120597120597
120597120597119905119905+ 119906119906 120597120597
120597120597119909119909+ 119907119907 120597120597
120597120597119910119910 + w 120597120597
120597120597119911119911
equiv 120597120597120597120597119905119905
+ 119958119958 ∙ 120513120513 (6-1)
ldquoTime rate of change of some characteristic of a particular element of fluid which in general is changing its positionrdquo Eulerian derivative (after Euler 1707-1783)
120597120597120597120597119905119905 fixed point
ldquoTime rate of change of some characteristic at a fixed point in space but with constantly changing fluid element because the fluid is movingrdquo
119958119958 = (uvw) velocity vector 120513120513 equiv 120597120597
120597120597119909119909 120597120597
120597120597119910119910 120597120597
120597120597119911119911 gradient operator
called ldquonablardquo
Nomenclature summary
bull δ Greek small delta infinitesimal small size eg δV= small air volume bull partial derivative of a quantity with respect to a coordinate eg x bull total differential derivative of a quantity with respect to all dependent coordinates bull Lagrangian (or total) derivative
10
xpartpart
dxd
2 Basics
Trajectories started 850 hPa - top between 2604-29041986 - bottom between 3004-05051986
Example nuclear reactor accident Chernobyl on 26041986 0130 am Trajectories (=flight path) are used for the prediction of air movement
11
2 Basics
Kraus (2004)
Differentiation following the motion - Lagrangian perspective
Think about examples in your everyday life for Eulerrsquos and Lagrangersquos differentiation Give three examples
Lagrange and Euler differentiation
12
2 Basics
22 Basics - Mathematical add on
- Vector operations - Nabla operator
13
22 Basics - Mathematical add on September 16 2014
Modified slides from Clemens Simmer University Bonn
Vector operations - Multiplication with a scalar a -
=
====
z
y
x
z
y
x
ii
avavav
vvv
aavavavva
bull With a scalar multiplication the vector stays a vector
bull Each element of a vector is multiplied individually with a scalar a
bull The vector extends (or shortens) itself by the factor a bull Convention With the multiplication scalar ndash vector we
donrsquot use a point (like with scalar ndash scalar)
22 Mathematical add on
14
Vector operations - Scalar product -
iii
iiiizzyyxx
z
y
x
z
y
x
vfvfvffvfvfvfff
vvv
vffv sum =equiv=++=
sdot
=sdot=sdot
bull The scalar product of two vectors is a scalar bull It is multiplied component-wise then added bull Convention The scalar product is marked by a multiplication sign (middot) bull It is at a maximum with parallel vectors and disappears when the
vectors are perpendicular to each other
f
α
v
αcosfvfv
=sdot
22 Mathematical add on
15
Vector operations ndash vector product
bcacbabac
babababababa
babakbabajbabaibbbaaakji
axbbac
xyyx
zxxz
yzzy
xyyxzxxzyzzy
zyx
zyx
perpandperp=times=
minusminusminus
=
minus+minusminusminus==minus=times=
sin
)()()(
α
a b
bull The vector product of two vectors is again a vector bull Convention The vector product or cross product is marked by an ldquoxrdquo bull If is turned to on the fastest route possible the vector (from the vector product) points in the direction in which a right-hand screw would move (right hand rule) bull With parallel vectors it disappears and it reaches its maximum when the
two vectors are at right angles to each other
16
22 Mathematical add on
Graphically - Pooling of the spatial gradients towards the spatial coordinate axes rarr vector
- Gradient points towards the largest increase of the quantity
- Amount is the magnitude of the derivative pointing towards the largest increase
Nabla operator ndash spatial gradient
with
Operator -Nabla
equivpart=nabla=nabla
nabla
partpartpartpartpartpart
z
y
x
i
17
bull Nabla is a (vector) operator that works to the right
bull It only results in a value if it stands left to an arithmetic expression bull If it stands to the right of an arithmetic expression it keeps its operator function (and ldquowaitsrdquo for application)
22 Mathematical add on
TTgradT
z
y
x
zTyTxT
=
=equivnabla
partpartpartpartpartpart
partpartpartpartpartpart
zw
yv
xu
wvu
udivu
z
y
x
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpart
minuspartpart
partpart
minuspartpart
partpart
minuspartpart
==
times
equivequivtimesnabla part
partpartpart
partpart
partpartpartpartpartpart
yu
xv
xw
zu
zv
yw
wvu
kji
wvu
urotu zyx
z
y
x
Vector product ldquoxrdquo vector
Scalar product ldquordquo scalar
Product with a Scalar vector
Nabla operator 22 Mathematical add on
18
Nabla operator ndash algorithms
=nablanenabla=
=
partpartpartpartpartpart
partpartpartpartpartpart
partpartpartpartpartpart
z
y
x
z
y
x
zTyTxT
TTT
TTT
zw
yv
xu
wvu
u
zw
yv
xu
wvu
udivu
z
y
x
z
y
x
partpart
+partpart
+partpart
=
sdot
=nablasdot
ne
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpartpartpartpartpart
22 Mathematical add on
Note Nabla is a (vector) operator ie the sequence must not be changed here
19
State variables for atmosphere and ocean
u Wind = (uvw) in ms T Temperature in degC or K p Pressure in Nm2 = 1 Pa (1 hPa = 102 Pa = 1 mbar) Specifically for the atmosphere q Specific humidity in kg water vapourkg moist air Specifically for the ocean S Salinity in kg saltkg seawater (often the salt content is expressed in practical salinity unit psu = kg salt1000 kg seawater) 20
2 Basics
I Laws of motion for a fluid parcel in x-y-z- directions (applying Newtons 1 and 2 law)
II Conservation of mass
III Law of thermodynamics (including motion) rarr5 equations for the evolution of the fluid (5 unknowns)
21
3 Equations of motion - nonrotating fluid 3 Equations of motion - nonrotating fluid
Momentum conservation The conservation of momentum is represented by
Newtonrsquos first law (ldquoLex primardquo) (momentum = mass middot velocity)
momentum = const at absence of forces
Momentum sentence (ldquoLex secundardquo) temporal change in momentum = Force (mass middot acceleration = Force)
22
Fu=
dtdM t time M mass u velocity vector F Force vector
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Forces on a fluid parcel
23
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119958119958119863119863119905119905
= F (Eq 6-2)
120588120588 density (120575120575119909119909 120575120575119910119910 120575120575119911119911) fluid parcel with infinitesimal dimensions 120575120575119872119872 mass of the parcel 120575120575119872119872 = 120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119958119958 parcel velocity F net force
119863119863119958119958119863119863119905119905
= 120597120597119958119958120597120597119905119905
+ u 120597120597119958119958120597120597119909119909
+v 120597120597119958119958120597120597119910119910
+w 120597120597119958119958120597120597119911119911
= 120597120597119958119958120597120597119905119905
+ (119958119958 ∙ 120571120571) 119958119958
Gravitational force
24
Fgravity= g 120575120575119872119872
Fgravity= minusg 120588120588119963119963 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-3)
Fgravity Gravitational force M mass g gravity acceleration g = const 120588120588 density 119963119963 unit vector in upward direction
Marshall and Plumb (2008)
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Pressure gradient force
25
Fpressure= 119865119865119909119909 119865119865119910119910 119865119865119911119911
=minus 120597120597119901119901120597120597119909119909
120597120597119901119901120597120597119910119910
120597120597119901119901120597120597119911119911
120575120575119909119909 120575120575119910119910 120575120575119911119911
=minus 120571120571119901119901 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-4)
F(A) = p ∙ 119860119860 = 119901119901 (x minus 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911
F(B) = p ∙ 119861119861 = minus119901119901 (x + 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911 Fx = F(A) + F(B)
Apply Taylor expansion (A21) at midpoint of parcel and neglect small terms (see book)
119901119901 pressure
Friction force
26
3 Equations of motion - nonrotating fluid
- Friction force operates on the earthrsquos surface
- the greater the surface roughness the higher the friction force
- the greater the wind velocity the higher the friction force
- friction force takes effect in vertical distances of ~ 100 m up to 1000 m (rarr atmospheric boundary layer Chapter 7)
Ffriction = 120588120588 ℱ 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-5)
ℱ frictional force per unit mass 120588120588 density
Equations of motion bull All three forces together in Eq 6-2 gives
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119854119854119863119863119905119905
= F119901119901119901119901119891119891119901119901119901119901119906119906119901119901119891119891 + F119892119892119901119901119901119901119907119907119891119891119905119905119910119910 + F119891119891119901119901119891119891119901119901119905119905119891119891119891119891119891119891
bull Re-arranging leads to equation of motion for fluid parcels
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = ℱ (Equation 6-6)
27
3 Equations of motion - nonrotating fluid
120588120588 density u velocity vector F Force vector ℱ frictional force per unit mass p pressure g gravity acceleration 119963119963 unit vector in z-direction
Equations of motion ndash component form
(Eq 6-6) in Cartesian coordinates
bull 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
bull 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
bull 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
28
3 Equations of motion - nonrotating fluid
119906119906 zonal velocity 119907119907meridional velocity w vertical velocity 120588120588 density ℱ frictional force per unit mass p pressure g gravity acceleration
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Nomenclature summary
bull δ Greek small delta infinitesimal small size eg δV= small air volume bull partial derivative of a quantity with respect to a coordinate eg x bull total differential derivative of a quantity with respect to all dependent coordinates bull Lagrangian (or total) derivative
10
xpartpart
dxd
2 Basics
Trajectories started 850 hPa - top between 2604-29041986 - bottom between 3004-05051986
Example nuclear reactor accident Chernobyl on 26041986 0130 am Trajectories (=flight path) are used for the prediction of air movement
11
2 Basics
Kraus (2004)
Differentiation following the motion - Lagrangian perspective
Think about examples in your everyday life for Eulerrsquos and Lagrangersquos differentiation Give three examples
Lagrange and Euler differentiation
12
2 Basics
22 Basics - Mathematical add on
- Vector operations - Nabla operator
13
22 Basics - Mathematical add on September 16 2014
Modified slides from Clemens Simmer University Bonn
Vector operations - Multiplication with a scalar a -
=
====
z
y
x
z
y
x
ii
avavav
vvv
aavavavva
bull With a scalar multiplication the vector stays a vector
bull Each element of a vector is multiplied individually with a scalar a
bull The vector extends (or shortens) itself by the factor a bull Convention With the multiplication scalar ndash vector we
donrsquot use a point (like with scalar ndash scalar)
22 Mathematical add on
14
Vector operations - Scalar product -
iii
iiiizzyyxx
z
y
x
z
y
x
vfvfvffvfvfvfff
vvv
vffv sum =equiv=++=
sdot
=sdot=sdot
bull The scalar product of two vectors is a scalar bull It is multiplied component-wise then added bull Convention The scalar product is marked by a multiplication sign (middot) bull It is at a maximum with parallel vectors and disappears when the
vectors are perpendicular to each other
f
α
v
αcosfvfv
=sdot
22 Mathematical add on
15
Vector operations ndash vector product
bcacbabac
babababababa
babakbabajbabaibbbaaakji
axbbac
xyyx
zxxz
yzzy
xyyxzxxzyzzy
zyx
zyx
perpandperp=times=
minusminusminus
=
minus+minusminusminus==minus=times=
sin
)()()(
α
a b
bull The vector product of two vectors is again a vector bull Convention The vector product or cross product is marked by an ldquoxrdquo bull If is turned to on the fastest route possible the vector (from the vector product) points in the direction in which a right-hand screw would move (right hand rule) bull With parallel vectors it disappears and it reaches its maximum when the
two vectors are at right angles to each other
16
22 Mathematical add on
Graphically - Pooling of the spatial gradients towards the spatial coordinate axes rarr vector
- Gradient points towards the largest increase of the quantity
- Amount is the magnitude of the derivative pointing towards the largest increase
Nabla operator ndash spatial gradient
with
Operator -Nabla
equivpart=nabla=nabla
nabla
partpartpartpartpartpart
z
y
x
i
17
bull Nabla is a (vector) operator that works to the right
bull It only results in a value if it stands left to an arithmetic expression bull If it stands to the right of an arithmetic expression it keeps its operator function (and ldquowaitsrdquo for application)
22 Mathematical add on
TTgradT
z
y
x
zTyTxT
=
=equivnabla
partpartpartpartpartpart
partpartpartpartpartpart
zw
yv
xu
wvu
udivu
z
y
x
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpart
minuspartpart
partpart
minuspartpart
partpart
minuspartpart
==
times
equivequivtimesnabla part
partpartpart
partpart
partpartpartpartpartpart
yu
xv
xw
zu
zv
yw
wvu
kji
wvu
urotu zyx
z
y
x
Vector product ldquoxrdquo vector
Scalar product ldquordquo scalar
Product with a Scalar vector
Nabla operator 22 Mathematical add on
18
Nabla operator ndash algorithms
=nablanenabla=
=
partpartpartpartpartpart
partpartpartpartpartpart
partpartpartpartpartpart
z
y
x
z
y
x
zTyTxT
TTT
TTT
zw
yv
xu
wvu
u
zw
yv
xu
wvu
udivu
z
y
x
z
y
x
partpart
+partpart
+partpart
=
sdot
=nablasdot
ne
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpartpartpartpartpart
22 Mathematical add on
Note Nabla is a (vector) operator ie the sequence must not be changed here
19
State variables for atmosphere and ocean
u Wind = (uvw) in ms T Temperature in degC or K p Pressure in Nm2 = 1 Pa (1 hPa = 102 Pa = 1 mbar) Specifically for the atmosphere q Specific humidity in kg water vapourkg moist air Specifically for the ocean S Salinity in kg saltkg seawater (often the salt content is expressed in practical salinity unit psu = kg salt1000 kg seawater) 20
2 Basics
I Laws of motion for a fluid parcel in x-y-z- directions (applying Newtons 1 and 2 law)
II Conservation of mass
III Law of thermodynamics (including motion) rarr5 equations for the evolution of the fluid (5 unknowns)
21
3 Equations of motion - nonrotating fluid 3 Equations of motion - nonrotating fluid
Momentum conservation The conservation of momentum is represented by
Newtonrsquos first law (ldquoLex primardquo) (momentum = mass middot velocity)
momentum = const at absence of forces
Momentum sentence (ldquoLex secundardquo) temporal change in momentum = Force (mass middot acceleration = Force)
22
Fu=
dtdM t time M mass u velocity vector F Force vector
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Forces on a fluid parcel
23
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119958119958119863119863119905119905
= F (Eq 6-2)
120588120588 density (120575120575119909119909 120575120575119910119910 120575120575119911119911) fluid parcel with infinitesimal dimensions 120575120575119872119872 mass of the parcel 120575120575119872119872 = 120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119958119958 parcel velocity F net force
119863119863119958119958119863119863119905119905
= 120597120597119958119958120597120597119905119905
