AOSS 401, Fall 2006Lecture 9

September 26, 2007

Richard B. Rood (Room 2525, SRB)rbrood@umich.edu

734-647-3530Derek Posselt (Room 2517D, SRB)

dposselt@umich.edu734-936-0502

Class News

• Contract with class.– First exam October 10.

• Will make it completely through Chapter 3.

Weather

• National Weather Service– http://www.nws.noaa.gov/– Model forecasts:

http://www.hpc.ncep.noaa.gov/basicwx/day0-7loop.html

• Weather Underground– http://www.wunderground.com/cgi-bin/findweather/getForecast?

query=ann+arbor

– Model forecasts: http://www.wunderground.com/modelmaps/maps.asp?model=NAM&domain=US

Outline

• Thermal Wind

• Equations of motion in pressure coordinates

• Maps

Full equations of motion

1 and

)1

(

)()cos(21v

)v()sin(21v)tan(v

)()cos(2)sin(v21)vtan(

222

22

2

RTp

JDt

Dp

Dt

DTc

Dt

D

wΩugz

p

a

u

Dt

Dw

Ωuy

p

a

w

a

u

Dt

D

uΩwΩx

p

a

uw

a

u

Dt

Du

v

u Tangential coordinate system.z, height, as a vertical coordinate.zonal, meridional, vertical

• What are the three major balances we have discussed?– What are the assumptions of those balances?

The geostrophic balance

y

pfu

z

x

pfv

z

1

1

Take a vertical derivative of the equation.

The geostrophic balance

y

T

fT

g

z

u

x

T

fT

g

z

v

z

T

T

u

y

T

fT

g

z

u

z

T

T

v

x

T

fT

g

z

v

Use equation of state to eliminate density.

Thermal wind relationship in height (z) coordinates

moving fluid

Shear? (3)

• Shear is a word used to describe that velocity varies in space.

more slowly moving fluid

wind.zonal ofshear verticalz

u

z

The geostrophic balance

y

T

fT

g

z

u

y

T

fT

g

z

u

What does this equation tell us?

Zonal wind a a level is a function of average meridional temperature BELOW.

Thermal wind relationship in height (z) coordinates

An estimate of the July mean zonal wind

northsummer

southwinter

note the jet streams

Now we return to our march to pressure coordinates.

Pressure altitude

Under virtually all conditions pressure (and density) decreases with height. ∂p/∂z < 0. That’s why it is a good vertical coordinate. If ∂p/∂z = 0, then utility as a vertical coordinate falls apart. What does this look like?

Use pressure as a vertical coordinate?

• What do we need.– Pressure gradient force in pressure

coordinates.– Way to express derivatives in pressure

coordinates.– Way to express vertical velocity in pressure

coordinates.

Expressing pressure gradient force

Integrate in altitude

gz

p

z

z

gdzzp

gdzzpp

)(

)()(

Pressure at height z is force (weight) of air above height z.

Concept of geopotential

kF g

gdz

d

Define a variable such that the gradient of is equal to g. This is called a potential function.

We have assumed here that is a function of only z.

Integrating with height

gdz

d

z

z

gdzz

gdzz

gdzd

0

0

)(

0)0(

)0()(

What is geopotential?

• Potential energy that a parcel would have if it was lifted from surface to the height z.

• It is analogous to the height of a pressure surface.– We seek to have an analogue for pressure on

a height surface, which will be height on a pressure surface.

xx

zg

x

p

x

zg

x

p

1

1

Implicit that this is on a constant z surface

Implicit that this is on a constant p surface

Horizontal pressure gradient force in pressure coordinate is the gradient of geopotential

pz

pz

yy

p

xx

p

1

1

Our horizontal momentum equation(rotating coordinate system)

p

pp

pp

fDt

D

fuyDt

D

fxDt

Du

uku

)()v

(

v)()(

Assume no viscosity

Other assumptions in these equations?

2.4) Homework question

• Below the transformation of some scalar Ψ from the vertical coordinate z to the vertical coordinate p, pressure, is expressed symbolically. Write expressions for the horizontal (x and y) derivatives of Ψ and the vertical derivative (z) of Ψ in the pressure coordinate system.

Ψ (x, y, z, t) = Ψ (x, y, p(x, y, z, t), t)

What happens if ∂p/∂z = 0?

2.4) Answer

Ψ (x, y, z, t) = Ψ (x, y, p(x, y, z, t), t)

z

p

pz

y

p

pyy

x

p

pxx

pz

pz

If ∂p/∂z=0 then the transform cannot be made, because p does not depend on z. It is not a monotonic function.

What do we do with the material derivative?

Tt

T

Dt

DT

z

Tw

y

Tv

x

Tu

t

T

Dt

DT

Dt

Dz

z

T

Dt

Dy

y

T

Dt

Dx

x

T

t

T

Dt

DT

U

By definition: wDt

Dzv

Dt

Dyu

Dt

Dx ,,

Implicit that horizontal and time derivatives at constant z

What do we do with the material derivative?

