aoss 401, fall 2007 lecture 22 november 02 , 2007

AOSS 401, Fall 2007Lecture 22

November 02, 2007

Richard B. Rood (Room 2525, SRB)[email protected]

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

[email protected]

mailto:[email protected]

mailto:[email protected]

Class News November 02, 2007

• Homework 5 (Due Monday)– Posted to web– Computing assignment posted to ctools under

the Homework section of Resources

• Next Test: November 16

Seminars Today

• Professor Cecilia Bitz is giving the Dept. of Geological Sciences’ Smith Lecture on Friday (tomorrow, Nov. 2) from 4-5pm. The lecture is held in room 1528 in C.C. Little and is followed by a reception. Cecilia is an expert in high-latitude climate, climate change and variability. The title of her lecture will be:– “Future thermohaline collapse and its impact

are unlike the past”

Seminars Today

• Dr. Guy Brasseur - Professor and Associate Director, National Center for Atmospheric Research and Director of the Earth and Sun Systems Laboratory will visit U of M on Thursday/Friday. – Impact of solar variability and anthropogenic

forcing on the whole atmosphere: Simulations with the HAMMONIA Model

• Friday, November 2, 3:30 pm -- refreshments at 3 pm North Campus AOSS Auditorium, Room #2246

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

http://www.nws.noaa.gov/



http://www.wunderground.com/cgi-bin/findweather/getForecast?query=ann+arbor

http://www.wunderground.com/cgi-bin/findweather/getForecast?query=ann+arbor

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






Material from Chapter 6

• Quasi-geostrophic theory

• Quasi-geostrophic vorticity– Relation between vorticity and geopotential

Going way back

Mathematics

• Remember the idea that mathematics is a language to use to help us explore a complex system.– Verb: equal, – Qualification: not equal, greater than, less than,

approximately

• We own the equations and can do to them what we want, as long as we remember equal and not equal.

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

Approximate horizontal momentum equation

gagg yf

Dt

DVkVk

V 0

This equation states that the time rate of change of the geostrophic wind is related to1. the coriolis force due to the ageostrophic

wind and 2. the part of the coriolis force due to the

variability of the coriolis force with latitude and the geostrophic wind.

Both of these terms are smaller than the geostrophic wind itself.

Derived a vorticity equation

• Provides a “suitable” prognostic equation because need to include div(ageostrophic wind) in the prognostics.

• Remember the importance of divergence in vorticity equations.

Scaled horizontal momentum in pressure coordinates

0

0

0

0

0

gagg

gagg

gagg

yuufDt

vD

yvvfDt

uD

yfDt

D

VkVkV

Use definition of vorticity vorticity equation

gaagg

gagg

gagg

vy

v

x

uf

Dt

D

yuufDt

vD

x

yvvfDt

uD

y

)(

0)(

0)(

0

0

0

One interesting way to rewrite this equation

gaa

ggg

gaagg

vy

f

y

v

x

uf

t

y

f

vy

v

x

uf

Dt

D

)(

)(

0

0

V

Expand material derivative


)(0

0

fp

ft

vy

f

pf

t

ggg

gggg

V

V

Equation of continuity

Understand how this is equivalent


)(0 fp

ft ggg

V

Advection of vorticity


ggggg vf VV )(


Advection of relative vorticity

Advection of planetary vorticity

Let’s take this to the atmosphere

Geopotential Map (Northern Hemisphere)

Φ0 - ΔΦ

Φ0 + ΔΦ

Φ0

ΔΦ > 0

A

B

C

٠

٠

٠

x, east

y, north

L LH

Where is geostrophic approximation valid?What other force balance is important?What is the sign of the geostrophic wind?

Geostrophic wind

Φ0 - ΔΦ

Φ0 + ΔΦ

Φ0

ΔΦ > 0

A

B

C

٠

٠

٠

x, east

y, north

L LH

vg > 0 vg < 0

Geopotential Map (Northern Hemisphere)

Φ0 - ΔΦ

Φ0 + ΔΦ

Φ0

ΔΦ > 0

A

B

C

٠

٠

٠

x, east

y, north

L LH

What is the sign of planetary vorticity?What is the sign of the relative vorticity?