+ u 120597120597119958119958120597120597119909119909
+v 120597120597119958119958120597120597119910119910
+w 120597120597119958119958120597120597119911119911
= 120597120597119958119958120597120597119905119905
+ (119958119958 ∙ 120571120571) 119958119958
Gravitational force
24
Fgravity= g 120575120575119872119872
Fgravity= minusg 120588120588119963119963 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-3)
Fgravity Gravitational force M mass g gravity acceleration g = const 120588120588 density 119963119963 unit vector in upward direction
Marshall and Plumb (2008)
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Pressure gradient force
25
Fpressure= 119865119865119909119909 119865119865119910119910 119865119865119911119911
=minus 120597120597119901119901120597120597119909119909
120597120597119901119901120597120597119910119910
120597120597119901119901120597120597119911119911
120575120575119909119909 120575120575119910119910 120575120575119911119911
=minus 120571120571119901119901 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-4)
F(A) = p ∙ 119860119860 = 119901119901 (x minus 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911
F(B) = p ∙ 119861119861 = minus119901119901 (x + 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911 Fx = F(A) + F(B)
Apply Taylor expansion (A21) at midpoint of parcel and neglect small terms (see book)
119901119901 pressure
Friction force
26
3 Equations of motion - nonrotating fluid
- Friction force operates on the earthrsquos surface
- the greater the surface roughness the higher the friction force
- the greater the wind velocity the higher the friction force
- friction force takes effect in vertical distances of ~ 100 m up to 1000 m (rarr atmospheric boundary layer Chapter 7)
Ffriction = 120588120588 ℱ 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-5)
ℱ frictional force per unit mass 120588120588 density
Equations of motion bull All three forces together in Eq 6-2 gives
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119854119854119863119863119905119905
= F119901119901119901119901119891119891119901119901119901119901119906119906119901119901119891119891 + F119892119892119901119901119901119901119907119907119891119891119905119905119910119910 + F119891119891119901119901119891119891119901119901119905119905119891119891119891119891119891119891
bull Re-arranging leads to equation of motion for fluid parcels
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = ℱ (Equation 6-6)
27
3 Equations of motion - nonrotating fluid
120588120588 density u velocity vector F Force vector ℱ frictional force per unit mass p pressure g gravity acceleration 119963119963 unit vector in z-direction
Equations of motion ndash component form
(Eq 6-6) in Cartesian coordinates
bull 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
bull 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
bull 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
28
3 Equations of motion - nonrotating fluid
119906119906 zonal velocity 119907119907meridional velocity w vertical velocity 120588120588 density ℱ frictional force per unit mass p pressure g gravity acceleration
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Trajectories started 850 hPa - top between 2604-29041986 - bottom between 3004-05051986
Example nuclear reactor accident Chernobyl on 26041986 0130 am Trajectories (=flight path) are used for the prediction of air movement
11
2 Basics
Kraus (2004)
Differentiation following the motion - Lagrangian perspective
Think about examples in your everyday life for Eulerrsquos and Lagrangersquos differentiation Give three examples
Lagrange and Euler differentiation
12
2 Basics
22 Basics - Mathematical add on
- Vector operations - Nabla operator
13
22 Basics - Mathematical add on September 16 2014
Modified slides from Clemens Simmer University Bonn
Vector operations - Multiplication with a scalar a -
=
====
z
y
x
z
y
x
ii
avavav
vvv
aavavavva
bull With a scalar multiplication the vector stays a vector
bull Each element of a vector is multiplied individually with a scalar a
bull The vector extends (or shortens) itself by the factor a bull Convention With the multiplication scalar ndash vector we
donrsquot use a point (like with scalar ndash scalar)
22 Mathematical add on
14
Vector operations - Scalar product -
iii
iiiizzyyxx
z
y
x
z
y
x
vfvfvffvfvfvfff
vvv
vffv sum =equiv=++=
sdot
=sdot=sdot
bull The scalar product of two vectors is a scalar bull It is multiplied component-wise then added bull Convention The scalar product is marked by a multiplication sign (middot) bull It is at a maximum with parallel vectors and disappears when the
vectors are perpendicular to each other
f
α
v
αcosfvfv
=sdot
22 Mathematical add on
15
Vector operations ndash vector product
bcacbabac
babababababa
babakbabajbabaibbbaaakji
axbbac
xyyx
zxxz
yzzy
xyyxzxxzyzzy
zyx
zyx
perpandperp=times=
minusminusminus
=
minus+minusminusminus==minus=times=
sin
)()()(
α
a b
bull The vector product of two vectors is again a vector bull Convention The vector product or cross product is marked by an ldquoxrdquo bull If is turned to on the fastest route possible the vector (from the vector product) points in the direction in which a right-hand screw would move (right hand rule) bull With parallel vectors it disappears and it reaches its maximum when the
two vectors are at right angles to each other
16
22 Mathematical add on
Graphically - Pooling of the spatial gradients towards the spatial coordinate axes rarr vector
- Gradient points towards the largest increase of the quantity
- Amount is the magnitude of the derivative pointing towards the largest increase
Nabla operator ndash spatial gradient
with
Operator -Nabla
equivpart=nabla=nabla
nabla
partpartpartpartpartpart
z
y
x
i
17
bull Nabla is a (vector) operator that works to the right
bull It only results in a value if it stands left to an arithmetic expression bull If it stands to the right of an arithmetic expression it keeps its operator function (and ldquowaitsrdquo for application)
22 Mathematical add on
TTgradT
z
y
x
zTyTxT
=
=equivnabla
partpartpartpartpartpart
partpartpartpartpartpart
zw
yv
xu
wvu
udivu
z
y
x
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpart
minuspartpart
partpart
minuspartpart
partpart
minuspartpart
==
times
equivequivtimesnabla part
partpartpart
partpart
partpartpartpartpartpart
yu
xv
xw
zu
zv
yw
wvu
kji
wvu
urotu zyx
z
y
x
Vector product ldquoxrdquo vector
Scalar product ldquordquo scalar
Product with a Scalar vector
Nabla operator 22 Mathematical add on
18
Nabla operator ndash algorithms
=nablanenabla=
=
partpartpartpartpartpart
partpartpartpartpartpart
partpartpartpartpartpart
z
y
x
z
y
x
zTyTxT
TTT
TTT
zw
yv
xu
wvu
u
zw
yv
xu
wvu
udivu
z
y
x
z
y
x
partpart
+partpart
+partpart
=
sdot
=nablasdot
ne
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpartpartpartpartpart
22 Mathematical add on
Note Nabla is a (vector) operator ie the sequence must not be changed here
19
State variables for atmosphere and ocean
u Wind = (uvw) in ms T Temperature in degC or K p Pressure in Nm2 = 1 Pa (1 hPa = 102 Pa = 1 mbar) Specifically for the atmosphere q Specific humidity in kg water vapourkg moist air Specifically for the ocean S Salinity in kg saltkg seawater (often the salt content is expressed in practical salinity unit psu = kg salt1000 kg seawater) 20
2 Basics
I Laws of motion for a fluid parcel in x-y-z- directions (applying Newtons 1 and 2 law)
II Conservation of mass
III Law of thermodynamics (including motion) rarr5 equations for the evolution of the fluid (5 unknowns)
21
3 Equations of motion - nonrotating fluid 3 Equations of motion - nonrotating fluid
Momentum conservation The conservation of momentum is represented by
Newtonrsquos first law (ldquoLex primardquo) (momentum = mass middot velocity)
momentum = const at absence of forces
Momentum sentence (ldquoLex secundardquo) temporal change in momentum = Force (mass middot acceleration = Force)
22
Fu=
dtdM t time M mass u velocity vector F Force vector
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Forces on a fluid parcel
23
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119958119958119863119863119905119905
= F (Eq 6-2)
120588120588 density (120575120575119909119909 120575120575119910119910 120575120575119911119911) fluid parcel with infinitesimal dimensions 120575120575119872119872 mass of the parcel 120575120575119872119872 = 120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119958119958 parcel velocity F net force
119863119863119958119958119863119863119905119905
= 120597120597119958119958120597120597119905119905
+ u 120597120597119958119958120597120597119909119909
+v 120597120597119958119958120597120597119910119910
+w 120597120597119958119958120597120597119911119911
= 120597120597119958119958120597120597119905119905
+ (119958119958 ∙ 120571120571) 119958119958
Gravitational force
24
Fgravity= g 120575120575119872119872
Fgravity= minusg 120588120588119963119963 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-3)
Fgravity Gravitational force M mass g gravity acceleration g = const 120588120588 density 119963119963 unit vector in upward direction
Marshall and Plumb (2008)
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Pressure gradient force
25
Fpressure= 119865119865119909119909 119865119865119910119910 119865119865119911119911
=minus 120597120597119901119901120597120597119909119909
120597120597119901119901120597120597119910119910
120597120597119901119901120597120597119911119911
120575120575119909119909 120575120575119910119910 120575120575119911119911
=minus 120571120571119901119901 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-4)
F(A) = p ∙ 119860119860 = 119901119901 (x minus 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911
F(B) = p ∙ 119861119861 = minus119901119901 (x + 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911 Fx = F(A) + F(B)
Apply Taylor expansion (A21) at midpoint of parcel and neglect small terms (see book)
119901119901 pressure
Friction force
26
3 Equations of motion - nonrotating fluid
- Friction force operates on the earthrsquos surface
- the greater the surface roughness the higher the friction force
- the greater the wind velocity the higher the friction force
- friction force takes effect in vertical distances of ~ 100 m up to 1000 m (rarr atmospheric boundary layer Chapter 7)
Ffriction = 120588120588 ℱ 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-5)
ℱ frictional force per unit mass 120588120588 density
Equations of motion bull All three forces together in Eq 6-2 gives
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119854119854119863119863119905119905
= F119901119901119901119901119891119891119901119901119901119901119906119906119901119901119891119891 + F119892119892119901119901119901119901119907119907119891119891119905119905119910119910 + F119891119891119901119901119891119891119901119901119905119905119891119891119891119891119891119891
bull Re-arranging leads to equation of motion for fluid parcels
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = ℱ (Equation 6-6)
27
3 Equations of motion - nonrotating fluid
120588120588 density u velocity vector F Force vector ℱ frictional force per unit mass p pressure g gravity acceleration 119963119963 unit vector in z-direction
Equations of motion ndash component form
(Eq 6-6) in Cartesian coordinates
bull 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
bull 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
bull 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
28
3 Equations of motion - nonrotating fluid
119906119906 zonal velocity 119907119907meridional velocity w vertical velocity 120588120588 density ℱ frictional force per unit mass p pressure g gravity acceleration
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Think about examples in your everyday life for Eulerrsquos and Lagrangersquos differentiation Give three examples
Lagrange and Euler differentiation
12
2 Basics
22 Basics - Mathematical add on
- Vector operations - Nabla operator
13
22 Basics - Mathematical add on September 16 2014
Modified slides from Clemens Simmer University Bonn
Vector operations - Multiplication with a scalar a -
=
====
z
y
x
z
y
x
ii
avavav
vvv
aavavavva
bull With a scalar multiplication the vector stays a vector
bull Each element of a vector is multiplied individually with a scalar a
bull The vector extends (or shortens) itself by the factor a bull Convention With the multiplication scalar ndash vector we
donrsquot use a point (like with scalar ndash scalar)
22 Mathematical add on
14
Vector operations - Scalar product -
iii
iiiizzyyxx
z
y
x
z
y
x
vfvfvffvfvfvfff
vvv
vffv sum =equiv=++=
sdot
=sdot=sdot
bull The scalar product of two vectors is a scalar bull It is multiplied component-wise then added bull Convention The scalar product is marked by a multiplication sign (middot) bull It is at a maximum with parallel vectors and disappears when the
vectors are perpendicular to each other
f
α
v
αcosfvfv
=sdot
22 Mathematical add on
15
Vector operations ndash vector product
bcacbabac
babababababa
babakbabajbabaibbbaaakji
axbbac
xyyx
zxxz
yzzy
xyyxzxxzyzzy
zyx
zyx
perpandperp=times=
minusminusminus
=
minus+minusminusminus==minus=times=
sin
)()()(
α
a b
bull The vector product of two vectors is again a vector bull Convention The vector product or cross product is marked by an ldquoxrdquo bull If is turned to on the fastest route possible the vector (from the vector product) points in the direction in which a right-hand screw would move (right hand rule) bull With parallel vectors it disappears and it reaches its maximum when the
two vectors are at right angles to each other
16
22 Mathematical add on
Graphically - Pooling of the spatial gradients towards the spatial coordinate axes rarr vector
- Gradient points towards the largest increase of the quantity
- Amount is the magnitude of the derivative pointing towards the largest increase
Nabla operator ndash spatial gradient
with
Operator -Nabla
equivpart=nabla=nabla
nabla
partpartpartpartpartpart
z
y
x
i
17
bull Nabla is a (vector) operator that works to the right
bull It only results in a value if it stands left to an arithmetic expression bull If it stands to the right of an arithmetic expression it keeps its operator function (and ldquowaitsrdquo for application)
22 Mathematical add on
TTgradT
z
y
x
zTyTxT
=
=equivnabla
partpartpartpartpartpart
partpartpartpartpartpart
zw
yv
xu
wvu
udivu
z
y
x
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpart
minuspartpart
partpart
minuspartpart
partpart
minuspartpart
==
times
equivequivtimesnabla part
partpartpart
partpart
partpartpartpartpartpart
yu
xv
xw
zu
zv
yw
wvu
kji
wvu
urotu zyx
z
y
x
Vector product ldquoxrdquo vector
Scalar product ldquordquo scalar
Product with a Scalar vector
Nabla operator 22 Mathematical add on
18
Nabla operator ndash algorithms
=nablanenabla=
=
partpartpartpartpartpart
partpartpartpartpartpart
partpartpartpartpartpart
z
y
x
z
y
x
zTyTxT
TTT
TTT
zw
yv
xu
wvu
u
zw
yv
xu
wvu
udivu
z
y
x
z
y
x
partpart
+partpart
+partpart
=
sdot
=nablasdot
ne
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpartpartpartpartpart
22 Mathematical add on
Note Nabla is a (vector) operator ie the sequence must not be changed here
19
State variables for atmosphere and ocean
u Wind = (uvw) in ms T Temperature in degC or K p Pressure in Nm2 = 1 Pa (1 hPa = 102 Pa = 1 mbar) Specifically for the atmosphere q Specific humidity in kg water vapourkg moist air Specifically for the ocean S Salinity in kg saltkg seawater (often the salt content is expressed in practical salinity unit psu = kg salt1000 kg seawater) 20
