AOSS 401, Fall 2007Lecture 23

November 05, 2007

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

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

dposselt@umich.edu734-936-0502

Class News November 05, 2007

• Homework 6 (Posted this evening)– Due Next Monday

• Important Dates: – November 16: Next Exam (Review on 14th)– November 21: No Class– December 10: Final Exam

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

Couple of Links you should know about

• http://www.lib.umich.edu/ejournals/– Library electronic journals

• http://portal.isiknowledge.com/portal.cgi?Init=Yes&SID=4Ajed7dbJbeGB3KcpBh– Web o’ Science

Material from Chapter 6

• Quasi-geostrophic theory

• Quasi-geostrophic vorticity– Relation between vorticity and geopotential

• Geopotential prognostic equation

• Relationship to mid-latitude cyclones

One interesting way to rewrite this equation

)(0 fp

ft ggg

V

Advection of vorticity

Let’s take this to the atmosphere

Advection of planetary vorticity

Φ0 - ΔΦ

Φ0 + ΔΦ

Φ0

ΔΦ > 0

A

B

C

٠

٠

٠

x, east

y, north

L LH

ζ < 0; anticyclonic

ζ > 0; cyclonicζ > 0; cyclonic

vg > 0 ; β > 0 vg < 0 ; β > 0

Advection of planetary vorticity

Φ0 - ΔΦ

Φ0 + ΔΦ

Φ0

ΔΦ > 0

A

B

C

٠

٠

٠

x, east

y, north

L LH

ζ < 0; anticyclonic

ζ > 0; cyclonicζ > 0; cyclonic

-vg β < 0 -vg β > 0

Advection of relative vorticity

Φ0 - ΔΦ

Φ0 + ΔΦ

Φ0

ΔΦ > 0

A

B

C

٠

٠

٠

x, east

y, north

L LH

ζ < 0; anticyclonic

ζ > 0; cyclonicζ > 0; cyclonic

Advection of ζ> 0

Advection of ζ< 0

Advection of vorticity

Φ0 - ΔΦ

Φ0 + ΔΦ

Φ0

ΔΦ > 0

A

B

C

٠

٠

٠

x, east

y, north

L LH

ζ < 0; anticyclonic

ζ > 0; cyclonicζ > 0; cyclonic

Advection of ζ> 0

Advection of f< 0

Advection of ζ< 0

Advection of f> 0

Summary: Vorticity Advection in Wave

• Planetary and relative vorticity advection in a wave oppose each other.

• This is consistent with the balance that we intuitively derived from the conservation of absolute vorticity over the mountain.

Advection of vorticity

Φ0 - ΔΦ

Φ0 + ΔΦ

Φ0

ΔΦ > 0

A

B

C

٠

٠

٠

x, east

y, north

L LH

ζ < 0; anticyclonic

ζ > 0; cyclonicζ > 0; cyclonic

Advection of ζ tries to propagate the wave this way

Advection of f tries to propagate the wave this way

Geopotential Nuanced

Assume that the geopotential is a wave

yx Lland

Lk

ay

lykxpAfypUfpyx

2

2

)(

cossin)()()(),(

0

000

Remember the relation to geopotential

)cossin)()()((1

)cossin)()()((11

;

windcgeostrophi of Definition

0000

00000

00

lykxpAfypUfpyf

u

lykxpAfypUfpxfxf

v

yuf

xvf

g

g

gg

Remember the relation to geopotential

'

0000

'

0

00000

sinsin)()(

)cossin)()()((1

coscos)(1

)cossin)()()((11

gg

g

gg

g

uUlykxplApUu

lykxpAfypUfpyf

u

vlykxpkAxf

v

lykxpAfypUfpxfxf

v

Advection of relative vorticity

lykxpAlkUkx

U

yv

xuU

g

gg

gggg

coscos)()(

)(

22

''

V

Advection of planetary vorticity

lykxpkAvg coscos)(

Compare advection of planetary and relative vorticity

))2

()2

((

coscos)()(

coscos)(

22

22

yx

gg

g

gg

g

LLU

v

lykxpAlkUk

lykxpkAv

V

V

Advection of vorticity

Φ0 - ΔΦ

Φ0 + ΔΦ

Φ0

ΔΦ > 0

A

B

C

٠

٠

٠

x, east

y, north

L LH

ζ < 0; anticyclonic

ζ > 0; cyclonicζ > 0; cyclonic

Advection of ζ tries to propagate the wave this way

Advection of f tries to propagate the wave this way

Compare advection of planetary and relative vorticity

))2

()2

(( 22

yx

gg

g

LLU

v

V

Short waves, advection of relative vorticity is larger

Long waves, advection of planetary vorticity is larger

Advection of vorticity

Φ0 - ΔΦ

Φ0 + ΔΦ

Φ0

ΔΦ > 0

A

B

C

٠

٠

٠

x, east

y, north

L LH

ζ < 0; anticyclonic

ζ > 0; cyclonicζ > 0; cyclonic

Short waves

Long waves

Go to the real atmosphere

An estimate of the January mean zonal wind

northwinter

southsummer

--u

Advection of relative vorticity for our idealized wave

lykxpAlkUkx

U

yv

xuU

g

gg

gggg

coscos)()(

)(

22

''

