09 - Math and Equations

8/8/2019 09 - Math and Equations

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Math Review and the EquationsMath Review and the Equations

of Motionof MotionATMS-303

Fall 2010


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Atmospheric VariablesAtmospheric Variables

y Most atmospheric variables (temperature,pressure, moisture, etc) vary in all threespatial dimensions (x, y, z) and time (t)

y

Example: Temperature We write this as T(x, y, z, t)

Or T(x, y, p, t) if pressure is vertical coordinate

y Cannot be treated as a constant in

differentiation or integration unlesssimplifying assumptions are made Variables assumed constant in one more

directions or in time


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Example: Temperature (x, y)Example: Temperature (x, y)

XXyy


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Example: Temperature (z)Example: Temperature (z)

zz


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Example: Temperature (t)Example: Temperature (t)

TimeTime


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DifferentiationDifferentiation

yWe can differentiate these variables with

respect to (w.r.t.) x, y, z, or t

y A partial derivative expresses how Tchanges in one dimension with all other

dimensions kept constant

yWritten as

x

T

x

x

y

T

x

x

z

T

x

x

t

T

x

x


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Example: Temperature (x, y)Example: Temperature (x, y)

xxx

y : y, z, t remain

constant How temperature

changes in x directiononly

y : x, z, t remainconstant

xx

x

yx

x

yx

x


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Time DerivativesTime Derivatives

y Partial derivative :

Keep x, y, z constant

How does temperature change at a fixed point?

Also called Eulerian derivative

y But what if you are moving?

Example: Driving from Champaign to Chicago

Temperature can change due to weather, solarheating, etc

Also expect temperature change betweenChampaign and Chicago at any fixed time

t

T

x

x

t

T

x

x

y

T

x

x


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Total Time DerivativeTotal Time Derivative

y Must use total time derivative to expresstime rate of change of temperature followinga moving object Examples: A car, an air parcel

yWritten as ory Expressed as

y u, v, w are components of velocity vectory Remember, the total derivative is w.r.t. time

only!y Also called Lagrangian derivative

dtdT

DtDT

z

Tw

y

Tv

x

Tu

t

T

dt

dT

x

x

x

x

x

x

x

x!


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Chain RuleChain Rule

y Remember to apply the chain rule (productrule) when differentiating products ofvariables

y Example: Take z-derivative of Ideal Gas Law

y

Often simplify this be assuming that densityis constant Called incompressibility or Boussinesq

Approximation

RTz

pz

Vx

x!

x

x

z

TR

zRT

z

p

x

x

x

x!

x

xV

V


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VectorsVectors

y Vectors define quantities that have bothmagnitude and direction

y Example: Position (x, y, z) x = East-west (zonal) direction

y = North-south (meridional) direction z = Up-down (vertical) direction

y Example: Velocity (u, v, w) u = Zonal velocity

v = Meridional velocity w = Vertical velocity

y Scalar: A quantity with magnitude but no direction Examples: Pressure, temperature, moisture


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Vector MagnitudeVector Magnitude

y For any vector a, its magnitude is given by

y Can calculate direction of 2-

Dvectors withinverse tangent function

y Be careful of signs and angles Draw it out Tangent has period of 180 Wind direction is direction wind is blowing from

y Can also obtain x and y components ofvectors given magnitude and angle

2

3

2

2

2

1aaaa !

T


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Adding or Subtracting VectorsAdding or Subtracting Vectors

y Add vectors head to tail

y Must split vectors into components

x, y, z, components

Add components if pointing in same direction

Subtract components if pointing in oppositedirections


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Unit VectorsUnit Vectors

y Often use unit vectors (i, j, k) to denote

direction

i = x direction

j = y direction

k = z direction

y Magnitudes of unit vectors are one, so

they do not affect magnitudey Example: Velocity vector

kwjviuv !T


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Dot ProductDot Product

y Consider two vectors a and b

y Can define dot product ab as

y Result is scalar

Dot product of a unit vector with itself is oney Dot product of two perpendicular

vectors is always zero. Why?

kajaiaa 321

!T

kbjbibb 321

!T

332211babababa !

TT


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Dot ProductDot Product

y The dot product is also given by

y Note:E is angle between two vectors

y Note: A scalar times a vector is just the

scalar times each component of the

vector

EcosbabaTTTT

!


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Cross ProductCross Product

y Consider two vectors a and b

y Can define cross product a x b as

y Result is a vectory Cross product of two parallel vectors is

always zeroy Cross product of any two vectors is always

perpendicular to both vectorsy Remember right-hand rule!

kajaiaa 321

!T

kbjbibb 321

!T

kbabajbabaibababa )()()(122131132332

!vTT


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Cross ProductCross Product

y Easy way to remember: 3 x 3 determinant!

y Can write first two columns to right of

third

y Down and right Positive

y Down and left Negative

321

321

bbb

aaa

kji

ba !vTT


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Vector DifferentiationVector Differentiation

y To differentiate a vector, take the

appropriate derivative of eachcomponent:

y Remember, unit vectors are constant

kx

wj

x

vi

x

u

x

v x

x

x

x

x

x!

x

xT


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The Del Operator GradientThe Del Operator Gradient

y The vector operator is defined by

y

This is an operator; it has no value (justlike differentials (d/dt)

y Gradient: Del operator applied to a scalar

Result is always a vector, directedperpendicular to isopleths toward highervalues

kz

jy

ix

x

x

x

x

x

x|

kz

jy

ix

x

x

x

x

x

x!


