AOSS 401, Fall 2007Lecture 21

October 31, 2007

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

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

dposselt@umich.edu734-936-0502

Class News October 31, 2007

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

the Homework section of Resources

• Next Test November 16

• Thanks for the comments on the Midterms evaluations

A Diversion

• Santa Ana Winds

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

Material from Chapter 6

• Quasi-geostrophic theory

• Quasi-geostrophic vorticity– Relation between vorticity and geopotential

Quasi-Geostrophic

• Derive a system of equations that is close to geostrophic and hydrostatic balance, but includes the effects of ageostrophic wind

• Comes from scale analysis of equations of motion in pressure coordinates

• Scale analysis = make assumptions(where do these assumptions break down?)

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

From the scale analysis we introduced the non-dimensional Rossby number

114

15

00

0

1010

10

Number Rossby

s

s

Lf

U

f

RoLf

U

A measure of planetary vorticity compared to relative vorticity.A measure of the importance of rotation.

Scale analysis of equations in pressure coordinates

• Start:– horizontal flow is approximately geostrophic– vertical velocity much smaller than horizontal

velocity

We will scale the material derivative

yv

xu

tDt

( )D

Dt

( )D

tDt

D( )

Dt

Dp

ptDt

D( )

ggg

gpgp

pp

)()

; )()

V

V

Ignore // small

This is for use in the advection of temperature and momentum.

ω comes from div(ageostrophic wind)

Variation of Coriolis parameter

• L, length scale, is small compared to the radius of the Earth

• In the calculation of geostrophic wind, assume f is constant; f = f0

• We cannot assume f is constant in the Coriolis terms…

...0 ydy

dfff

Variation of Coriolis parameter

0

0

00

0

at 0

cos2)

...

0

y

ady

df

yfydy

dfff

ydy

dfff

Variation of Coriolis parameter

1sin

cos

0

0

0

00

Roa

L

f

L

yfydy

dfff

Scale of first two terms.

Continuity equation becomes

0

0)(

py

v

x

u

ppy

v

x

u

aa

p

V

Thermodynamic equation

pTS

c

JST

t

TS

y

Tv

x

Tu

t

T

p

ppp

ln

V

geostrophic wind can be used here.

static stability, Sp, is large; ω cannot be ignored

Thermodynamic equation(use the fact that atmosphere is near hydrostatic balance)

p

T

dp

dT

tpyxTpTtpyxT

pTS

c

JST

t

TS

y

Tv

x

Tu

t

T

tot

p

ppgpgg

0

0 ),,,()(),,,(

ln

V

split temperature into basic state plus deviation

Thermodynamic equation(and with the hydrostatic equation)

pg

pg

pg

c

R

p

J

pt

c

J

R

p

ptR

p

dp

d

p

RT

p

RT

p

c

J

R

pT

t

T

;

ln ; 00

V

V

V

note the inverse relation of heating with pressure

The Momentum Equation

horizontal flow is approximately geostrophic

• L, length scale, is small compared to the radius of the Earth

• In the calculation of geostrophic wind, assume f is constant; f = f0

windicageostroph

1

0

a

ag

g f

V

VVV

kV

horizontal flow is approximately geostrophic

110

Rog

a

ag

V

V

VV

Forcing terms in momentum equation

gag

ag

ag

fyf

yf

ff

fDt

D

VkVVk

VVk

VVkVk

VkV

00

0

)()(

)()(

)(

approx of coriolis parameter Use definition of geostrophic wind

in the pressure gradient force

def’n of the full wind

Forcing terms in momentum equation

ga

gag

ag

ag

yf

fyf

yf

ff

VkVk

VkVVk

VVk

VVkVk

0

00

0

)()(

)()(

)(

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.

A Point

• All of the terms in the equation for the CHANGE in the geostrophic wind, which is really a measure of the difference from geostrophic balance, are order Ro (Rossby number).

– Again, reflects the importance of rotation to the dynamics of the atmosphere and ocean

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

What is the point?

• Set of equations that describes synoptic-scale motions and includes the effects of ageostrophic wind (vertical motion)

Scale Analysis = Make Assumptions

Quasi-geostrophic system is good for:

• Synoptic scales

• Middle latitudes

• Situations in which Va is important

• Flows in approximate geostrophic and hydrostatic balance

• Mid-latitude cyclones

Scale Analysis = Make Assumptions

Quasi-geostrophic system is not good for:

• Very small or very large scales

• Flows with large vertical velocities

• Situations in which Va ≈ Vg• Flows not in approximate geostrophic and

hydrostatic balance

• Thunderstorms/convection, boundary layer, tropics, etc…

What will we do next?

