AOSS 401, Fall 2007Lecture 27

November 28, 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 28, 2007

• Homework 7 (Posted tomorrow, Thursday)– Due Next Friday

• Important Dates: – December 10: Final Exam– December 7:

• Go over homework• Review session

– December 5: Hurricanes• Joint with AOSS 451

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 11

• Tropics– Tropics versus middle latitudes– Features of the tropical circulation

• Tropical scale analysis

• Tropical waves– Kelvin waves– Equatorial Rossby Waves– Mixed Rossby-gravity Waves

Picture of Earth: What can you say about this figure?

What are the differences between the tropics and the middle latitudes on Earth?

• Tropics:– The area of the tropics – say + and – 30

degrees latitude is half the area of the Earth.– Might say the tropics is + and – 20 degrees of

latitude, and subtropics are between 20 and 30 degrees of latitude.

• The importance of rotation, the Coriolis parameter.

• What else is different?

)sin(2constant

earth) theof radius=(a )-(=with

a

)cos(2=

:ionapproximat plane- eMidlatitud

)sin(2

00

0

0000

0

f

ay

yfyfyy

fff

f

Let’s think about the Coriolis parameter

0constant

earth) theof radius=(a 0 )-(=with

a

)cos(2=

:ionapproximat plane- Tropical

)sin(2

0

00

0

0

f

ay

yyyy

ff

f

Coriolis parameter in the tropics

What latitude is coriolis parameter, say, 10% of mid-latitude value?

Recall our idealized Rossby wave equation and middle latitude scale analysis

Perturbation equation: Barotropic Rossby wave at middle latitudes

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

Where does the coriolis parameter appear in this equation?

Coriolis force

• The coriolis parameter decreases to zero at the equator.– Approximated by X distance from the

equator.

• We see from our wave equation and the conservation of vorticity that is a parameter of central importance to the dynamics– Advection of planetary vorticity.

Recalling our simple wave solution and comparing

advection of planetary and relative vorticity.

))2

()2

((

coscos)()(

coscos)(

22

22

yx

gg

g

gg

g

LLU

v

lykxpAlkUk

lykxpkAv

V

V

Coriolis force

• Can we say that the advection of planetary vorticity is less important?– We will come back to this.

Differences between the tropics and middle latitudes

• What are other differences?

Differences between the tropics and middle latitudes

• The contrast between summer and winter is not as large as at middle and high latitudes.– There is lot of solar heating.

• There is a lot of water!– What is the “physical” difference between water and

land?– Sea surface temperature is important to dynamics.

• What happens to water when it is warm?

Let’s think about waves some more

• We assume that dependent variables like u and v can be represented by an average and deviation from the average.

vuvuvuvuuv

vvuuuv

vvvuuu

))((

and

Let’s think about waves some more

• Some fundamental ideas

)( average

linearized 0.01,order

ignore then ,1.0~say "small," are onsperturbati

))((

and

vuavevuave(uv)

vuvuvuuvvu

vuu

u

vuvuvuvuuv

vvuuuv

vvvuuu

Let’s think about waves some more

• Some fundamental ideas.– Waves have some sort of restoring force

• Buoyancy waves: gravity• Rossby waves: The gradient of planetary vorticity

– Think about the conservation of potential vorticity

– Waves tend to grow and decay at the expense of the “energy” in the mean state.

– Waves tend to respond to out of balance situations.– Waves tend to move things towards equilibrium– Waves propagate

• So they can communicate things happening in one part of the fluid to far away places.

In middle latitudes:

• How do the Rossby waves that cause weather, the synoptic waves, get their energy?

Energetics of Midlatitude Cyclone Development

• The jet stream is commonly associated with strong temperature gradients in the middle/lower troposphere (thermal wind relationship)

• Midlatitude cyclones develop along waves in the jet stream

• Midlatitude cyclones are always associated with fronts (Norwegian cyclone model)

• There is a link between temperature gradients and cyclone development…

Idealized vertical cross section

Two important definitions

• barotropic – density depends only on pressure. – By the ideal gas equation, surfaces of constant

pressure are surfaces of constant density are surfaces of constant temperature (idealized assumption). = (p)

• baroclinic – density depends on pressure and temperature (as in

the real world). = (p,T)

Barotropic/baroclinic atmosphere

Barotropic: pp + pp + 2p

pp + pp + 2p

T+2TT+TT

T

T+2TT+T

Baroclinic:

ENERGY HERE THAT IS CONVERTED TO MOTION

Barotropic/baroclinic atmosphere

Barotropic: pp + pp + 2p

pp + pp + 2p

T+2TT+TT

T

T+2TT+T

Baroclinic:

DIABATIC HEATING KEEPS BUILDING THIS UP

Lifting and sinking

There are massive amounts of available energy in these sloping surfaces.And what do waves want to do?