2 Basics
I Laws of motion for a fluid parcel in x-y-z- directions (applying Newtons 1 and 2 law)
II Conservation of mass
III Law of thermodynamics (including motion) rarr5 equations for the evolution of the fluid (5 unknowns)
21
3 Equations of motion - nonrotating fluid 3 Equations of motion - nonrotating fluid
Momentum conservation The conservation of momentum is represented by
Newtonrsquos first law (ldquoLex primardquo) (momentum = mass middot velocity)
momentum = const at absence of forces
Momentum sentence (ldquoLex secundardquo) temporal change in momentum = Force (mass middot acceleration = Force)
22
Fu=
dtdM t time M mass u velocity vector F Force vector
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Forces on a fluid parcel
23
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119958119958119863119863119905119905
= F (Eq 6-2)
120588120588 density (120575120575119909119909 120575120575119910119910 120575120575119911119911) fluid parcel with infinitesimal dimensions 120575120575119872119872 mass of the parcel 120575120575119872119872 = 120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119958119958 parcel velocity F net force
119863119863119958119958119863119863119905119905
= 120597120597119958119958120597120597119905119905
+ u 120597120597119958119958120597120597119909119909
+v 120597120597119958119958120597120597119910119910
+w 120597120597119958119958120597120597119911119911
= 120597120597119958119958120597120597119905119905
+ (119958119958 ∙ 120571120571) 119958119958
Gravitational force
24
Fgravity= g 120575120575119872119872
Fgravity= minusg 120588120588119963119963 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-3)
Fgravity Gravitational force M mass g gravity acceleration g = const 120588120588 density 119963119963 unit vector in upward direction
Marshall and Plumb (2008)
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Pressure gradient force
25
Fpressure= 119865119865119909119909 119865119865119910119910 119865119865119911119911
=minus 120597120597119901119901120597120597119909119909
120597120597119901119901120597120597119910119910
120597120597119901119901120597120597119911119911
120575120575119909119909 120575120575119910119910 120575120575119911119911
=minus 120571120571119901119901 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-4)
F(A) = p ∙ 119860119860 = 119901119901 (x minus 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911
F(B) = p ∙ 119861119861 = minus119901119901 (x + 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911 Fx = F(A) + F(B)
Apply Taylor expansion (A21) at midpoint of parcel and neglect small terms (see book)
119901119901 pressure
Friction force
26
3 Equations of motion - nonrotating fluid
- Friction force operates on the earthrsquos surface
- the greater the surface roughness the higher the friction force
- the greater the wind velocity the higher the friction force
- friction force takes effect in vertical distances of ~ 100 m up to 1000 m (rarr atmospheric boundary layer Chapter 7)
Ffriction = 120588120588 ℱ 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-5)
ℱ frictional force per unit mass 120588120588 density
Equations of motion bull All three forces together in Eq 6-2 gives
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119854119854119863119863119905119905
= F119901119901119901119901119891119891119901119901119901119901119906119906119901119901119891119891 + F119892119892119901119901119901119901119907119907119891119891119905119905119910119910 + F119891119891119901119901119891119891119901119901119905119905119891119891119891119891119891119891
bull Re-arranging leads to equation of motion for fluid parcels
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = ℱ (Equation 6-6)
27
3 Equations of motion - nonrotating fluid
120588120588 density u velocity vector F Force vector ℱ frictional force per unit mass p pressure g gravity acceleration 119963119963 unit vector in z-direction
Equations of motion ndash component form
(Eq 6-6) in Cartesian coordinates
bull 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
bull 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
bull 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
28
3 Equations of motion - nonrotating fluid
119906119906 zonal velocity 119907119907meridional velocity w vertical velocity 120588120588 density ℱ frictional force per unit mass p pressure g gravity acceleration
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
22 Basics - Mathematical add on
- Vector operations - Nabla operator
13
22 Basics - Mathematical add on September 16 2014
Modified slides from Clemens Simmer University Bonn
Vector operations - Multiplication with a scalar a -
=
====
z
y
x
z
y
x
ii
avavav
vvv
aavavavva
bull With a scalar multiplication the vector stays a vector
bull Each element of a vector is multiplied individually with a scalar a
bull The vector extends (or shortens) itself by the factor a bull Convention With the multiplication scalar ndash vector we
donrsquot use a point (like with scalar ndash scalar)
22 Mathematical add on
14
Vector operations - Scalar product -
iii
iiiizzyyxx
z
y
x
z
y
x
vfvfvffvfvfvfff
vvv
vffv sum =equiv=++=
sdot
=sdot=sdot
bull The scalar product of two vectors is a scalar bull It is multiplied component-wise then added bull Convention The scalar product is marked by a multiplication sign (middot) bull It is at a maximum with parallel vectors and disappears when the
vectors are perpendicular to each other
f
α
v
αcosfvfv
=sdot
22 Mathematical add on
15
Vector operations ndash vector product
bcacbabac
babababababa
babakbabajbabaibbbaaakji
axbbac
xyyx
zxxz
yzzy
xyyxzxxzyzzy
zyx
zyx
perpandperp=times=
minusminusminus
=
minus+minusminusminus==minus=times=
sin
)()()(
α
a b
bull The vector product of two vectors is again a vector bull Convention The vector product or cross product is marked by an ldquoxrdquo bull If is turned to on the fastest route possible the vector (from the vector product) points in the direction in which a right-hand screw would move (right hand rule) bull With parallel vectors it disappears and it reaches its maximum when the
two vectors are at right angles to each other
16
22 Mathematical add on
Graphically - Pooling of the spatial gradients towards the spatial coordinate axes rarr vector
- Gradient points towards the largest increase of the quantity
- Amount is the magnitude of the derivative pointing towards the largest increase
Nabla operator ndash spatial gradient
with
Operator -Nabla
equivpart=nabla=nabla
nabla
partpartpartpartpartpart
z
y
x
i
17
bull Nabla is a (vector) operator that works to the right
bull It only results in a value if it stands left to an arithmetic expression bull If it stands to the right of an arithmetic expression it keeps its operator function (and ldquowaitsrdquo for application)
22 Mathematical add on
TTgradT
z
y
x
zTyTxT
=
=equivnabla
partpartpartpartpartpart
partpartpartpartpartpart
zw
yv
xu
wvu
udivu
z
y
x
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpart
minuspartpart
partpart
minuspartpart
partpart
minuspartpart
==
times
equivequivtimesnabla part
partpartpart
partpart
partpartpartpartpartpart
yu
xv
xw
zu
zv
yw
wvu
kji
wvu
urotu zyx
z
y
x
Vector product ldquoxrdquo vector
Scalar product ldquordquo scalar
Product with a Scalar vector
Nabla operator 22 Mathematical add on
18
Nabla operator ndash algorithms
=nablanenabla=
=
partpartpartpartpartpart
partpartpartpartpartpart
partpartpartpartpartpart
z
y
x
z
y
x
zTyTxT
TTT
TTT
zw
yv
xu
wvu
u
zw
yv
xu
wvu
udivu
z
y
x
z
y
x
partpart
+partpart
+partpart
=
sdot
=nablasdot
ne
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpartpartpartpartpart
22 Mathematical add on
Note Nabla is a (vector) operator ie the sequence must not be changed here
19
State variables for atmosphere and ocean
u Wind = (uvw) in ms T Temperature in degC or K p Pressure in Nm2 = 1 Pa (1 hPa = 102 Pa = 1 mbar) Specifically for the atmosphere q Specific humidity in kg water vapourkg moist air Specifically for the ocean S Salinity in kg saltkg seawater (often the salt content is expressed in practical salinity unit psu = kg salt1000 kg seawater) 20
2 Basics
I Laws of motion for a fluid parcel in x-y-z- directions (applying Newtons 1 and 2 law)
II Conservation of mass
III Law of thermodynamics (including motion) rarr5 equations for the evolution of the fluid (5 unknowns)
21
3 Equations of motion - nonrotating fluid 3 Equations of motion - nonrotating fluid
Momentum conservation The conservation of momentum is represented by
Newtonrsquos first law (ldquoLex primardquo) (momentum = mass middot velocity)
momentum = const at absence of forces
Momentum sentence (ldquoLex secundardquo) temporal change in momentum = Force (mass middot acceleration = Force)
22
Fu=
dtdM t time M mass u velocity vector F Force vector
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Forces on a fluid parcel
23
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119958119958119863119863119905119905
= F (Eq 6-2)
120588120588 density (120575120575119909119909 120575120575119910119910 120575120575119911119911) fluid parcel with infinitesimal dimensions 120575120575119872119872 mass of the parcel 120575120575119872119872 = 120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119958119958 parcel velocity F net force
119863119863119958119958119863119863119905119905
= 120597120597119958119958120597120597119905119905
+ u 120597120597119958119958120597120597119909119909
+v 120597120597119958119958120597120597119910119910
+w 120597120597119958119958120597120597119911119911
= 120597120597119958119958120597120597119905119905
+ (119958119958 ∙ 120571120571) 119958119958
Gravitational force
24
Fgravity= g 120575120575119872119872
Fgravity= minusg 120588120588119963119963 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-3)
Fgravity Gravitational force M mass g gravity acceleration g = const 120588120588 density 119963119963 unit vector in upward direction
Marshall and Plumb (2008)
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Pressure gradient force
25
Fpressure= 119865119865119909119909 119865119865119910119910 119865119865119911119911
=minus 120597120597119901119901120597120597119909119909
120597120597119901119901120597120597119910119910
120597120597119901119901120597120597119911119911
120575120575119909119909 120575120575119910119910 120575120575119911119911
=minus 120571120571119901119901 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-4)
F(A) = p ∙ 119860119860 = 119901119901 (x minus 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911
F(B) = p ∙ 119861119861 = minus119901119901 (x + 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911 Fx = F(A) + F(B)
Apply Taylor expansion (A21) at midpoint of parcel and neglect small terms (see book)
119901119901 pressure
Friction force
26
3 Equations of motion - nonrotating fluid
- Friction force operates on the earthrsquos surface
- the greater the surface roughness the higher the friction force
- the greater the wind velocity the higher the friction force
- friction force takes effect in vertical distances of ~ 100 m up to 1000 m (rarr atmospheric boundary layer Chapter 7)
Ffriction = 120588120588 ℱ 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-5)
ℱ frictional force per unit mass 120588120588 density
Equations of motion bull All three forces together in Eq 6-2 gives
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119854119854119863119863119905119905
= F119901119901119901119901119891119891119901119901119901119901119906119906119901119901119891119891 + F119892119892119901119901119901119901119907119907119891119891119905119905119910119910 + F119891119891119901119901119891119891119901119901119905119905119891119891119891119891119891119891
bull Re-arranging leads to equation of motion for fluid parcels
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = ℱ (Equation 6-6)
27
3 Equations of motion - nonrotating fluid
120588120588 density u velocity vector F Force vector ℱ frictional force per unit mass p pressure g gravity acceleration 119963119963 unit vector in z-direction
Equations of motion ndash component form
(Eq 6-6) in Cartesian coordinates
bull 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
bull 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
bull 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
28
3 Equations of motion - nonrotating fluid
119906119906 zonal velocity 119907119907meridional velocity w vertical velocity 120588120588 density ℱ frictional force per unit mass p pressure g gravity acceleration
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Vector operations - Multiplication with a scalar a -
=
====
z
y
x
z
y
x
ii
avavav
vvv
aavavavva
bull With a scalar multiplication the vector stays a vector
bull Each element of a vector is multiplied individually with a scalar a
bull The vector extends (or shortens) itself by the factor a bull Convention With the multiplication scalar ndash vector we
donrsquot use a point (like with scalar ndash scalar)
22 Mathematical add on
14
Vector operations - Scalar product -
iii
iiiizzyyxx
z
y
x
z
y
x
vfvfvffvfvfvfff
vvv
vffv sum =equiv=++=
sdot
=sdot=sdot
bull The scalar product of two vectors is a scalar bull It is multiplied component-wise then added bull Convention The scalar product is marked by a multiplication sign (middot) bull It is at a maximum with parallel vectors and disappears when the
vectors are perpendicular to each other
f
α
v
αcosfvfv
=sdot
22 Mathematical add on
15
Vector operations ndash vector product
bcacbabac
babababababa
babakbabajbabaibbbaaakji
axbbac
xyyx
zxxz
yzzy
xyyxzxxzyzzy
zyx
zyx
perpandperp=times=
minusminusminus
=
minus+minusminusminus==minus=times=
sin
)()()(
α
a b
bull The vector product of two vectors is again a vector bull Convention The vector product or cross product is marked by an ldquoxrdquo bull If is turned to on the fastest route possible the vector (from the vector product) points in the direction in which a right-hand screw would move (right hand rule) bull With parallel vectors it disappears and it reaches its maximum when the
two vectors are at right angles to each other
16
22 Mathematical add on
Graphically - Pooling of the spatial gradients towards the spatial coordinate axes rarr vector
- Gradient points towards the largest increase of the quantity
- Amount is the magnitude of the derivative pointing towards the largest increase
Nabla operator ndash spatial gradient
with
Operator -Nabla
equivpart=nabla=nabla
nabla
partpartpartpartpartpart
z
y
x
i
17
bull Nabla is a (vector) operator that works to the right
bull It only results in a value if it stands left to an arithmetic expression bull If it stands to the right of an arithmetic expression it keeps its operator function (and ldquowaitsrdquo for application)
22 Mathematical add on
TTgradT
z
y
x
zTyTxT
=
=equivnabla
partpartpartpartpartpart
partpartpartpartpartpart
zw
yv
xu
wvu
udivu
z
y
x
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpart
minuspartpart
partpart
minuspartpart
partpart
minuspartpart
==
times
equivequivtimesnabla part
partpartpart
partpart
partpartpartpartpartpart
yu
xv
xw
zu
zv
yw
wvu
kji
wvu
urotu zyx
z
y
x
Vector product ldquoxrdquo vector
Scalar product ldquordquo scalar
Product with a Scalar vector
Nabla operator 22 Mathematical add on
18
Nabla operator ndash algorithms
=nablanenabla=
=
partpartpartpartpartpart
partpartpartpartpartpart
partpartpartpartpartpart