V

An estimate of the January mean zonal wind

northwinter

southsummer

What is the difference in

the advection of vorticity at

the two levels?

An estimate of the January mean zonal wind

lykxpAlkUkx

U g coscos)()( 22

Vertical Structure

• The waves propagate at different speeds at different altitudes.

• The waves do not align perfectly in the vertical.

• (This example shows that there is vertical structure, but it is only a (small) part of the story.)

A more general equation for geopotential

An equation for geopotential tendency

gg

gaag

ggg

gaagg

vfp

fDt

D

vfy

v

x

uf

Dt

D

fy

u

x

v

vy

v

x

uf

Dt

D

02

02

02

02

2

0

0

)(

1

)(

Another interesting way to rewrite vorticity equation

)1

(1

)1

(1

2

00

2

0

2

00

2

0

ffp

ftf

ffp

fft

g

g

V

V

(Flirting with) An equation for geopotential tendencyAn equation in geopotential and omega. (2 unknowns, 1 equation)

Quasi-geostrophic

)1

(1

)1

(1

2

00

2

0

2

00

2

0

ffp

ftf

ffp

fft

g

g

V

V

Geostrophic

ageostrophic

Previous analysis

• In our discussion of the advection of vorticity, we completely ignored the term that had the vertical velocity.

• Go back to our original vorticity equation– Tilting– Divergence– Thermodynamic ... (solenoidal, baroclinic)

• Which still exist after our scaling and assumptions?

We used these equations to get previous equation for

geopotential tendency

pg

aa

g

gagg

c

R

p

J

pt

py

v

x

u

f

yfDt

D

;

0

1

0

0

V

kV

VkVkV

Now let’s use this equation

pg

aa

g

gagg

c

R

p

J

pt

py

v

x

u

f

yfDt

D

;

0

1

0

0

V

kV

VkVkV

Rewrite the thermodynamic equation to get geopotential

tendency

p

J

ptp

p

J

ppt

p

J

pt

g

g

g

V

V

V

Rewrite this equation to relate to our first equation for

geopotential tendency.

p

J

pf

pf

p

f

ptp

f

p

p

Jff

p

f

tp

f

p

J

ptp

g

g

g

0000

00

00

)()( V

V

V

Scaled equations of motion in pressure coordinates

)1

(1

)()()(

2

00

2

0

0000

ffp

ftf

p

J

pf

pf

p

f

ptp

f

p

g

g

V

V

Note this is, through continuity, related to the divergence of the ageostrophic wind

Note that it is the divergence of the horizontal wind, which is related to the vertical wind, that links the momentum (vorticity equation) to the thermodynamic equation

Scaled equations of motion in pressure coordinates

)1

(1

)()()(

2

00

2

0

0000

ffp

ftf

p

J

pf

pf

p

f

ptp

f

p

g

g

V

V

Note that this looks something like the time rate of change of static stability

Explore this a bit.

)1

()1

()(

)()()(

000

0000

t

T

Spf

p

T

tpRf

tp

f

p

p

RT

p

p

J

pf

pf

p

f

ptp

f

p

p

g

V

So this is a measure of how far the atmosphere moves away from its background equilibrium state

Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency

(assume J=0)

))(()1

())((

)1

(1

)()()(

202

00

202

2

00

2

0

0000

p

f

pf

ff

tp

f

p

ffp

ftf

p

J

pf

pf

p

f

ptp

f

p

gg

g

g

VV

V

V

Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency

(assume J=0)

))(()1

())((

)1

(1

)()()(

202

00

202

2

00

2

0

0000

p

f

pf

ff

tp

f

p

ffp

ftf

p

J

pf

pf

p

f

ptp

f

p

gg

g

g

VV

V

V

Vorticity Advection

Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency

(assume J=0)

))(()1

())((

)1

(1

)()()(

202

00

202

2

00

2

0

0000

p

f

pf

ff

tp

f

p

ffp

ftf

p

J

pf

pf

p

f

ptp

f

p

gg

g

g

VV

V

V

Thickness Advection

How do you interpret this figure in terms of geopotential?