Dt

Dp

p

T

Dt

Dy

y

T

Dt

Dx

x

T

t

T

Dt

DT

Dt

Dpv

Dt

Dyu

Dt

Dx,,By definition:

Implicit that horizontal and time derivatives at constant p

p

T

y

Tv

x

Tu

t

T

Dt

DT

Continuity equation

0)(

py

v

x

u

pyv

xu

t

Dt

D

p

u

u

Think about this derivation!

Thermodynamic equation

Jp

T

y

Tv

x

Tu

t

Tc

Rcc

JDt

Dp

Dt

DTRc

JDt

Dp

Dt

DTc

p

vp

v

v

)(

)(

Thermodynamic equation

pp

pp

p

c

J

pc

RT

p

T

y

Tv

x

Tu

t

T

c

J

cp

T

y

Tv

x

Tu

t

T

Jp

T

y

Tv

x

Tu

t

Tc

)(

)(

)(

Equation of state

Thermodynamic equation

pp

pp

pp

c

JS

y

Tv

x

Tu

t

T

pc

RT

p

TS

c

J

pc

RT

p

T

y

Tv

x

Tu

t

T

)(

)(

Sp is the static stability parameter.What is static stability?

Equations of motion in pressure coordinates(plus hydrostatic and equation of state)

pp

p

p

c

JS

y

Tv

x

Tu

t

T

py

v

x

u

fDt

D

0)(

uku

Tangential coordinate system.p, pressure, as a vertical coordinate.zonal, meridional, vertical

Full equations of motion

1 and

)1

(

)()cos(21v

)v()sin(21v)tan(v

)()cos(2)sin(v21)vtan(

222

22

2

RTp

JDt

Dp

Dt

DTc

Dt

D

wΩugz

p

a

u

Dt

Dw

Ωuy

p

a

w

a

u

Dt

D

uΩwΩx

p

a

uw

a

u

Dt

Du

v

u Tangential coordinate system.z, height, as a vertical coordinate.zonal, meridional, vertical

In the derivation

• Have used the conservation principle.

• Have relied heavily on the hydrostatic assumption.

• Require that the conservation principle holds in all coordinate systems.

• Plus we did some implicit scaling.

Let’s move this to a chart

Geopotential, Φ, in upper troposphere

eastwest

Φ0+ΔΦ

Φ0+3ΔΦ

Φ0

Φ0+2ΔΦ

ΔΦ > 0

south

north

Geopotential, Φ, in upper troposphere

eastwest

Φ0+ΔΦ

Φ0+3ΔΦ

Φ0

Φ0+2ΔΦ

ΔΦ > 0

south

north

Δy

Geopotential, Φ, in upper troposphere

eastwest

Φ0+ΔΦ

Φ0+3ΔΦ

Φ0

Φ0+2ΔΦ

ΔΦ > 0

south

north

Δy

δΦ = Φ0 – (Φ0+2ΔΦ)

Geopotential, Φ, in upper troposphere

eastwest

Φ0+ΔΦ

Φ0+3ΔΦ

Φ0

Φ0+2ΔΦ

ΔΦ > 0

south

north

Δy

yy

2

The horizontal momentum equation

p

pp

pp

fDt

D

fuydt

d

fxdt

du

uku

)()v

(

v)()(

Assume no viscosity

Geostrophic approximation

gp

p

fuy

fx

)(

v)( g

Geopotential, Φ, in upper troposphere

eastwest

Φ0+ΔΦ

Φ0+3ΔΦ

Φ0

Φ0+2ΔΦ

ΔΦ > 0

south

north

Δy

yfug

2

Think a second

• This is the i (east-west, x) component of the geostrophic wind.

• We have estimated the derivatives based on finite differences.– Does this seem like a reverse engineering of the

methods we used to derive the equations?

• There is a consistency– The direction comes out correctly! (towards east)– The strength is proportional to the gradient.

Geopotential, Φ, in upper troposphere

eastwest

Φ0+ΔΦ

Φ0+3ΔΦ

Φ0

Φ0+2ΔΦ

ΔΦ > 0

south

north

Δy

yfug

Return toGeopotential, Φ, in upper troposphere

eastwest

Φ0+ΔΦ

Φ0+3ΔΦ

Φ0

Φ0+2ΔΦ

ΔΦ > 0

south

northDo not assume geostrophic.Are the winds parallel to the height contours?

Weather

• National Weather Service– http://www.nws.noaa.gov/– Model forecasts:

http://www.hpc.ncep.noaa.gov/basicwx/day0-7loop.html

• Weather Underground– http://www.wunderground.com/cgi-bin/findweather/getForecast?

query=ann+arbor

– Model forecasts: http://www.wunderground.com/modelmaps/maps.asp?model=NAM&domain=US

Return toGeopotential, Φ, in upper troposphere

eastwest

Φ0+3ΔΦ

Φ0

Φ0+2ΔΦ

ΔΦ > 0

south

northDo not assume geostrophic.A qualitative velocity contour. Not the same as geopotential, but usually close.

Next Time

• Natural Coordinates

• Balanced Flows