z
y
x
z
y
x
zTyTxT
TTT
TTT
zw
yv
xu
wvu
u
zw
yv
xu
wvu
udivu
z
y
x
z
y
x
partpart
+partpart
+partpart
=
sdot
=nablasdot
ne
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpartpartpartpartpart
22 Mathematical add on
Note Nabla is a (vector) operator ie the sequence must not be changed here
19
State variables for atmosphere and ocean
u Wind = (uvw) in ms T Temperature in degC or K p Pressure in Nm2 = 1 Pa (1 hPa = 102 Pa = 1 mbar) Specifically for the atmosphere q Specific humidity in kg water vapourkg moist air Specifically for the ocean S Salinity in kg saltkg seawater (often the salt content is expressed in practical salinity unit psu = kg salt1000 kg seawater) 20
2 Basics
I Laws of motion for a fluid parcel in x-y-z- directions (applying Newtons 1 and 2 law)
II Conservation of mass
III Law of thermodynamics (including motion) rarr5 equations for the evolution of the fluid (5 unknowns)
21
3 Equations of motion - nonrotating fluid 3 Equations of motion - nonrotating fluid
Momentum conservation The conservation of momentum is represented by
Newtonrsquos first law (ldquoLex primardquo) (momentum = mass middot velocity)
momentum = const at absence of forces
Momentum sentence (ldquoLex secundardquo) temporal change in momentum = Force (mass middot acceleration = Force)
22
Fu=
dtdM t time M mass u velocity vector F Force vector
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Forces on a fluid parcel
23
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119958119958119863119863119905119905
= F (Eq 6-2)
120588120588 density (120575120575119909119909 120575120575119910119910 120575120575119911119911) fluid parcel with infinitesimal dimensions 120575120575119872119872 mass of the parcel 120575120575119872119872 = 120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119958119958 parcel velocity F net force
119863119863119958119958119863119863119905119905
= 120597120597119958119958120597120597119905119905
+ u 120597120597119958119958120597120597119909119909
+v 120597120597119958119958120597120597119910119910
+w 120597120597119958119958120597120597119911119911
= 120597120597119958119958120597120597119905119905
+ (119958119958 ∙ 120571120571) 119958119958
Gravitational force
24
Fgravity= g 120575120575119872119872
Fgravity= minusg 120588120588119963119963 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-3)
Fgravity Gravitational force M mass g gravity acceleration g = const 120588120588 density 119963119963 unit vector in upward direction
Marshall and Plumb (2008)
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Pressure gradient force
25
Fpressure= 119865119865119909119909 119865119865119910119910 119865119865119911119911
=minus 120597120597119901119901120597120597119909119909
120597120597119901119901120597120597119910119910
120597120597119901119901120597120597119911119911
120575120575119909119909 120575120575119910119910 120575120575119911119911
=minus 120571120571119901119901 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-4)
F(A) = p ∙ 119860119860 = 119901119901 (x minus 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911
F(B) = p ∙ 119861119861 = minus119901119901 (x + 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911 Fx = F(A) + F(B)
Apply Taylor expansion (A21) at midpoint of parcel and neglect small terms (see book)
119901119901 pressure
Friction force
26
3 Equations of motion - nonrotating fluid
- Friction force operates on the earthrsquos surface
- the greater the surface roughness the higher the friction force
- the greater the wind velocity the higher the friction force
- friction force takes effect in vertical distances of ~ 100 m up to 1000 m (rarr atmospheric boundary layer Chapter 7)
Ffriction = 120588120588 ℱ 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-5)
ℱ frictional force per unit mass 120588120588 density
Equations of motion bull All three forces together in Eq 6-2 gives
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119854119854119863119863119905119905
= F119901119901119901119901119891119891119901119901119901119901119906119906119901119901119891119891 + F119892119892119901119901119901119901119907119907119891119891119905119905119910119910 + F119891119891119901119901119891119891119901119901119905119905119891119891119891119891119891119891
bull Re-arranging leads to equation of motion for fluid parcels
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = ℱ (Equation 6-6)
27
3 Equations of motion - nonrotating fluid
120588120588 density u velocity vector F Force vector ℱ frictional force per unit mass p pressure g gravity acceleration 119963119963 unit vector in z-direction
Equations of motion ndash component form
(Eq 6-6) in Cartesian coordinates
bull 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
bull 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
bull 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
28
3 Equations of motion - nonrotating fluid
119906119906 zonal velocity 119907119907meridional velocity w vertical velocity 120588120588 density ℱ frictional force per unit mass p pressure g gravity acceleration
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Vector operations - Scalar product -
iii
iiiizzyyxx
z
y
x
z
y
x
vfvfvffvfvfvfff
vvv
vffv sum =equiv=++=
sdot
=sdot=sdot
bull The scalar product of two vectors is a scalar bull It is multiplied component-wise then added bull Convention The scalar product is marked by a multiplication sign (middot) bull It is at a maximum with parallel vectors and disappears when the
vectors are perpendicular to each other
f
α
v
αcosfvfv
=sdot
22 Mathematical add on
15
Vector operations ndash vector product
bcacbabac
babababababa
babakbabajbabaibbbaaakji
axbbac
xyyx
zxxz
yzzy
xyyxzxxzyzzy
zyx
zyx
perpandperp=times=
minusminusminus
=
minus+minusminusminus==minus=times=
sin
)()()(
α
a b
bull The vector product of two vectors is again a vector bull Convention The vector product or cross product is marked by an ldquoxrdquo bull If is turned to on the fastest route possible the vector (from the vector product) points in the direction in which a right-hand screw would move (right hand rule) bull With parallel vectors it disappears and it reaches its maximum when the
two vectors are at right angles to each other
16
22 Mathematical add on
Graphically - Pooling of the spatial gradients towards the spatial coordinate axes rarr vector
- Gradient points towards the largest increase of the quantity
- Amount is the magnitude of the derivative pointing towards the largest increase
Nabla operator ndash spatial gradient
with
Operator -Nabla
equivpart=nabla=nabla
nabla
partpartpartpartpartpart
z
y
x
i
17
bull Nabla is a (vector) operator that works to the right
bull It only results in a value if it stands left to an arithmetic expression bull If it stands to the right of an arithmetic expression it keeps its operator function (and ldquowaitsrdquo for application)
22 Mathematical add on
TTgradT
z
y
x
zTyTxT
=
=equivnabla
partpartpartpartpartpart
partpartpartpartpartpart
zw
yv
xu
wvu
udivu
z
y
x
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpart
minuspartpart
partpart
minuspartpart
partpart
minuspartpart
==
times
equivequivtimesnabla part
partpartpart
partpart
partpartpartpartpartpart
yu
xv
xw
zu
zv
yw
wvu
kji
wvu
urotu zyx
z
y
x
Vector product ldquoxrdquo vector
Scalar product ldquordquo scalar
Product with a Scalar vector
Nabla operator 22 Mathematical add on
18
Nabla operator ndash algorithms
=nablanenabla=
=
partpartpartpartpartpart
partpartpartpartpartpart
partpartpartpartpartpart
z
y
x
z
y
x
zTyTxT
TTT
TTT
zw
yv
xu
wvu
u
zw
yv
xu
wvu
udivu
z
y
x
z
y
x
partpart
+partpart
+partpart
=
sdot
=nablasdot
ne
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpartpartpartpartpart
22 Mathematical add on
Note Nabla is a (vector) operator ie the sequence must not be changed here
19
State variables for atmosphere and ocean
u Wind = (uvw) in ms T Temperature in degC or K p Pressure in Nm2 = 1 Pa (1 hPa = 102 Pa = 1 mbar) Specifically for the atmosphere q Specific humidity in kg water vapourkg moist air Specifically for the ocean S Salinity in kg saltkg seawater (often the salt content is expressed in practical salinity unit psu = kg salt1000 kg seawater) 20
2 Basics
I Laws of motion for a fluid parcel in x-y-z- directions (applying Newtons 1 and 2 law)
II Conservation of mass
III Law of thermodynamics (including motion) rarr5 equations for the evolution of the fluid (5 unknowns)
21
3 Equations of motion - nonrotating fluid 3 Equations of motion - nonrotating fluid
Momentum conservation The conservation of momentum is represented by
Newtonrsquos first law (ldquoLex primardquo) (momentum = mass middot velocity)
momentum = const at absence of forces
Momentum sentence (ldquoLex secundardquo) temporal change in momentum = Force (mass middot acceleration = Force)
22
Fu=
dtdM t time M mass u velocity vector F Force vector
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Forces on a fluid parcel
23
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119958119958119863119863119905119905
= F (Eq 6-2)
120588120588 density (120575120575119909119909 120575120575119910119910 120575120575119911119911) fluid parcel with infinitesimal dimensions 120575120575119872119872 mass of the parcel 120575120575119872119872 = 120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119958119958 parcel velocity F net force
119863119863119958119958119863119863119905119905
= 120597120597119958119958120597120597119905119905
+ u 120597120597119958119958120597120597119909119909
+v 120597120597119958119958120597120597119910119910
+w 120597120597119958119958120597120597119911119911
= 120597120597119958119958120597120597119905119905
+ (119958119958 ∙ 120571120571) 119958119958
Gravitational force
24
Fgravity= g 120575120575119872119872
Fgravity= minusg 120588120588119963119963 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-3)
Fgravity Gravitational force M mass g gravity acceleration g = const 120588120588 density 119963119963 unit vector in upward direction
Marshall and Plumb (2008)
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Pressure gradient force
25
Fpressure= 119865119865119909119909 119865119865119910119910 119865119865119911119911
=minus 120597120597119901119901120597120597119909119909
120597120597119901119901120597120597119910119910
120597120597119901119901120597120597119911119911
120575120575119909119909 120575120575119910119910 120575120575119911119911
=minus 120571120571119901119901 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-4)
F(A) = p ∙ 119860119860 = 119901119901 (x minus 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911
F(B) = p ∙ 119861119861 = minus119901119901 (x + 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911 Fx = F(A) + F(B)
Apply Taylor expansion (A21) at midpoint of parcel and neglect small terms (see book)
119901119901 pressure
Friction force
26
3 Equations of motion - nonrotating fluid
- Friction force operates on the earthrsquos surface
- the greater the surface roughness the higher the friction force
- the greater the wind velocity the higher the friction force
- friction force takes effect in vertical distances of ~ 100 m up to 1000 m (rarr atmospheric boundary layer Chapter 7)
Ffriction = 120588120588 ℱ 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-5)
ℱ frictional force per unit mass 120588120588 density
Equations of motion bull All three forces together in Eq 6-2 gives
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119854119854119863119863119905119905
= F119901119901119901119901119891119891119901119901119901119901119906119906119901119901119891119891 + F119892119892119901119901119901119901119907119907119891119891119905119905119910119910 + F119891119891119901119901119891119891119901119901119905119905119891119891119891119891119891119891
bull Re-arranging leads to equation of motion for fluid parcels
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = ℱ (Equation 6-6)
27
3 Equations of motion - nonrotating fluid
120588120588 density u velocity vector F Force vector ℱ frictional force per unit mass p pressure g gravity acceleration 119963119963 unit vector in z-direction
Equations of motion ndash component form
(Eq 6-6) in Cartesian coordinates
bull 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
bull 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
bull 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
28
3 Equations of motion - nonrotating fluid
119906119906 zonal velocity 119907119907meridional velocity w vertical velocity 120588120588 density ℱ frictional force per unit mass p pressure g gravity acceleration
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Vector operations ndash vector product
bcacbabac
babababababa
babakbabajbabaibbbaaakji
axbbac
xyyx
zxxz
yzzy
xyyxzxxzyzzy
zyx
zyx
perpandperp=times=
minusminusminus
=
minus+minusminusminus==minus=times=
sin
)()()(
α
a b
bull The vector product of two vectors is again a vector bull Convention The vector product or cross product is marked by an ldquoxrdquo bull If is turned to on the fastest route possible the vector (from the vector product) points in the direction in which a right-hand screw would move (right hand rule) bull With parallel vectors it disappears and it reaches its maximum when the
two vectors are at right angles to each other
16
22 Mathematical add on
Graphically - Pooling of the spatial gradients towards the spatial coordinate axes rarr vector
- Gradient points towards the largest increase of the quantity
- Amount is the magnitude of the derivative pointing towards the largest increase
Nabla operator ndash spatial gradient
with
Operator -Nabla
equivpart=nabla=nabla
nabla
partpartpartpartpartpart
z
y
x
i
17
bull Nabla is a (vector) operator that works to the right
bull It only results in a value if it stands left to an arithmetic expression bull If it stands to the right of an arithmetic expression it keeps its operator function (and ldquowaitsrdquo for application)
22 Mathematical add on
TTgradT
z
y
x
zTyTxT
=
=equivnabla
partpartpartpartpartpart
partpartpartpartpartpart
zw
yv
xu
wvu
udivu
z
y
x
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpart
minuspartpart
partpart
minuspartpart
partpart
minuspartpart
==
times
equivequivtimesnabla part
partpartpart
partpart
partpartpartpartpartpart
yu
xv
xw
zu
zv
yw
wvu
kji
wvu
urotu zyx
z
y
x
Vector product ldquoxrdquo vector
Scalar product ldquordquo scalar
Product with a Scalar vector
Nabla operator 22 Mathematical add on
18
Nabla operator ndash algorithms
=nablanenabla=
=
partpartpartpartpartpart
partpartpartpartpartpart
partpartpartpartpartpart
z
y
x
z
y
x
zTyTxT
TTT
TTT
zw
yv
xu
wvu
u
zw
yv
xu
wvu
udivu
z
y
x
z
y
x
partpart
+partpart
+partpart
=
sdot
=nablasdot
ne
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpartpartpartpartpart
22 Mathematical add on
Note Nabla is a (vector) operator ie the sequence must not be changed here
19
State variables for atmosphere and ocean
u Wind = (uvw) in ms T Temperature in degC or K p Pressure in Nm2 = 1 Pa (1 hPa = 102 Pa = 1 mbar) Specifically for the atmosphere q Specific humidity in kg water vapourkg moist air Specifically for the ocean S Salinity in kg saltkg seawater (often the salt content is expressed in practical salinity unit psu = kg salt1000 kg seawater) 20
2 Basics
I Laws of motion for a fluid parcel in x-y-z- directions (applying Newtons 1 and 2 law)
II Conservation of mass
III Law of thermodynamics (including motion) rarr5 equations for the evolution of the fluid (5 unknowns)