Φ0 - ΔΦ

Φ0 + ΔΦ

Φ0

ΔΦ > 0

A

B

C

٠

٠

٠

x, east

y, north

L LH

ζ < 0; anticyclonic

ζ > 0; cyclonicζ > 0; cyclonic

Short waves

Long waves

2

0

1

fg

Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency

(assume J=0)

))(()1

())((2

02

00

202

p

f

pf

ff

tp

f

p gg

VV

This is, in fact, an equation that given a geopotential distribution at a given time, then it is a linear partial

differential equation for geopotential tendency.

Right hand side is like a forcing.

You now have a real equation for forecasting the height (the pressure field), and we know that the pressure

gradient force is really the key, the initiator, of motion.

Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency

(assume J=0)

))(()1

())((2

02

00

202

p

f

pf

ff

tp

f

p gg

VV

An equation like this was very important for weather forecasting before we had comprehensive numerical models. It is still important for field forecasting, and

knowing how to adapt a forecast to a particular region given, for instance, local information.

Think about thickness advection

))(()1

())((

)1

(1

)()()(

202

00

202

2

00

2

0

0000

p

f

pf

ff

tp

f

p

ffp

ftf

p

J

pf

pf

p

f

ptp

f

p

gg

g

g

VV

V

V

Thickness Advection

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

Cold and warm advection

cold

warm

Question

• What happens when warm air is advected towards cool air?

COOL WARM

Question

• What happens when warm air is advected towards cool air?

COOL WARM

Question

• What happens the warm air?– Tell me at least two things.

COOL

WARM

Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency

(assume J=0)

))(()1

())((

)1

(1

)()()(

202

00

202

2

00

2

0

0000

p

f

pf

ff

tp

f

p

ffp

ftf

p

J

pf

pf

p

f

ptp

f

p

gg

g

g

VV

V

V

Thickness Advection

Lifting and sinking

Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency

(assume J=0)

))(()1

())((

)1

(1

)()()(

202

00

202

2

00

2

0

0000

p

f

pf

ff

tp

f

p

ffp

ftf

p

J

pf

pf

p

f

ptp

f

p

gg

g

g

VV

V

V

Thickness Advection

A nice schematic

• http://atschool.eduweb.co.uk/kingworc/departments/geography/nottingham/atmosphere/pages/depressionsalevel.html

More in the atmosphere(northern hemisphere)

SouthNorth

WarmCool

Temperature

What can you say about the wind?

Idealized vertical cross section

Increasing the pressure gradient force

Relationship between upper troposphere and surface

divergence over low enhances surface low

//increases vorticity

Relationship between upper troposphere and surface

vertical stretching //

increases vorticity

Relationship between upper troposphere and surface

vorticity advection

thickness advection

Relationship between upper troposphere and surface

note tilt with height

Mid-latitude cyclones: Norwegian Cyclone Model

Fronts and Precipitation

CloudSat Radar

Norwegian Cyclone Model

What’s at work here?

Mid-latitude cyclone development

Mid-latitude cyclones: Norwegian Cyclone Model

• http://www.srh.weather.gov/jetstream/synoptic/cyclone.htm

Below

• Basic Background Material

Tangential coordinate system

Ω

R

Earth

Place a coordinate system on the surface.

x = east – west (longitude)y = north – south (latitude)

z = local vertical orp = local vertical

Φ

a

R=acos()

Tangential coordinate system

Ω

R

Earth

Relation between latitude, longitude and x and y

dx = acos() dis longitudedy = ad is latitude

dz = drr is distance from center of a “spherical earth”

Φ

a

f=2Ωsin()

=2Ωcos()/a

Equations of motion in pressure coordinates(using Holton’s notation)

written)explicitlynot (often

pressureconstant at sderivative horizontal and time

; )()

re temperatupotential ; velocity horizontal

ln ;

0)(

Dt

Dp

ptDt

D( )

vu

pTS

p

RT

p

c

JST

t

TS

y

Tv

x

Tu

t

T

ppy

v

x

u

fDt

D

pp

p

ppp

p

V

jiV

V

V

VkV

Scale factors for “large-scale” mid-latitude

s 10 /

m 10

m 10

! s cm 1

s m 10

5

4

6

1-

-1

UL

H

L

unitsW

U

1-1-11-

14-0

2

3-

sm10

10

10/

m kg 1

hPa 10

y

f

sf

P

Scaled equations of motion in pressure coordinates

pg

aa

gagg

g

c

R

p

J

pt

py

v

x

u

yfDt

D

f

;

0

1

0

0

V

VkVkV

kV Definition of geostrophic wind

Momentum equation

Continuity equation

ThermodynamicEnergy equation