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The Del Operator GradientThe Del Operator Gradient


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The Del Operator DivergenceThe Del Operator Divergence

y Divergence:

Del operator applied to a vector

Result is always a scalar

y The divergence of the wind field yields a

scalar known as the divergence

Note:Divergence = -1 x Convergence

y Important: The del operator isdistributive, but not commutative

zw

yv

xuv

x

xx

xx

x!T


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The Total DerivativeThe Total Derivative

y Recall that the total derivative is

y Can rewrite this using the dot product

and del operators as

y Can think of v dot del as operator:

Not equal to divergence!

z

Tw

y

Tv

x

Tu

t

T

dt

dT

x

x

x

x

x

x

x

x!

Tvt

T

dt

dT

x

x!

Tvv

t

v

dt

vd TTTT

x

x!

zw

yv

xuv

x

x

x

x

x

x!

T


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AdvectionAdvection

y Advection The transport of a quantityby the wind If the wind is zero, advection is zero

If the spatial derivative is zero, advection iszero

y Advection =

y Moisture advection =

y Advection in x direction =

y Total time derivative = Local timederivative + Advection

zw

yv

xuv

x

x

x

x

x

x!

T

z

qw

y

qv

x

quqv

x

x

x

x

x

x!

T

xqux

x


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TheThe LaplacianLaplacian 22

y Two consecutive applications of the deloperator

y Scalars: The divergence of the gradient

Always returns a scalar

y

Vectors: The gradient of the divergence Always returns a vector

2

2

2

2

2

2

2

z

T

y

T

x

TT

x

x

x

x

x

x!

kz

w

y

v

x

u

zj

z

w

y

v

x

u

yi

z

w

y

v

x

u

xv

2

!

T


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The Del Operator CurlThe Del Operator Curl

yWritten out this is

wv

zyx

kji

vx

x

x

x

x

x!v

T

kyx

vj

xzi

z

v

yv

x

x

x

x

x

x

x

x

x

x

x

x!vT


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Momentum Equation: Vector FormMomentum Equation: Vector Form

y LHS: Acceleration of wind

y RHS1: Pressure gradienty RHS2: Coriolis acceleration

y RHS3: Gravity

y RHS4: Friction

y RHS is the four fundamental forces at workin the atmosphere

Fkgvkfpdt

vdTT

T

v! )(1

V


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Momentum Equations:ComponentMomentum Equations:Component

FormForm

y Coriolis parameter:y

F = frictiony p = pressurey V = densityy g = gravity

xFfvx

p

dt

du

x

x!

V

1

yFfuy

p

dt

dv

x

x!

V

1

zFgz

p

dt

dw

x

x!

V

1

-14s10sin2

};! Nf

xFfvx

puv

t

u

x

x!

x

x

V

1T

yFfuy

pvv

t

v

x

x!

x

x

V

1T

zFgz

pwv

t

w

x

x!

x

x

V

1T


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The Four ForcesThe Four Forces

y Pressure gradient force (PGF)

y Coriolis force Apparent force due to rotation of earth

y Frictiony Gravity

y Note:Centripetal force is not a unique force.It is the force needed for an object to movein a circular path. The force must be suppliedby a physical force.


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Pressure Gradient ForcePressure Gradient Force

y Pressure gradient force always directedperpendicular to isobars toward lowerpressure Why? Negative sign

y Unitsy Air molecules want to flow from where

there is greater pressure to where there isless pressure

Nature abhors a vacuum!y Increasing the pressure gradient yields dv/dt

>0 and faster flow


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Rotation of the EarthRotation of the Earth

y Viewed from NorthPole, earth appears to

rotate counter-clockwise

y Analogous to merry-

go-round


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MerryMerry--GoGo--Round ExampleRound Example

y If merry-go-round is NOT moving, ball doesnot appear to be deflected

y If merry-go-around is moving, ball will still

travel in straight path as seen from abovey Ball will appear to be deflected to its right

from rotating platform Catcher sees ball move to his left appears to be

due to external force Catcher actually rotates to his right out of the

way of ball

y Same effect occurs on rotating earth


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The Coriolis ForceThe Coriolis Force


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CoriolisCoriolis Force and LatitudeForce and Latitude


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TheThe CoriolisCoriolis ForceForce


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CoriolisCoriolis ForceForce SummarySummary

y Deflects objects to the RIGHT in NH

y Deflects objects to the LEFT in SH

y Changes the direction, not the speed of

motion

y Force is proportional to speedy Maximum at Poles, zero at Equator

Distance from axis of rotation does not change atEquator


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FrictionFriction

y Molecular friction

Between air molecules and the earths surface

Between air molecules (viscosity)

Not very important

y Eddies

Small circulations that mix the effect of

friction in the atmosphere Boundary layer


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Eddy ViscosityEddy Viscosity


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GravityGravity

y Force acting toward the center of the

earth

y Small components in x and y directions

Earth is not a perfect sphere

Centrifugal force

y For our purposes, treat gravity as a

constant force acting downward Vertical equation of motion only

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09 - Math and Equations