21
3 Equations of motion - nonrotating fluid 3 Equations of motion - nonrotating fluid
Momentum conservation The conservation of momentum is represented by
Newtonrsquos first law (ldquoLex primardquo) (momentum = mass middot velocity)
momentum = const at absence of forces
Momentum sentence (ldquoLex secundardquo) temporal change in momentum = Force (mass middot acceleration = Force)
22
Fu=
dtdM t time M mass u velocity vector F Force vector
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Forces on a fluid parcel
23
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119958119958119863119863119905119905
= F (Eq 6-2)
120588120588 density (120575120575119909119909 120575120575119910119910 120575120575119911119911) fluid parcel with infinitesimal dimensions 120575120575119872119872 mass of the parcel 120575120575119872119872 = 120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119958119958 parcel velocity F net force
119863119863119958119958119863119863119905119905
= 120597120597119958119958120597120597119905119905
+ u 120597120597119958119958120597120597119909119909
+v 120597120597119958119958120597120597119910119910
+w 120597120597119958119958120597120597119911119911
= 120597120597119958119958120597120597119905119905
+ (119958119958 ∙ 120571120571) 119958119958
Gravitational force
24
Fgravity= g 120575120575119872119872
Fgravity= minusg 120588120588119963119963 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-3)
Fgravity Gravitational force M mass g gravity acceleration g = const 120588120588 density 119963119963 unit vector in upward direction
Marshall and Plumb (2008)
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Pressure gradient force
25
Fpressure= 119865119865119909119909 119865119865119910119910 119865119865119911119911
=minus 120597120597119901119901120597120597119909119909
120597120597119901119901120597120597119910119910
120597120597119901119901120597120597119911119911
120575120575119909119909 120575120575119910119910 120575120575119911119911
=minus 120571120571119901119901 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-4)
F(A) = p ∙ 119860119860 = 119901119901 (x minus 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911
F(B) = p ∙ 119861119861 = minus119901119901 (x + 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911 Fx = F(A) + F(B)
Apply Taylor expansion (A21) at midpoint of parcel and neglect small terms (see book)
119901119901 pressure
Friction force
26
3 Equations of motion - nonrotating fluid
- Friction force operates on the earthrsquos surface
- the greater the surface roughness the higher the friction force
- the greater the wind velocity the higher the friction force
- friction force takes effect in vertical distances of ~ 100 m up to 1000 m (rarr atmospheric boundary layer Chapter 7)
Ffriction = 120588120588 ℱ 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-5)
ℱ frictional force per unit mass 120588120588 density
Equations of motion bull All three forces together in Eq 6-2 gives
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119854119854119863119863119905119905
= F119901119901119901119901119891119891119901119901119901119901119906119906119901119901119891119891 + F119892119892119901119901119901119901119907119907119891119891119905119905119910119910 + F119891119891119901119901119891119891119901119901119905119905119891119891119891119891119891119891
bull Re-arranging leads to equation of motion for fluid parcels
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = ℱ (Equation 6-6)
27
3 Equations of motion - nonrotating fluid
120588120588 density u velocity vector F Force vector ℱ frictional force per unit mass p pressure g gravity acceleration 119963119963 unit vector in z-direction
Equations of motion ndash component form
(Eq 6-6) in Cartesian coordinates
bull 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
bull 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
bull 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
28
3 Equations of motion - nonrotating fluid
119906119906 zonal velocity 119907119907meridional velocity w vertical velocity 120588120588 density ℱ frictional force per unit mass p pressure g gravity acceleration
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Graphically - Pooling of the spatial gradients towards the spatial coordinate axes rarr vector
- Gradient points towards the largest increase of the quantity
- Amount is the magnitude of the derivative pointing towards the largest increase
Nabla operator ndash spatial gradient
with
Operator -Nabla
equivpart=nabla=nabla
nabla
partpartpartpartpartpart
z
y
x
i
17
bull Nabla is a (vector) operator that works to the right
bull It only results in a value if it stands left to an arithmetic expression bull If it stands to the right of an arithmetic expression it keeps its operator function (and ldquowaitsrdquo for application)
22 Mathematical add on
TTgradT
z
y
x
zTyTxT
=
=equivnabla
partpartpartpartpartpart
partpartpartpartpartpart
zw
yv
xu
wvu
udivu
z
y
x
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpart
minuspartpart
partpart
minuspartpart
partpart
minuspartpart
==
times
equivequivtimesnabla part
partpartpart
partpart
partpartpartpartpartpart
yu
xv
xw
zu
zv
yw
wvu
kji
wvu
urotu zyx
z
y
x
Vector product ldquoxrdquo vector
Scalar product ldquordquo scalar
Product with a Scalar vector
Nabla operator 22 Mathematical add on
18
Nabla operator ndash algorithms
=nablanenabla=
=
partpartpartpartpartpart
partpartpartpartpartpart
partpartpartpartpartpart
z
y
x
z
y
x
zTyTxT
TTT
TTT
zw
yv
xu
wvu
u
zw
yv
xu
wvu
udivu
z
y
x
z
y
x
partpart
+partpart
+partpart
=
sdot
=nablasdot
ne
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpartpartpartpartpart
22 Mathematical add on
Note Nabla is a (vector) operator ie the sequence must not be changed here
19
State variables for atmosphere and ocean
u Wind = (uvw) in ms T Temperature in degC or K p Pressure in Nm2 = 1 Pa (1 hPa = 102 Pa = 1 mbar) Specifically for the atmosphere q Specific humidity in kg water vapourkg moist air Specifically for the ocean S Salinity in kg saltkg seawater (often the salt content is expressed in practical salinity unit psu = kg salt1000 kg seawater) 20
2 Basics
I Laws of motion for a fluid parcel in x-y-z- directions (applying Newtons 1 and 2 law)
II Conservation of mass
III Law of thermodynamics (including motion) rarr5 equations for the evolution of the fluid (5 unknowns)
21
3 Equations of motion - nonrotating fluid 3 Equations of motion - nonrotating fluid
Momentum conservation The conservation of momentum is represented by
Newtonrsquos first law (ldquoLex primardquo) (momentum = mass middot velocity)
momentum = const at absence of forces
Momentum sentence (ldquoLex secundardquo) temporal change in momentum = Force (mass middot acceleration = Force)
22
Fu=
dtdM t time M mass u velocity vector F Force vector
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Forces on a fluid parcel
23
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119958119958119863119863119905119905
= F (Eq 6-2)
120588120588 density (120575120575119909119909 120575120575119910119910 120575120575119911119911) fluid parcel with infinitesimal dimensions 120575120575119872119872 mass of the parcel 120575120575119872119872 = 120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119958119958 parcel velocity F net force
119863119863119958119958119863119863119905119905
= 120597120597119958119958120597120597119905119905
+ u 120597120597119958119958120597120597119909119909
+v 120597120597119958119958120597120597119910119910
+w 120597120597119958119958120597120597119911119911
= 120597120597119958119958120597120597119905119905
+ (119958119958 ∙ 120571120571) 119958119958
Gravitational force
24
Fgravity= g 120575120575119872119872
Fgravity= minusg 120588120588119963119963 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-3)
Fgravity Gravitational force M mass g gravity acceleration g = const 120588120588 density 119963119963 unit vector in upward direction
Marshall and Plumb (2008)
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Pressure gradient force
25
Fpressure= 119865119865119909119909 119865119865119910119910 119865119865119911119911
=minus 120597120597119901119901120597120597119909119909
120597120597119901119901120597120597119910119910
120597120597119901119901120597120597119911119911
120575120575119909119909 120575120575119910119910 120575120575119911119911
=minus 120571120571119901119901 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-4)
F(A) = p ∙ 119860119860 = 119901119901 (x minus 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911
F(B) = p ∙ 119861119861 = minus119901119901 (x + 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911 Fx = F(A) + F(B)
Apply Taylor expansion (A21) at midpoint of parcel and neglect small terms (see book)
119901119901 pressure
Friction force
26
3 Equations of motion - nonrotating fluid
- Friction force operates on the earthrsquos surface
- the greater the surface roughness the higher the friction force
- the greater the wind velocity the higher the friction force
- friction force takes effect in vertical distances of ~ 100 m up to 1000 m (rarr atmospheric boundary layer Chapter 7)
Ffriction = 120588120588 ℱ 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-5)
ℱ frictional force per unit mass 120588120588 density
Equations of motion bull All three forces together in Eq 6-2 gives
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119854119854119863119863119905119905
= F119901119901119901119901119891119891119901119901119901119901119906119906119901119901119891119891 + F119892119892119901119901119901119901119907119907119891119891119905119905119910119910 + F119891119891119901119901119891119891119901119901119905119905119891119891119891119891119891119891
bull Re-arranging leads to equation of motion for fluid parcels
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = ℱ (Equation 6-6)
27
3 Equations of motion - nonrotating fluid
120588120588 density u velocity vector F Force vector ℱ frictional force per unit mass p pressure g gravity acceleration 119963119963 unit vector in z-direction
Equations of motion ndash component form
(Eq 6-6) in Cartesian coordinates
bull 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
bull 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
bull 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
28
3 Equations of motion - nonrotating fluid
119906119906 zonal velocity 119907119907meridional velocity w vertical velocity 120588120588 density ℱ frictional force per unit mass p pressure g gravity acceleration
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
TTgradT
z
y
x
zTyTxT
=
=equivnabla
partpartpartpartpartpart
partpartpartpartpartpart
zw
yv
xu
wvu
udivu
z
y
x
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpart
minuspartpart
partpart
minuspartpart
partpart
minuspartpart
==
times
equivequivtimesnabla part
partpartpart
partpart
partpartpartpartpartpart
yu
xv
xw
zu
zv
yw
wvu
kji
wvu
urotu zyx
z
y
x
Vector product ldquoxrdquo vector
Scalar product ldquordquo scalar
Product with a Scalar vector
Nabla operator 22 Mathematical add on
18
Nabla operator ndash algorithms
=nablanenabla=
=
partpartpartpartpartpart
partpartpartpartpartpart
partpartpartpartpartpart
z
y
x
z
y
x
zTyTxT
TTT
TTT
zw
yv
xu
wvu
u
zw
yv
xu
wvu
udivu
z
y
x
z
y
x
partpart
+partpart
+partpart
=
sdot
=nablasdot
ne
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpartpartpartpartpart
22 Mathematical add on
Note Nabla is a (vector) operator ie the sequence must not be changed here
19
State variables for atmosphere and ocean
u Wind = (uvw) in ms T Temperature in degC or K p Pressure in Nm2 = 1 Pa (1 hPa = 102 Pa = 1 mbar) Specifically for the atmosphere q Specific humidity in kg water vapourkg moist air Specifically for the ocean S Salinity in kg saltkg seawater (often the salt content is expressed in practical salinity unit psu = kg salt1000 kg seawater) 20
2 Basics
I Laws of motion for a fluid parcel in x-y-z- directions (applying Newtons 1 and 2 law)
II Conservation of mass
III Law of thermodynamics (including motion) rarr5 equations for the evolution of the fluid (5 unknowns)
21
3 Equations of motion - nonrotating fluid 3 Equations of motion - nonrotating fluid
Momentum conservation The conservation of momentum is represented by
Newtonrsquos first law (ldquoLex primardquo) (momentum = mass middot velocity)
momentum = const at absence of forces
Momentum sentence (ldquoLex secundardquo) temporal change in momentum = Force (mass middot acceleration = Force)
22
Fu=
dtdM t time M mass u velocity vector F Force vector
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Forces on a fluid parcel
23
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119958119958119863119863119905119905
= F (Eq 6-2)
120588120588 density (120575120575119909119909 120575120575119910119910 120575120575119911119911) fluid parcel with infinitesimal dimensions 120575120575119872119872 mass of the parcel 120575120575119872119872 = 120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119958119958 parcel velocity F net force
119863119863119958119958119863119863119905119905
= 120597120597119958119958120597120597119905119905
+ u 120597120597119958119958120597120597119909119909
+v 120597120597119958119958120597120597119910119910
+w 120597120597119958119958120597120597119911119911
= 120597120597119958119958120597120597119905119905
+ (119958119958 ∙ 120571120571) 119958119958
Gravitational force
24
Fgravity= g 120575120575119872119872
Fgravity= minusg 120588120588119963119963 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-3)
Fgravity Gravitational force M mass g gravity acceleration g = const 120588120588 density 119963119963 unit vector in upward direction
Marshall and Plumb (2008)
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Pressure gradient force
25
Fpressure= 119865119865119909119909 119865119865119910119910 119865119865119911119911
=minus 120597120597119901119901120597120597119909119909
120597120597119901119901120597120597119910119910
120597120597119901119901120597120597119911119911
120575120575119909119909 120575120575119910119910 120575120575119911119911
=minus 120571120571119901119901 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-4)
F(A) = p ∙ 119860119860 = 119901119901 (x minus 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911
F(B) = p ∙ 119861119861 = minus119901119901 (x + 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911 Fx = F(A) + F(B)
Apply Taylor expansion (A21) at midpoint of parcel and neglect small terms (see book)
119901119901 pressure
Friction force
26
3 Equations of motion - nonrotating fluid
- Friction force operates on the earthrsquos surface
- the greater the surface roughness the higher the friction force
- the greater the wind velocity the higher the friction force
- friction force takes effect in vertical distances of ~ 100 m up to 1000 m (rarr atmospheric boundary layer Chapter 7)
Ffriction = 120588120588 ℱ 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-5)
ℱ frictional force per unit mass 120588120588 density
Equations of motion bull All three forces together in Eq 6-2 gives
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119854119854119863119863119905119905
= F119901119901119901119901119891119891119901119901119901119901119906119906119901119901119891119891 + F119892119892119901119901119901119901119907119907119891119891119905119905119910119910 + F119891119891119901119901119891119891119901119901119905119905119891119891119891119891119891119891