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

β > 0 β > 0


Φ0 - ΔΦ

Φ0 + ΔΦ

Φ0

ΔΦ > 0

A

B

C

٠

٠

٠

x, east

y, north

L LH



vg > 0 ; β > 0 vg < 0 ; β > 0


Φ0 - ΔΦ

Φ0 + ΔΦ

Φ0

ΔΦ > 0

A

B

C

٠

٠

٠

x, east

y, north

L LH



-vg β < 0 -vg β > 0


Φ0 - ΔΦ

Φ0 + ΔΦ

Φ0

ΔΦ > 0

A

B

C

٠

٠

٠

x, east

y, north

L LH



ζ from >0 to <0vg > 0; ug > 0

ζ from <0 to >0vg < 0; ug > 0


Φ0 - ΔΦ

Φ0 + ΔΦ

Φ0

ΔΦ > 0

A

B

C

٠

٠

٠

x, east

y, north

L LH



Advection of ζ> 0

Advection of ζ< 0


Φ0 - ΔΦ

Φ0 + ΔΦ

Φ0

ΔΦ > 0

A

B

C

٠

٠

٠

x, east

y, north

L LH



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.

Summary: Vorticity Advection in Wave

• What does this do to the wave.


Φ0 - ΔΦ

Φ0 + ΔΦ

Φ0

ΔΦ > 0

A

B

C

٠

٠

٠

x, east

y, north

L LH



Advection of ζ tries to propagate the wave this way

Advection of f tries to propagate the wave this way

Remember the relation to geopotential

2

02

2

2

2

0

00

1)(

1

;

windcgeostrophi of Definition

fyxfy

u

x

v

yuf

xvf

gg

gg

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

)(

Barotropic fluid

gg

gg

vfDt

D

vfp

fDt

D

02

02

02

barotropic

Perturbation equation

xxu

t

vvuuu

vfDt

D

g

ggggg

gg

2

02

)(

equation of formon perturbati

;

Wave like solutions

0))((

)Re(

)(

22

)(0

2

klkuk

e

xxu

t

g

tlykxi

g

Dispersion relation. Relates frequency and wave number to flow. Must be true for waves.

Stationary wave

g

g

tlykxi

ulk

klkuk

e

22

22

)(0

0))((

0

)Re(

Wind must be positive, from the west, for a wave.

Geopotential Nuanced

Assume that the geopotential is a wave

yx Lland

Lk

ay

lykxpAfypUfpyx

2

2

)(

cossin)()()(),(

0

000


)cossin)()()((1

)cossin)()()((11

;

windcgeostrophi of Definition

0000

00000

00

lykxpAfypUfpyf

u

lykxpAfypUfpxfxf

v

yuf

xvf

g

g

gg


'

0000

'

0

00000

sinsin)()(

)cossin)()()((1

coscos)(1

)cossin)()()((11

gg

g

gg

g

uUlykxplApUu

lykxpAfypUfpyf

u

vlykxpkAxf

v

lykxpAfypUfpxfxf

v


lykxpAlkUkx

U

yv

xuU

g

gg

gggg

coscos)()(

)(

22

''

V


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


Φ0 - ΔΦ

Φ0 + ΔΦ

Φ0

ΔΦ > 0

A

B

C

٠

٠

٠

x, east

y, north

L LH



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


Φ0 - ΔΦ

Φ0 + ΔΦ

Φ0

ΔΦ > 0

A

B

C

٠

٠

٠

x, east

y, north

L LH



Short waves

Long waves

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

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


)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


)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


(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


(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



Short waves

Long waves


(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.


(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 adapt a forecast to a particular region

given, for instance, local information.

• See you Monday

aoss 401, fall 2007 lecture 22 november 02 , 2007

Documents

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highlatitude climate