Barotropic/baroclinic atmosphere

• Energetics:– Baroclinic = temperature contrast = density

contrast = available potential energy– Extratropical cyclones intensify through

conversion of available potential energy to kinetic energy

Energetics in the atmosphere

Ability to convert potential energy to kinetic energy directly related to tilt with height (offset) of low/high pressure

Tropics and middle latitudes

• In middle latitudes the waves grow from the energy available in the baroclinic atmosphere.– horizontal temperature gradients

• scale is large• latent heat release is on scales small compared to

baroclinic energy convergence.

• In the tropics the horizontal temperature gradients are small.

An estimate of the January mean temperature

northwinter

southsummer

tropopause

stratopause

mesosphere

stratosphere

troposphere

note where the

horizontal temperature gradients are

large

Tropics and middle latitudes

• Baroclinicity is less important in the tropics– Latent heat release is generally most

important.

– What does this mean?

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

What does importance of latent heat release mean.

• Diabatic processes are more important in the tropics.

• Hence, vertical velocity is more strongly related to diabatic heating than to temperature advection.– What about divergence?

• The scale of the forcing of motions is small– Related to the phase change of water.

Picture of Earth: What can you say about this figure?

Inter-tropical Convergence Zone(ITCZ)

ITCZ Circulation

A couple of things to note

• The winds at the surface in the tropics are, on average, easterly, from the east, towards the west.

• Go back to our mid-latitude wave:– What does this say about waves in the

tropics?– Well, it says, they are different!

• (Not that they don’t exist!)

ITCZ: Seasonal differences

ITCZ: Seasonal differences

What is happening here and here?

South American Seasonal Cycle

CONVECTION GOES WILD

Cloud Liquid Water: Average

NOTE: Remarkable areas with no clouds! No rain!

Vertical circulation around the ITCZ

What is the direction of the zonal wind?

ITCZ: Seasonal differences

What is happening here and here?

Monsoonal Flow

Lets return to our simple problem

Earth’s surface

warmcore

coldcore

PGF HL

LH PGF

DIVERGENCE

CONVERGENCE

CONVERGENCE

DIVERGENCE

LANDOCEANSUMMER TIME

Lets return to our simple problem

Earth’s surface

warming

cooling

PGF H

L

LH PGF

Earth’s surfaceLANDOCEAN

SUMMER TIME

Monsoonal Circulation

• Driven by land-sea temperature contrast.

• Reversal of flow from summer to winter.

• Tremendously important to precipitation.

• South and East Asian monsoon among most important of circulation features.

Circulation features of the tropics

• Inter-tropical convergence zone– Hadley circulation

• Monsoonal circulations

• Madden-Julian Oscillation

• African easterly waves

• Walker circulation

• El Nino and La Nina

Madden-Julian Oscillation

OLR = outgoing longwave radiation.Cold is the top of the clouds.Cold is the top of the hot towers.

African easterly waves

African easterly waves

What is happening here and here?

That’s north Africa. It’s summer.

African easterly waves

• The Sahara gets SO HOT, that the meridional temperature gradient is important.– But it is reversed over our normal thinking!

So let’s think about these scales of motion.

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

Introduce another vertical coordinate.

Dt

Dzw

H

z

p

pHz

s

s

**

)*

exp(

)ln(*

0

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

*

*)()

velocity horizontal

* ;

*

**

0*

*1)(

**

2

2

0

0*

zw

tDt

D( )

vu

Hc

RT

z

T

H

RN

H

RT

z

Hc

RJwN

zyv

xu

t

z

w

y

v

x

u

fDt

D

zz

p

p

z

V

jiV

VkV

Scale factors for “large-scale” tropics

LHUW

UL

H

L

U

/

s 10 /

m 10

m 10

s m 10

5

4

6

-1

y

f

sf

15-0 10

Scaling: Momentum Equation

/)/(

1Ro/1/)/(

1)/()/(*

*

1)/(

/

2

2

U

UfLf

UHWLz

w

t

LU

VV

VVVk

VVV

VVV

VV

Scaling: Momentum equation

• Geostrophic balance is not dominant.– How many km from the equator is geostrophic term no longer

small?– What about ?

• If the pressure gradient is balanced in the momentum equation, then ...

2U

This means something!

• For a similar scale disturbances in the tropics and middle latitudes the geopotential perturbation is a smaller by an order of magnitude in the tropics.

• What does this mean– for the scales of motion?– for the important physical terms?