bull Re-arranging leads to equation of motion for fluid parcels
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = ℱ (Equation 6-6)
27
3 Equations of motion - nonrotating fluid
120588120588 density u velocity vector F Force vector ℱ frictional force per unit mass p pressure g gravity acceleration 119963119963 unit vector in z-direction
Equations of motion ndash component form
(Eq 6-6) in Cartesian coordinates
bull 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
bull 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
bull 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
28
3 Equations of motion - nonrotating fluid
119906119906 zonal velocity 119907119907meridional velocity w vertical velocity 120588120588 density ℱ frictional force per unit mass p pressure g gravity acceleration
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Nabla operator ndash algorithms
=nablanenabla=
=
partpartpartpartpartpart
partpartpartpartpartpart
partpartpartpartpartpart
z
y
x
z
y
x
zTyTxT
TTT
TTT
zw
yv
xu
wvu
u
zw
yv
xu
wvu
udivu
z
y
x
z
y
x
partpart
+partpart
+partpart
=
sdot
=nablasdot
ne
partpart
+partpart
+partpart
=
sdot
equivequivsdotnabla
partpartpartpartpartpart
partpartpartpartpartpart
22 Mathematical add on
Note Nabla is a (vector) operator ie the sequence must not be changed here
19
State variables for atmosphere and ocean
u Wind = (uvw) in ms T Temperature in degC or K p Pressure in Nm2 = 1 Pa (1 hPa = 102 Pa = 1 mbar) Specifically for the atmosphere q Specific humidity in kg water vapourkg moist air Specifically for the ocean S Salinity in kg saltkg seawater (often the salt content is expressed in practical salinity unit psu = kg salt1000 kg seawater) 20
2 Basics
I Laws of motion for a fluid parcel in x-y-z- directions (applying Newtons 1 and 2 law)
II Conservation of mass
III Law of thermodynamics (including motion) rarr5 equations for the evolution of the fluid (5 unknowns)
21
3 Equations of motion - nonrotating fluid 3 Equations of motion - nonrotating fluid
Momentum conservation The conservation of momentum is represented by
Newtonrsquos first law (ldquoLex primardquo) (momentum = mass middot velocity)
momentum = const at absence of forces
Momentum sentence (ldquoLex secundardquo) temporal change in momentum = Force (mass middot acceleration = Force)
22
Fu=
dtdM t time M mass u velocity vector F Force vector
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Forces on a fluid parcel
23
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119958119958119863119863119905119905
= F (Eq 6-2)
120588120588 density (120575120575119909119909 120575120575119910119910 120575120575119911119911) fluid parcel with infinitesimal dimensions 120575120575119872119872 mass of the parcel 120575120575119872119872 = 120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119958119958 parcel velocity F net force
119863119863119958119958119863119863119905119905
= 120597120597119958119958120597120597119905119905
+ u 120597120597119958119958120597120597119909119909
+v 120597120597119958119958120597120597119910119910
+w 120597120597119958119958120597120597119911119911
= 120597120597119958119958120597120597119905119905
+ (119958119958 ∙ 120571120571) 119958119958
Gravitational force
24
Fgravity= g 120575120575119872119872
Fgravity= minusg 120588120588119963119963 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-3)
Fgravity Gravitational force M mass g gravity acceleration g = const 120588120588 density 119963119963 unit vector in upward direction
Marshall and Plumb (2008)
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Pressure gradient force
25
Fpressure= 119865119865119909119909 119865119865119910119910 119865119865119911119911
=minus 120597120597119901119901120597120597119909119909
120597120597119901119901120597120597119910119910
120597120597119901119901120597120597119911119911
120575120575119909119909 120575120575119910119910 120575120575119911119911
=minus 120571120571119901119901 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-4)
F(A) = p ∙ 119860119860 = 119901119901 (x minus 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911
F(B) = p ∙ 119861119861 = minus119901119901 (x + 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911 Fx = F(A) + F(B)
Apply Taylor expansion (A21) at midpoint of parcel and neglect small terms (see book)
119901119901 pressure
Friction force
26
3 Equations of motion - nonrotating fluid
- Friction force operates on the earthrsquos surface
- the greater the surface roughness the higher the friction force
- the greater the wind velocity the higher the friction force
- friction force takes effect in vertical distances of ~ 100 m up to 1000 m (rarr atmospheric boundary layer Chapter 7)
Ffriction = 120588120588 ℱ 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-5)
ℱ frictional force per unit mass 120588120588 density
Equations of motion bull All three forces together in Eq 6-2 gives
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119854119854119863119863119905119905
= F119901119901119901119901119891119891119901119901119901119901119906119906119901119901119891119891 + F119892119892119901119901119901119901119907119907119891119891119905119905119910119910 + F119891119891119901119901119891119891119901119901119905119905119891119891119891119891119891119891
bull Re-arranging leads to equation of motion for fluid parcels
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = ℱ (Equation 6-6)
27
3 Equations of motion - nonrotating fluid
120588120588 density u velocity vector F Force vector ℱ frictional force per unit mass p pressure g gravity acceleration 119963119963 unit vector in z-direction
Equations of motion ndash component form
(Eq 6-6) in Cartesian coordinates
bull 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
bull 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
bull 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
28
3 Equations of motion - nonrotating fluid
119906119906 zonal velocity 119907119907meridional velocity w vertical velocity 120588120588 density ℱ frictional force per unit mass p pressure g gravity acceleration
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
State variables for atmosphere and ocean
u Wind = (uvw) in ms T Temperature in degC or K p Pressure in Nm2 = 1 Pa (1 hPa = 102 Pa = 1 mbar) Specifically for the atmosphere q Specific humidity in kg water vapourkg moist air Specifically for the ocean S Salinity in kg saltkg seawater (often the salt content is expressed in practical salinity unit psu = kg salt1000 kg seawater) 20
2 Basics
I Laws of motion for a fluid parcel in x-y-z- directions (applying Newtons 1 and 2 law)
II Conservation of mass
III Law of thermodynamics (including motion) rarr5 equations for the evolution of the fluid (5 unknowns)
21
3 Equations of motion - nonrotating fluid 3 Equations of motion - nonrotating fluid
Momentum conservation The conservation of momentum is represented by
Newtonrsquos first law (ldquoLex primardquo) (momentum = mass middot velocity)
momentum = const at absence of forces
Momentum sentence (ldquoLex secundardquo) temporal change in momentum = Force (mass middot acceleration = Force)
22
Fu=
dtdM t time M mass u velocity vector F Force vector
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Forces on a fluid parcel
23
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119958119958119863119863119905119905
= F (Eq 6-2)
120588120588 density (120575120575119909119909 120575120575119910119910 120575120575119911119911) fluid parcel with infinitesimal dimensions 120575120575119872119872 mass of the parcel 120575120575119872119872 = 120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119958119958 parcel velocity F net force
119863119863119958119958119863119863119905119905
= 120597120597119958119958120597120597119905119905
+ u 120597120597119958119958120597120597119909119909
+v 120597120597119958119958120597120597119910119910
+w 120597120597119958119958120597120597119911119911
= 120597120597119958119958120597120597119905119905
+ (119958119958 ∙ 120571120571) 119958119958
Gravitational force
24
Fgravity= g 120575120575119872119872
Fgravity= minusg 120588120588119963119963 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-3)
Fgravity Gravitational force M mass g gravity acceleration g = const 120588120588 density 119963119963 unit vector in upward direction
Marshall and Plumb (2008)
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Pressure gradient force
25
Fpressure= 119865119865119909119909 119865119865119910119910 119865119865119911119911
=minus 120597120597119901119901120597120597119909119909
120597120597119901119901120597120597119910119910
120597120597119901119901120597120597119911119911
120575120575119909119909 120575120575119910119910 120575120575119911119911
=minus 120571120571119901119901 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-4)
F(A) = p ∙ 119860119860 = 119901119901 (x minus 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911
F(B) = p ∙ 119861119861 = minus119901119901 (x + 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911 Fx = F(A) + F(B)
Apply Taylor expansion (A21) at midpoint of parcel and neglect small terms (see book)
119901119901 pressure
Friction force
26
3 Equations of motion - nonrotating fluid
- Friction force operates on the earthrsquos surface
- the greater the surface roughness the higher the friction force
- the greater the wind velocity the higher the friction force
- friction force takes effect in vertical distances of ~ 100 m up to 1000 m (rarr atmospheric boundary layer Chapter 7)
Ffriction = 120588120588 ℱ 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-5)
ℱ frictional force per unit mass 120588120588 density
Equations of motion bull All three forces together in Eq 6-2 gives
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119854119854119863119863119905119905
= F119901119901119901119901119891119891119901119901119901119901119906119906119901119901119891119891 + F119892119892119901119901119901119901119907119907119891119891119905119905119910119910 + F119891119891119901119901119891119891119901119901119905119905119891119891119891119891119891119891
bull Re-arranging leads to equation of motion for fluid parcels
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = ℱ (Equation 6-6)
27
3 Equations of motion - nonrotating fluid
120588120588 density u velocity vector F Force vector ℱ frictional force per unit mass p pressure g gravity acceleration 119963119963 unit vector in z-direction
Equations of motion ndash component form
(Eq 6-6) in Cartesian coordinates
bull 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
bull 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
bull 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
28
3 Equations of motion - nonrotating fluid
119906119906 zonal velocity 119907119907meridional velocity w vertical velocity 120588120588 density ℱ frictional force per unit mass p pressure g gravity acceleration
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
I Laws of motion for a fluid parcel in x-y-z- directions (applying Newtons 1 and 2 law)
II Conservation of mass
III Law of thermodynamics (including motion) rarr5 equations for the evolution of the fluid (5 unknowns)
21
3 Equations of motion - nonrotating fluid 3 Equations of motion - nonrotating fluid
Momentum conservation The conservation of momentum is represented by
Newtonrsquos first law (ldquoLex primardquo) (momentum = mass middot velocity)
momentum = const at absence of forces
Momentum sentence (ldquoLex secundardquo) temporal change in momentum = Force (mass middot acceleration = Force)
22
Fu=
dtdM t time M mass u velocity vector F Force vector
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Forces on a fluid parcel
23
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119958119958119863119863119905119905
= F (Eq 6-2)
120588120588 density (120575120575119909119909 120575120575119910119910 120575120575119911119911) fluid parcel with infinitesimal dimensions 120575120575119872119872 mass of the parcel 120575120575119872119872 = 120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119958119958 parcel velocity F net force
119863119863119958119958119863119863119905119905
= 120597120597119958119958120597120597119905119905
+ u 120597120597119958119958120597120597119909119909
+v 120597120597119958119958120597120597119910119910
+w 120597120597119958119958120597120597119911119911
= 120597120597119958119958120597120597119905119905
+ (119958119958 ∙ 120571120571) 119958119958
Gravitational force
24
Fgravity= g 120575120575119872119872
Fgravity= minusg 120588120588119963119963 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-3)
Fgravity Gravitational force M mass g gravity acceleration g = const 120588120588 density 119963119963 unit vector in upward direction
Marshall and Plumb (2008)
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Pressure gradient force
25
Fpressure= 119865119865119909119909 119865119865119910119910 119865119865119911119911
=minus 120597120597119901119901120597120597119909119909
120597120597119901119901120597120597119910119910
120597120597119901119901120597120597119911119911
120575120575119909119909 120575120575119910119910 120575120575119911119911
=minus 120571120571119901119901 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-4)
F(A) = p ∙ 119860119860 = 119901119901 (x minus 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911
F(B) = p ∙ 119861119861 = minus119901119901 (x + 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911 Fx = F(A) + F(B)
Apply Taylor expansion (A21) at midpoint of parcel and neglect small terms (see book)
119901119901 pressure
Friction force
26
3 Equations of motion - nonrotating fluid
- Friction force operates on the earthrsquos surface
- the greater the surface roughness the higher the friction force
- the greater the wind velocity the higher the friction force
- friction force takes effect in vertical distances of ~ 100 m up to 1000 m (rarr atmospheric boundary layer Chapter 7)
Ffriction = 120588120588 ℱ 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-5)
ℱ frictional force per unit mass 120588120588 density
Equations of motion bull All three forces together in Eq 6-2 gives
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119854119854119863119863119905119905
= F119901119901119901119901119891119891119901119901119901119901119906119906119901119901119891119891 + F119892119892119901119901119901119901119907119907119891119891119905119905119910119910 + F119891119891119901119901119891119891119901119901119905119905119891119891119891119891119891119891
bull Re-arranging leads to equation of motion for fluid parcels
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = ℱ (Equation 6-6)
27
3 Equations of motion - nonrotating fluid
120588120588 density u velocity vector F Force vector ℱ frictional force per unit mass p pressure g gravity acceleration 119963119963 unit vector in z-direction
Equations of motion ndash component form
(Eq 6-6) in Cartesian coordinates
bull 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
bull 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
bull 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
28
3 Equations of motion - nonrotating fluid
119906119906 zonal velocity 119907119907meridional velocity w vertical velocity 120588120588 density ℱ frictional force per unit mass p pressure g gravity acceleration
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Momentum conservation The conservation of momentum is represented by
