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AOSS 401, Fall 2007Lecture 4
September 12, 2007
Richard B. Rood (Room 2525, SRB)[email protected]
734-647-3530Derek Posselt (Room 2517D, SRB)
Class News
• Posselt office hours: Tues/Thurs AM and after class– If you are coming from outside the building for office
hours (central or north campus), please email or call ahead
• Class cancelled Friday 14 September• No office hours Thursday 13 September
– I will be available during regular class time Friday• Homework 1 due today (Questions?)• Homework 2 posted by the end of the day
– Under “resources” in homework folder• Due Monday (September 17, 2007)
Weather
• NCAR Research Applications Program– http://www.rap.ucar.edu/weather/
• National Weather Service– http://www.nws.noaa.gov/dtx/
• Weather Underground– http://www.wunderground.com/cgi-bin/findweather/get
Forecast?query=ann+arbor– Model forecasts: http://
www.wunderground.com/modelmaps/maps.asp?model=NAM&domain=US
Outline
1. Review• Momentum equation(s)• Geopotential and atmospheric thickness• Transformation of vertical coordinates
2. Material Derivative• Lagrangian and Eulerian reference frames• Material (total, substantive) derivative• Mathematical tools needed for Homework 2
From last time
Our momentum equation
jikuu
fufvgpdt
d )(
1 2
Acceleration (change in
momentum)
Pressure Gradient Force: Initiates Motion
Friction/Viscosity: Opposes Motion
Gravity: Stratification
and buoyancy
Coriolis: Modifies Motion
Surface Body Apparent
Our momentum equation
jikuu
fufvgpdt
d )(
1 2
Surface Body Apparent
This equation is a statement of conservation of momentum.
We are more than half-way to forming a set of equations that can be used to describe and predict the motion of the atmosphere!
Once we add conservation of mass and energy, we will spend the rest of the course studying what we can learn from these equations.
Review: Vertical Structure and Pressure as Vertical Coordinate
Vertical Structure and Pressure as a Vertical Coordinate
• Remember, we defined the geopotential as
• And we were able to use hydrostatic balanceand the ideal gas law to show
zgdzzgdzd0
)(,
gdz
dp
pdRTp
RTdpdpdgdz ln
Vertical Structure and Pressure as a Vertical Coordinate
• Integrate from pressure p1 to p2 at heights z1 and z2
• From the definition of geopotential we get thickness and the fact that thickness is proportional to temperature
• So, hydrostatic balance and the ideal gas law form the basis for the relationship between and
2
1
ln12
p
p
pdTg
RZZ
2
1
ln)()(
ln
12
p
p
pTdRzz
pRTdd
p
Pressure Gradient in Pressure Coordinates
• Remember from Monday: horizontal pressure gradient force in pressure coordinates is the gradient of geopotential
• Remember, if we have hydrostatic balance:
pz
pz
yy
p
xx
p
1
1
dp
d
Pressure Coordinates: Why?
• From Holton, p2:“The general set of … equations governing the motion of the atmosphere is extremely complex; no general solutions are known to exist. …it is necessary to develop models based on systematic simplification of the fundamental governing equations.”
• Two goals of dynamic meteorology:
1. Understand atmospheric motions (diagnosis)2. Predict future atmospheric motions (prognosis)
• Use of pressure coordinates simplifies the equations of motion
Pressure Coordinates: Why?
fuy
p
dt
dv
fvx
p
dt
du
zz
zz
)1()(
)1()(
Horizontal momentum equations (u, v), no viscosity
Height (z) coordinates
fuydt
d
fxdt
du
pp
pp
)()v
(
v)()(
Pressure (p) coordinates
Density is no longer a part of the equations of motionHidden inside the geopotential… We will see that thissimplifies other relationships as well…
New Material: Holton Chapter 2
• Lagrangian and Eulerian Points of View
• Material (total) derivatives
• Review of key mathematical tools
Vector Momentum Equation(Conservation of Momentum)
jikuu
fufvgpdt
d )(
1 2
Vector Momentum Equation(Conservation of Momentum)
Coordinate system is defined as tangent to the Earth’s surface
jikuu
fufvgpdt
d )(
1 2
xiyj
z k
eastnorth
Local vertical
Velocity (u) = (ui + vj + wk)
Have entertained the possibility of several vertical coordinates z, p, …
Previously: Conservation of Momentum
Consider a fluid parcel moving along some trajectory.
Now we are going to think about fluids.
Consider a fluid parcel moving along some trajectory
(What is the primary force for moving the parcel around?)
CurvCorkuu
gpdt
d)(
1 2
Consider several trajectories
How would we quantify this?
Use a position vector that changes in time
Parcel position is a function of its starting point.The history of the parcel is known
Lagrangian Point of View
• This parcel-trajectory point of view, which follows a parcel, is known as the Lagrangian point of view.
• Benefits:– Useful for developing theory – Very powerful for visualizing fluid motion– The history of each fluid parcel is known
• Problems:– Requires considering a coordinate system for each parcel– How do you account for interactions of parcels with each other?– How do you know about the fluid where there are no parcels?– How do you know about the fluid if all of the parcels bunch
together?
Lagrangian Movie:Mt. Pinatubo, 1992
Consider a fluid parcel moving along some trajectory
Could sit in one place and watch parcels go by.
How would we quantify this?