• Derive a vorticity equation for these scaled equations.– Actually provides a “suitable” prognostic

equation because need to include div(ageostrophic wind) in the prognostics.

– Remember the importance of divergence in vorticity equations.

Derive a vorticity equation

• Going to spend some time with this.

Vorticity

fy

u

x

v

a

Uk

vorticityofcomponent Vertical

relative vorticityvelocity in (x,y) plane

shear of velocity suggests rotation

Vorticity

fy

u

x

v

a

Uk

vorticityofcomponent Vertical

view this as the definition of relative vorticity

If we want an equation for the conservation of vorticity, then

• We want an equation that represents the time rate of change of vorticity in terms of sources and sinks of vorticity.

Conservation (continuity) principle

• dM/dt = Production – Loss

Newton’s Law of Motion

mdt

d/F

v

Which is the vector form of the momentum equation.(Conservation of momentum)

Where F is the sum of forces acting on a parcel, m mass, v velocity

What are the forces?

• Total Force is the sum of all of these forces– Pressure gradient force– Gravitational force– Viscous force– Apparent forces

• Derived these forces from first principles

If we want an equation for the conservation of vorticity, then

• We could approach it the same way as momentum, define the sources and sinks of vorticity from first principles.– But that is hard to do. What are the first

principle sources of vorticity?

• Or we could use the conservation of momentum, and the definition of vorticity to derive the equation.

Newton’s Law of Motion(components)

y

u

x

v

mFdt

dv

mFdt

du

y

x

vorticityof definition and

/

/

Combine definition and conservation principle

y

u

x

v

mFdt

dv

mFdt

du

y

x

vorticityof definition and

/

/

Operate on momentum equation

)/()(

)/()(

mFxdt

dv

x

mFydt

du

y

y

x

Subtract

)/()/()()( mFy

mFxdt

du

ydt

dv

x xy

so a time rate of change of vorticity will come from here.

Subtract

)/()/()()( mFy

mFxdt

du

ydt

dv

x xy

so details will depend on d( )/dt.For an Eulerian fluid d( )/dt = D( )Dt, material derivative.

For a Lagrangian description could write immediately.

Expand derivative

)/()/(

)()(

)/()/()()(

)/()/()()(

mFy

mFx

uyt

u

yv

xt

v

x

mFy

mFx

ut

u

yv

t

v

x

mFy

mFxDt

Du

yDt

Dv

x

xy

xy

xy

vv

vv

Expand derivative

)/()/(

)/()/(

)()(

mFy

mFx

uy

uyt

u

y

vx

vxt

v

x

mFy

mFx

uyt

u

yv

xt

v

x

xy

xy

vv

vv

vv

Expand derivative

)/()/(

mFy

mFx

uy

uyt

u

y

vx

vxt

v

x

xy

vv

vv

Dζ/Dt comes from here.

Expand derivative

)/()/(

mFy

mFx

uy

uyt

u

y

vx

vxt

v

x

xy

vv

vv

Other things comes from here.

Collect terms

)/()/( mFy

mFx

uy

vxDt

Dxy

vv

Let’s return to our quasi-geostrophic formulation

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

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.

Study Questions

Study question 1Thermodynamic equation

pg

pg

pg

c

R

p

J

pt

c

J

R

p

ptR

p

p

RT

p

c

J

R

pT

t

T

; V

V

V

Why can we pull this p outside of the derivative operators?

Study Question 2Forcing terms in momentum equation

ga

gag

ag

ag

yf

fyf

yf

ff

VkVk

VkVVk

VVk

VVkVk

0

00

0

)()(

)()(

)(

Do the derivation of this approximation.

Study Question 3An estimate of the July mean zonal wind

northsummer

southwinter

Compare this figure to the similar figure for January. What is similar and different between the tropospheric jets? (magnitude, position). Without referring to the plots of temperature, what can you say about the temperature structure?

Study question 4

y

fv

Dt

Df

f

thatShow parameter. Coriolis theis

Study question 5

What is the scale of the horizontal divergence of the wind to the total vorticity in middle latitudes. Would you expect the same in the tropics?

What are the units of potential vorticity?

Study question 6

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

Carry out this derivation.

Scaled equations of motion in pressure coordinates

pg

aa

g

gagg

c

R

p

J

pt

py

v

x

u

f

yfDt

D

;

0

1

0

0

V

kV

VkVkV

Set up and start the derivation for the time rate of change of divergence of the horizontal divergence. Show how to extract D(div(uhorizontal)/Dt= ….