Newtonrsquos first law (ldquoLex primardquo) (momentum = mass middot velocity)
momentum = const at absence of forces
Momentum sentence (ldquoLex secundardquo) temporal change in momentum = Force (mass middot acceleration = Force)
22
Fu=
dtdM t time M mass u velocity vector F Force vector
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Forces on a fluid parcel
23
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119958119958119863119863119905119905
= F (Eq 6-2)
120588120588 density (120575120575119909119909 120575120575119910119910 120575120575119911119911) fluid parcel with infinitesimal dimensions 120575120575119872119872 mass of the parcel 120575120575119872119872 = 120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119958119958 parcel velocity F net force
119863119863119958119958119863119863119905119905
= 120597120597119958119958120597120597119905119905
+ u 120597120597119958119958120597120597119909119909
+v 120597120597119958119958120597120597119910119910
+w 120597120597119958119958120597120597119911119911
= 120597120597119958119958120597120597119905119905
+ (119958119958 ∙ 120571120571) 119958119958
Gravitational force
24
Fgravity= g 120575120575119872119872
Fgravity= minusg 120588120588119963119963 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-3)
Fgravity Gravitational force M mass g gravity acceleration g = const 120588120588 density 119963119963 unit vector in upward direction
Marshall and Plumb (2008)
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Pressure gradient force
25
Fpressure= 119865119865119909119909 119865119865119910119910 119865119865119911119911
=minus 120597120597119901119901120597120597119909119909
120597120597119901119901120597120597119910119910
120597120597119901119901120597120597119911119911
120575120575119909119909 120575120575119910119910 120575120575119911119911
=minus 120571120571119901119901 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-4)
F(A) = p ∙ 119860119860 = 119901119901 (x minus 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911
F(B) = p ∙ 119861119861 = minus119901119901 (x + 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911 Fx = F(A) + F(B)
Apply Taylor expansion (A21) at midpoint of parcel and neglect small terms (see book)
119901119901 pressure
Friction force
26
3 Equations of motion - nonrotating fluid
- Friction force operates on the earthrsquos surface
- the greater the surface roughness the higher the friction force
- the greater the wind velocity the higher the friction force
- friction force takes effect in vertical distances of ~ 100 m up to 1000 m (rarr atmospheric boundary layer Chapter 7)
Ffriction = 120588120588 ℱ 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-5)
ℱ frictional force per unit mass 120588120588 density
Equations of motion bull All three forces together in Eq 6-2 gives
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119854119854119863119863119905119905
= F119901119901119901119901119891119891119901119901119901119901119906119906119901119901119891119891 + F119892119892119901119901119901119901119907119907119891119891119905119905119910119910 + F119891119891119901119901119891119891119901119901119905119905119891119891119891119891119891119891
bull Re-arranging leads to equation of motion for fluid parcels
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = ℱ (Equation 6-6)
27
3 Equations of motion - nonrotating fluid
120588120588 density u velocity vector F Force vector ℱ frictional force per unit mass p pressure g gravity acceleration 119963119963 unit vector in z-direction
Equations of motion ndash component form
(Eq 6-6) in Cartesian coordinates
bull 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
bull 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
bull 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
28
3 Equations of motion - nonrotating fluid
119906119906 zonal velocity 119907119907meridional velocity w vertical velocity 120588120588 density ℱ frictional force per unit mass p pressure g gravity acceleration
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Forces on a fluid parcel
23
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119958119958119863119863119905119905
= F (Eq 6-2)
120588120588 density (120575120575119909119909 120575120575119910119910 120575120575119911119911) fluid parcel with infinitesimal dimensions 120575120575119872119872 mass of the parcel 120575120575119872119872 = 120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119958119958 parcel velocity F net force
119863119863119958119958119863119863119905119905
= 120597120597119958119958120597120597119905119905
+ u 120597120597119958119958120597120597119909119909
+v 120597120597119958119958120597120597119910119910
+w 120597120597119958119958120597120597119911119911
= 120597120597119958119958120597120597119905119905
+ (119958119958 ∙ 120571120571) 119958119958
Gravitational force
24
Fgravity= g 120575120575119872119872
Fgravity= minusg 120588120588119963119963 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-3)
Fgravity Gravitational force M mass g gravity acceleration g = const 120588120588 density 119963119963 unit vector in upward direction
Marshall and Plumb (2008)
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Pressure gradient force
25
Fpressure= 119865119865119909119909 119865119865119910119910 119865119865119911119911
=minus 120597120597119901119901120597120597119909119909
120597120597119901119901120597120597119910119910
120597120597119901119901120597120597119911119911
120575120575119909119909 120575120575119910119910 120575120575119911119911
=minus 120571120571119901119901 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-4)
F(A) = p ∙ 119860119860 = 119901119901 (x minus 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911
F(B) = p ∙ 119861119861 = minus119901119901 (x + 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911 Fx = F(A) + F(B)
Apply Taylor expansion (A21) at midpoint of parcel and neglect small terms (see book)
119901119901 pressure
Friction force
26
3 Equations of motion - nonrotating fluid
- Friction force operates on the earthrsquos surface
- the greater the surface roughness the higher the friction force
- the greater the wind velocity the higher the friction force
- friction force takes effect in vertical distances of ~ 100 m up to 1000 m (rarr atmospheric boundary layer Chapter 7)
Ffriction = 120588120588 ℱ 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-5)
ℱ frictional force per unit mass 120588120588 density
Equations of motion bull All three forces together in Eq 6-2 gives
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119854119854119863119863119905119905
= F119901119901119901119901119891119891119901119901119901119901119906119906119901119901119891119891 + F119892119892119901119901119901119901119907119907119891119891119905119905119910119910 + F119891119891119901119901119891119891119901119901119905119905119891119891119891119891119891119891
bull Re-arranging leads to equation of motion for fluid parcels
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = ℱ (Equation 6-6)
27
3 Equations of motion - nonrotating fluid
120588120588 density u velocity vector F Force vector ℱ frictional force per unit mass p pressure g gravity acceleration 119963119963 unit vector in z-direction
Equations of motion ndash component form
(Eq 6-6) in Cartesian coordinates
bull 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
bull 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
bull 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
28
3 Equations of motion - nonrotating fluid
119906119906 zonal velocity 119907119907meridional velocity w vertical velocity 120588120588 density ℱ frictional force per unit mass p pressure g gravity acceleration
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Gravitational force
24
Fgravity= g 120575120575119872119872
Fgravity= minusg 120588120588119963119963 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-3)
Fgravity Gravitational force M mass g gravity acceleration g = const 120588120588 density 119963119963 unit vector in upward direction
Marshall and Plumb (2008)
3 Equations of motion - nonrotating fluid
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Pressure gradient force
25
Fpressure= 119865119865119909119909 119865119865119910119910 119865119865119911119911
=minus 120597120597119901119901120597120597119909119909
120597120597119901119901120597120597119910119910
120597120597119901119901120597120597119911119911
120575120575119909119909 120575120575119910119910 120575120575119911119911
=minus 120571120571119901119901 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-4)
F(A) = p ∙ 119860119860 = 119901119901 (x minus 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911
F(B) = p ∙ 119861119861 = minus119901119901 (x + 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911 Fx = F(A) + F(B)
Apply Taylor expansion (A21) at midpoint of parcel and neglect small terms (see book)
119901119901 pressure
Friction force
26
3 Equations of motion - nonrotating fluid
- Friction force operates on the earthrsquos surface
- the greater the surface roughness the higher the friction force
- the greater the wind velocity the higher the friction force
- friction force takes effect in vertical distances of ~ 100 m up to 1000 m (rarr atmospheric boundary layer Chapter 7)
Ffriction = 120588120588 ℱ 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-5)
ℱ frictional force per unit mass 120588120588 density
Equations of motion bull All three forces together in Eq 6-2 gives
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119854119854119863119863119905119905
= F119901119901119901119901119891119891119901119901119901119901119906119906119901119901119891119891 + F119892119892119901119901119901119901119907119907119891119891119905119905119910119910 + F119891119891119901119901119891119891119901119901119905119905119891119891119891119891119891119891
bull Re-arranging leads to equation of motion for fluid parcels
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = ℱ (Equation 6-6)
27
3 Equations of motion - nonrotating fluid
120588120588 density u velocity vector F Force vector ℱ frictional force per unit mass p pressure g gravity acceleration 119963119963 unit vector in z-direction
Equations of motion ndash component form
(Eq 6-6) in Cartesian coordinates
bull 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
bull 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
bull 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
28
3 Equations of motion - nonrotating fluid
119906119906 zonal velocity 119907119907meridional velocity w vertical velocity 120588120588 density ℱ frictional force per unit mass p pressure g gravity acceleration
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
3 Equations of motion - nonrotating fluid
Marshall and Plumb (2008)
Pressure gradient force
25
Fpressure= 119865119865119909119909 119865119865119910119910 119865119865119911119911
=minus 120597120597119901119901120597120597119909119909
120597120597119901119901120597120597119910119910
120597120597119901119901120597120597119911119911
120575120575119909119909 120575120575119910119910 120575120575119911119911
=minus 120571120571119901119901 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-4)
F(A) = p ∙ 119860119860 = 119901119901 (x minus 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911
F(B) = p ∙ 119861119861 = minus119901119901 (x + 1205751205751199091199092
y z) 120575120575119910119910 120575120575119911119911 Fx = F(A) + F(B)
Apply Taylor expansion (A21) at midpoint of parcel and neglect small terms (see book)
119901119901 pressure
Friction force
26
3 Equations of motion - nonrotating fluid
- Friction force operates on the earthrsquos surface
- the greater the surface roughness the higher the friction force
- the greater the wind velocity the higher the friction force
- friction force takes effect in vertical distances of ~ 100 m up to 1000 m (rarr atmospheric boundary layer Chapter 7)
Ffriction = 120588120588 ℱ 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-5)
ℱ frictional force per unit mass 120588120588 density
Equations of motion bull All three forces together in Eq 6-2 gives
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119854119854119863119863119905119905
= F119901119901119901119901119891119891119901119901119901119901119906119906119901119901119891119891 + F119892119892119901119901119901119901119907119907119891119891119905119905119910119910 + F119891119891119901119901119891119891119901119901119905119905119891119891119891119891119891119891
bull Re-arranging leads to equation of motion for fluid parcels
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = ℱ (Equation 6-6)
27
3 Equations of motion - nonrotating fluid
120588120588 density u velocity vector F Force vector ℱ frictional force per unit mass p pressure g gravity acceleration 119963119963 unit vector in z-direction
Equations of motion ndash component form
(Eq 6-6) in Cartesian coordinates
bull 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
bull 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
bull 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
28
3 Equations of motion - nonrotating fluid
119906119906 zonal velocity 119907119907meridional velocity w vertical velocity 120588120588 density ℱ frictional force per unit mass p pressure g gravity acceleration
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Friction force
26
3 Equations of motion - nonrotating fluid
- Friction force operates on the earthrsquos surface
- the greater the surface roughness the higher the friction force
- the greater the wind velocity the higher the friction force
- friction force takes effect in vertical distances of ~ 100 m up to 1000 m (rarr atmospheric boundary layer Chapter 7)
Ffriction = 120588120588 ℱ 120575120575119909119909 120575120575119910119910 120575120575119911119911 (Eq 6-5)
ℱ frictional force per unit mass 120588120588 density
Equations of motion bull All three forces together in Eq 6-2 gives
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119854119854119863119863119905119905
= F119901119901119901119901119891119891119901119901119901119901119906119906119901119901119891119891 + F119892119892119901119901119901119901119907119907119891119891119905119905119910119910 + F119891119891119901119901119891119891119901119901119905119905119891119891119891119891119891119891
bull Re-arranging leads to equation of motion for fluid parcels
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = ℱ (Equation 6-6)
27
3 Equations of motion - nonrotating fluid
120588120588 density u velocity vector F Force vector ℱ frictional force per unit mass p pressure g gravity acceleration 119963119963 unit vector in z-direction
Equations of motion ndash component form
(Eq 6-6) in Cartesian coordinates
bull 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
bull 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
bull 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
28
3 Equations of motion - nonrotating fluid
119906119906 zonal velocity 119907119907meridional velocity w vertical velocity 120588120588 density ℱ frictional force per unit mass p pressure g gravity acceleration
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Equations of motion bull All three forces together in Eq 6-2 gives
120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911 119863119863119854119854119863119863119905119905