• In this case:• Our coordinate system does not change• We keep track of information about the atmosphere at a
number of (usually regularly spaced) points that are fixed relative to the Earth’s surface
Eulerian Point of View
• This point of view, where is observer sits at a point and watches the fluid go by, is known as the Eulerian point of view.
• Benefits:– Useful for developing theory– Requires considering only one coordinate system for all parcels– Easy to represent interactions of parcels through surface forces– Looks at the fluid as a field.– A value for each point in the field – no gaps or bundles of
“information.”
• Problems– More difficult to keep track of parcel history—not as useful for
applications such as pollutant dispersion…
An Eulerian Map
Why Consider Two Frames of Reference?
• Goal: understanding. Will allow us to derive simpler forms of the governing equations
• Basic principles still hold: the fundamental laws of conservation– Momentum– Mass– Energy
• are true no matter which reference frame we use
Movies Eulerian vs. Lagrangian
Eulerian Lagrangian
Why Lagrangian?
• Lagrangian reference frame leads to the material (total, substantive) derivative
• Useful for understanding atmospheric motion and for deriving mass continuity…
On to the Material Derivative…
Material Derivative
Δy
Δx
Consider a parcel with some property of the atmosphere, like temperature (T), that moves some distance in time Δt
x
y
Material Derivative
zz
Ty
y
Tx
x
Tt
t
TT
HigherOrderTerms
Assume increments over Δt are small, andignore Higher Order Terms
We would like to calculate the change in temperature over time Δt, following the parcel.
Expand the change in temperature in a Taylor series around the temperature at the initial position.
Material Derivative
t
z
z
T
t
y
y
T
t
x
x
T
t
T
t
T
Divide through by Δt
dt
dz
z
T
dt
dy
y
T
dt
dx
x
T
t
T
dt
dT
Take the limit for small Δt
Material Derivative
Dt
Dz
z
T
Dt
Dy
y
T
Dt
Dx
x
T
t
T
Dt
DT
Introduce the convention of d( )/dt ≡ D( )/Dt
This is the material derivative: the rate of change of T following the motion
Material Derivative
Tt
T
Dt
DT
z
Tw
y
Tv
x
Tu
t
T
Dt
DT
U
Remember, by definition:
wDt
Dzv
Dt
Dyu
Dt
Dx ,,
and the material derivative becomes
LagrangianEulerian
Material Derivative (Lagrangian)
Dt
DTMaterial derivative, T change following the parcel
Local Time Derivative (Eulerian)
t
TT change at a fixed point
Change Due to Advection TU Advection
COLD
WARM
A Closer Look at Advection
z
Tw
y
Tv
x
TuT
U
Expanding advection into its components, we have
Change Due to Advection TU Advection
y
Tv
x
Tu
Class Exercise: Gradients and Advection
• The temperature at a point 50 km north of a station is three degrees C cooler than at the station.
• If the wind is blowing from the north at 50 km h-1 and the air is being heated by radiation at the rate of 1 degree C h-1, what is the local temperature change at the station?
• Hints:– You should not need a calculator– Use the definition of the material derivative and of
advection
Material Derivative
TDt
DT
t
T
z
Tw
y
Tv
x
Tu
Dt
DT
t
T
z
Tw
y
Tv
x
Tu
t
T
Dt
DT
U
We will use this again later…
Can be rewritten in terms of the local change
Advection: A Recent ExampleSix-hour time temperature change at St. Cloud, MN
1100 UTC 1200 UTC 1300 UTC
1400 UTC 1500 UTC 1600 UTC
Return to the Momentum Equation
CurvCokuu
gpdt
d)(
1 2
Remember, we derived from force balances
This is in the Lagrangian reference frame
CurvCokuuuu
gpt
)(1 2
In the Eulerian reference frame, we have
Non-linearThis comes from Eulerian point of view
Homework 2:Mathematical Tools
• Problem 2 in homework 2 asks you to expand various vector operators
• A quick review of these follows
Gradient: Three-Dimensional Partial Spatial Derivative
• A vector operator defined as
• The gradient of a scalar (f) is a vector
kjizyx
kjiz
f
y
f
x
ff
Dot Product
• The divergence is the dot product of the gradient with another vector
• The dot product of two vectors A and B is
zzyyxx
zyx
zyx
BABABA
BBB
AAA
BA
kjiB
kjiA
Laplacian: Divergence of a Gradient
• Three-dimensional partial spatial second derivative.
• Since it is a dot-product, it is NOT a vector itself…
• The Laplacian of a scalar (f) is
2
2
2
2
2
22
z
f
y
f
x
fff
Curl (Cross-Product)
• The curl will be closely related to rotation—we will use this extensively when we cover vorticity
• The result of taking the curl is a vector that is perpendicular (orthogonal) to both of the original vectors
• The direction of the resulting vector depends on the order of operations…
• We will return to this in more detail later…ABBA
Curl (Cross-Product)
For vectors A and B
the curl is
)(
)(
)(
xyyx
zxxz
yzzy
BABA
BABA
BABA
k
j
i
BA
zyx
zyx
BBB
AAA
kji
BA
Same as the determinant
Next time
• Conservation of mass (the continuity equation) (Holton, 2.5.1, 2.5.2)
• Scale analysis (Holton, 2.4, 2.5.3)
• Reversing these (compared to Holton) – derivation of the continuity equation uses the
distinction between Eulerian and Lagrangian reference frames
– Do this while the material is relatively fresh…