= F119901119901119901119901119891119891119901119901119901119901119906119906119901119901119891119891 + F119892119892119901119901119901119901119907119907119891119891119905119905119910119910 + F119891119891119901119901119891119891119901119901119905119905119891119891119891119891119891119891
bull Re-arranging leads to equation of motion for fluid parcels
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = ℱ (Equation 6-6)
27
3 Equations of motion - nonrotating fluid
120588120588 density u velocity vector F Force vector ℱ frictional force per unit mass p pressure g gravity acceleration 119963119963 unit vector in z-direction
Equations of motion ndash component form
(Eq 6-6) in Cartesian coordinates
bull 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
bull 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
bull 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
28
3 Equations of motion - nonrotating fluid
119906119906 zonal velocity 119907119907meridional velocity w vertical velocity 120588120588 density ℱ frictional force per unit mass p pressure g gravity acceleration
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Equations of motion ndash component form
(Eq 6-6) in Cartesian coordinates
bull 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
bull 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
bull 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
28
3 Equations of motion - nonrotating fluid
119906119906 zonal velocity 119907119907meridional velocity w vertical velocity 120588120588 density ℱ frictional force per unit mass p pressure g gravity acceleration
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Hydrostatic balance
If friction ℱ119911119911 and vertical acceleration 119863119863119908119908119863119863119905119905
are negligible we derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)
120597120597119901119901120597120597119911119911
= - 120588120588g (Eq 6-8)
Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions
29
3 Equations of motion - nonrotating fluid
Balance between vertical pressure gradient and gravitational force
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
30
GEF1100ndashAutumn 2014 18092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
31 Addition not in book
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Conserved quantity A quantity of which the derivative equals zero
Conservation of mass for liquids is also called continuity equation
Conservation of mass ndash continuity equation
32
4 Conservation of mass
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Conservation of mass
33
4 Conservation of mass
Marshall and Plumb (2008)
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
4 Conservation of mass
34
Conservation of mass -1 Mass continuity requires
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = 120597120597120588120588120597120597119905119905
120575120575119909119909 120575120575119910119910 120575120575119911119911 = (net mass flux into the volume)
bull mass flux in x-direction per unit time into volume
120588120588119906119906 119909119909 minus12 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
bull mass flux in x-direction per unit time out of volume 120588120588119906119906 119909119909 + 1
2 120575120575119909119909 119910119910 119911119911 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
4 Conservation of mass
35
Conservation of mass -2
bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22) minus 120597120597
120597120597119909119909120588120588119906119906 120575120575119909119909 120575120575119910119910 120575120575119911119911
bull net mass flux in y- and z- direction accordingly (see book)
bull Net mass flux into the volume
minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911 bull substituting into mass continuity
120597120597120597120597119905119905
(120588120588 120575120575119909119909 120575120575119910119910 120575120575119911119911) = minus 120571120571 ∙ 120588120588119958119958 120575120575119909119909 120575120575119910119910 120575120575119911119911
Marshall and Plumb (2008)
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
4 Conservation of mass
36
Conservation of mass -3 bull leads to equation of continuity
120597120597120588120588
120597120597119905119905+ 120571120571 ∙ 120588120588119958119958 = 0 (Eq 6-9)
which has the form of conservation law 120597120597 119862119862119862119862119862119862119862119862119862119862119862119862119905119905119862119862119862119862119905119905119862119862119862119862119862119862
120597120597119905119905+ 120571120571 ∙ 119891119891119901119901119906119906119909119909 = 0
bull Using DDt (Eq 6-1) and 120571120571 ∙ 120588120588119958119958 =120588120588120571120571 ∙ 119958119958 + 119958119958 ∙ 120571120571120588120588 (see A22)
we can rewrite 119863119863120588120588
119863119863119905119905+ 120588120588120571120571 ∙ 119958119958 = 0 (Eq 6-10)
Marshall and Plumb (2008)
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Approximation for continuity equation incompressible - compressible flows
bull Incompressible flow (eg ocean)
120571120571 ∙ 119958119958 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119911119911
= 0 (Eq6-11)
bull Compressible flow (eg air) expressed in pressure coordinate p
120571120571119901119901 ∙ 119958119958119901119901 = 120597120597119906119906120597120597119909119909
+ 120597120597119907119907120597120597119910119910
+ 120597120597119908119908120597120597119901119901
= 0 (Eq6-12)
37
4 Conservation of mass
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Thermodynamic equation bull Temperature evolution can be derived from first law of
thermodynamics (Ch4 4-2)
119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (Eq 6-13)
bull If the heating rate is zero (119863119863119863119863119863119863119905119905
= 0 Ch 431) then
119863119863119863119863119863119863119905119905
= 1120588120588119862119862119901119901
119863119863119901119901119863119863119905119905
38
5 Thermodynamic equation
The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure
rarr Introduction of potential temperature 120579120579 (Eq 4-17)
120579120579 = 119879119879 1199011199010
119901119901
119876119876 heat 119863119863119863119863119863119863119905119905
diabatic heating rate per unit mass 119901119901119901119901 specific heat at constant pressure T temperature
120588120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Thermodynamic equation - potential temperature
bull Eq 6-13 for potential temperature 120579120579 (see book for details)
119863119863120579120579119863119863119905119905
= 1199011199011199011199010
minusκ
119863119863119863119863119863119863119863119863119862119862119901119901
(Eq 6-14)
Note For adiabatic motions 120575120575119876119876 = 0 120579120579 is conserved
39
5 Thermodynamic equation
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
40
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Coordinate systems in meteorology
bull Cartesian coordinates (xyzt) bull Spherical coordinates (λϕzt) bull Height coordinates geometrical altitude (z) pressure (p) and potential temperature (θ)
3 Equations of motion - nonrotating fluid
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Ia) 120597120597119906119906120597120597119905119905
+ 119906119906 120597120597119906119906120597120597119909119909
+ 119907119907 120597120597119906119906120597120597119910119910
+ w 120597120597119906119906120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119909119909
= ℱx (6-7a)
Ib) 120597120597119907119907120597120597119905119905
+ 119906119906 120597120597119907119907120597120597119909119909
+ 119907119907 120597120597119907119907120597120597119910119910
+ w 120597120597119907119907120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119910119910
= ℱ119910119910 (6minus7b)
Ic) 120597120597119908119908120597120597119905119905
+ 119906119906 120597120597119908119908120597120597119909119909
+ 119907119907 120597120597119908119908120597120597119910119910
+ w 120597120597119908119908120597120597119911119911
+ 1120588120588
120597120597119901119901120597120597119911119911
+ g = ℱz (6-7c)
II) 120597120597120588120588120597120597119905119905
+ 120571120571 ∙ 120588120588119958119958 = 0 (6-9 or 6-116-12)
III) 119863119863119863119863119863119863119905119905
= 119901119901119901119901119863119863119863119863119863119863119905119905
minus 1120588120588 119863119863119901119901
119863119863119905119905 (6-13 or 6-14)
42
Summary Summary 3-5)
rarr5 equations for the evolution of the fluid (5 unknowns)
Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Forces on rotating sphere Fictitious forces
- occur in revolvingaccelerating system (eg car in curve) - occur on the earthrsquos surface (Fixed system on earthrsquos surface forms an accelerated system
because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)
rarr Coriolis force
rarr Centrifugal force 43
5 Equations of motion - rotating fluid
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 44
Inertial - rotating frames
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
45
5 Equations of motion - rotating fluid
Transformation into rotating coordinates Consider figure 69
bull 119958119958119862119862119862119862 = 119958119958119901119901119891119891119905119905 + Ω times 119955119955 (Eq 6-24)
bull 119863119863119912119912119863119863119905119905 119862119862119862119862
= 119863119863119912119912119863119863119905119905 119862119862119862119862119905119905
+ Ω times A (Eq 6-26)
Setting A =r and A rarr 119958119958119862119862119862119862 in Eq 6-24 and 6-26 we can derive
119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862= 119863119863
119863119863119905119905 119862119862119862119862119905119905+ Ω times 119958119958119901119901119891119891119905119905 + Ω times 119955119955
= 119863119863119958119958119862119862119862119862119905119905
119863119863119905119905 119862119862119862119862119905119905+ 2Ω times 119958119958119901119901119891119891119905119905 + Ω times Ω times 119955119955 (Eq 6-27)
119863119863119955119955119863119863119905119905 119862119862119862119862119905119905
= 119958119958119901119901119891119891119905119905 119955119955 position vector A any vector Ω rotation vector
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Rotating equations of motion
Substituting 119863119863119958119958119862119862119862119862
119863119863119905119905 119862119862119862119862 from Eq 6-27 into Eq 6-6 (inertial frame
equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + g119963119963 = minus 2Ω times 119958119958 + minusΩ times Ω times 119955119955 + ℱ (Eq 6-28)
46
5 Equations of motion - rotating fluid
Coriolis acceleration
Centrifugal acceleration
Friction acceleration
Pressure gradient accel
Gravi- tational accel
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
bull bull
Centripetal force FCp Centrifugal force FC
Angular velocity Ω
Body (mass M) Centre
FCp = - FC
Centrifugal force ndash Centripetal force
r
47
FC = minusM Ω x (Ω x r)
bull Directed radially outward
bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Centrifugal accel ndash modified gravitational potential
Combine the gradient of Centrifugal potential minusΩ times Ω times 119955119955 = 120571120571 Ω2r2
2 and
Gravitational potential g119963119963 = 120571120571(gz) in Eq 6-28
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 = minus 2Ω times 119958119958 + ℱ Eq 6minus29
bull 120567120567 = g119911119911 minus Ω2r2
2 Eq 6minus30
120567120567(Greek ldquoPhildquo) is modified gravitational potential on Earth
48
Coriolis acceleration
Friction acceleration
Pressure gradient accel
Gravitational +Centrifugal accelerations
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Marshall and Plumb (2008) 49
5 Equations of motion - rotating fluid
On the sphere Shallow atmosphere approx a+z ≃ a r=(a+z)cos120593120593 ≃ a cos120593120593 Eq 6-30 becomes
120567120567 = g119911119911 minusΩ21199011199012cos2120593120593
2
Definition of geopotential surfaces
119911119911lowast = 119911119911 + Ω21198621198622cos21205931205932g (Eq 6-40)
Ω= 727 x 10-5 s-1 a = 637 x 106 m
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
GEF 1100 ndash Klimasystemet
Chapter 6 The equations of fluid motion
50
GEF1100ndashAutumn 2014 30092014
Prof Dr Kirstin Kruumlger (MetOs UiO)
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
1 Motivation
2 Basics 21 Lagrange - Euler 22 Mathematical add on
3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance
4 Conversation of mass
5 Thermodynamic equation
6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion
7 Take home messages
Lect
ure
Out
line
ndash Ch
6 Ch 6 - The equations of fluid motions
51 Addition not in book
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
5 Equations of motion - rotating fluid
Marshall and Plumb (2008)
52
Coriolis acceleration
119863119863119854119854119863119863119905119905
= minus 2120512120512 times 119958119958 Eq 6-31
NH (viewed from above the NP) Ω gt 0 rotation anticlockwise SH (viewed from above the SP) Ω lt 0 rotation clockwise Absence of other forces
120512120512 =0 ΩyΩz
=0
Ω 119901119901119891119891119901119901φΩ sin φ
Angular velocity
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
5 Equations of motion - rotating fluid
Marshall and Plumb (2008) 53
Assuming that 1) Ωz is negligible as Ωultltg 2) wltltuh leads to
minus2120512120512 times 119958119958 ≃(minus2Ω sin φ 119907119907 2Ω sin φ 1199061199060) (Eq 6-41) = 119891119891119963119963 times 119958119958 Coriolis parameter 119943119943
119891119891 = 2Ω sin 120593120593 (Eq 6-42)
Note Vertical component of the Earths rotation rate which matters
Coriolis force on the sphere ndash Coriolis parameter
Ω= 727 x 10-5 s-1 a= 637 x 106 m
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Coriolis force In direction of movement in NH deflection of air particles to the right
SH Deflection of air particles to the left takes place
Directional movement
54
Schematic picture for Coriolis force
5 Equations of motion - rotating fluid
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
Equations of motion ndash Coriolis parameter
119863119863119854119854119863119863119905119905
+ 1 120588120588
120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ (Eq 6minus43)
bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911119911
compared with gravity
119863119863u119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119909119909
minus 119891119891119907119907 = ℱ119909119909 (Eq 6minus44)
119863119863119907119907119863119863119905119905
+ 1 120588120588
120597120597119901119901120597120597119910119910
+ 119891119891119906119906 = ℱ119910119910
1 120588120588
120597120597119901119901120597120597119911119911
+ g = 0
55
5 Equations of motion - rotating fluid
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)
bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + g119963119963 = ℱ
bull Equations of motion on a rotating fluid Pressure gradient force modified gravitational potential (gravitational and centrifugal force) Coriolis force and friction force act 119863119863119854119854
119863119863119905119905 + 1
120588120588120571120571119901119901 + 120571120571120567120567 +f 119963119963 times 119958119958 = ℱ
Take home message
56
![Higher Equations of Motion Questions - Larbert High School25381]16._H...Higher Equations of Motion Questions 1. a) State the three equations of motion. b) List the five quantities](https://img.pdfslide.net/doc/110x75/5f048e337e708231d40e8d1a/higher-equations-of-motion-questions-larbert-high-school-2538116h-higher.jpg)
