AOSS 401, Fall 2006Lecture 18

October 24, 2007Richard B. Rood (Room 2525, SRB)

rbrood@umich.edu734-647-3530

Derek Posselt (Room 2517D, SRB)dposselt@umich.edu

734-936-0502

Class News

• Final exam will be last day of class

• Derek and I decided to think about good homework problems for another day.– No homework posted today.

Material from Chapter 4

• Vorticity, Vorticity, Vorticity

– Relative and planetary vorticity– Mid-latitude disturbances– Vorticity, divergence, in 3-D

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

Two important definitions

• barotropic – density depends only on pressure. And by the ideal gas equation, surfaces of constant pressure, are surfaces of constant density, are surfaces of constant temperature.

• baroclinic – density depends on pressure and temperature.

Absolute (or total) Vorticity

fyu

xveartha

vorticityabsolute

vorticityrelativevorticityplanetary vorticityabsolute

ukukUk

Relative and planetary vorticity

• Planetary vorticity is cyclonic is positive vorticity

• Planetary vorticity, in middle latitudes, is usually larger than relative vorticity

We derived the vorticity equation

xp

yyp

x

zu

yw

zv

xw

yv

xuf

yfv

t

11

)(

))((u

TERMS

DIVERGENCE

TILTING

SOLENOIDAL or

BAROCLINIC

Comments on the terms

• There are important dynamical features in the atmosphere where all of these terms are important.

• Baroclinic terms are due to there being gradients of temperature on pressure surfaces. (Are they explicitly there in pressure coordinates?) Like a thermodynamic “source” of rotation.

Tilting Term

rotation in, say, (y, z)

plane, “vorticity” in x

plane

as the wheel is turned there is a

component of “vorticity” in the z

plane

Divergence influence on vorticity

Divergence influence on vorticity

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

yfsf

P

Assume balance among terms of 10-10s-2

)(

0)(

yv

xuf

yfv

yv

xu

t

yfv

yv

xuf

yv

xu

t

A nuance on vorticity and the scaled equation: potential vorticity

A simple version of potential vorticity

0)( Hf

DtDhorizontal

Integrate with height,z1 z2 over a layer of depth H.

A simple version of potential vorticity

vorticitypotentialHf

This is the potential vorticity under the set of assumptions that we used to derive the equation. Constant density, constant temperature so only in a shallow layer might this be relevant to the atmosphere.

Potential vorticity is a measure of absolute vorticity relative to the depth of the vortex.

Relative vorticity with change of depth

Vorticity and depth

• We can see that there is a relationship between depth and vorticity.

• As the depth of the vortex changes, the relative vorticity has to change in order to conserve the potential vorticity.

• This is the play between relative and planetary vorticity.

Scaled vorticity equation

))(()(yv

xuf

DtfDhorizontal

An observation

• The vorticity is dominated by the geostrophic component of the wind.

• The divergence requires the wind to be away from geostrophic balance.

• Generally vg/va >= 10

Let’s explicitly map these ideas to the Earth

Local vertical / planetary vorticity

relative vorticity/planetary vorticity

relative vorticity

planetary vorticity

Compare relative vorticity to planetary vorticity

NumberRossby Ro

10

10

10

0

1

0

140

15

LfUf

sf

sLU

planetary vorticity

is usually larger than

relative vorticity

for large-scale and

middle latitudes

Relative and planetary vorticity

• Planetary vorticity is cyclonic is positive vorticity• Planetary vorticity, in middle latitudes, is usually

larger than relative vorticity• A growing cyclone “adds to” the planetary

vorticity.– Lows intense

• A growing anticyclone “opposes” the planetary vorticity.– Highs less intense

Compare relative vorticity to planetary vorticity and

to divergence

100

10

10

0

0

yv

xufyv

xu

f

Flow is rotationally dominated, but divergence is crucial to understanding

flow.

Consider our simple form of potential vorticity

vorticitypotential

0)(

Hf

Hf

DtDhorizontal

From scaled equation, with assumption of constant density and temperature.

Fluid of changing depth

Two things that we have learned about vorticity.

• Convergence and divergence in a column of fluid, impacts the vorticity throughout the column.– Specifically, divergence above causes low

pressure at the surface.• Stretching and shrinking of a column of

vorticity will change the relative vorticity.

Possible development of a surface low.

Earth’s surface

pressure surfaces

Lets return to our simple problem

Earth’s surface

pressure surfaces

warming

cooling

Lets return to our simple problem

Earth’s surface

pressure / height

surfacesrisewarming

cooling

pressure / height

surfacessink

Lets return to our simple problem

Earth’s surface

warming

cooling

PGF H

L

Lets return to our simple problem

Earth’s surface

warming

cooling

PGF H

L mass leaves

column / low forms at groundL

Lets return to our simple problem

Earth’s surface

warming

cooling

PGF H

L mass leaves

column / low forms at groundL

mass enters column / high

forms at ground

H

Lets return to our simple problem

Earth’s surface

warming

cooling

PGF H

L mass leaves

column / low forms at groundL

mass enters column / high

forms at ground

H PGF

Mass continuity?

• What are the implications of mass continuity?

• What is your law, your equation, your tool to answer that question?

Temperature

• Assuming the air moves isentropically, what happens to the temperature?

• What is your law, your equation, your tool to answer that question?

Lets return to our simple problem

Earth’s surface

warming

cooling

PGF H

L

LH PGF

Simple Thermal Circulation

• There is the sense of the air moves to counter the heating.

• If the heating ended, then the circulation would end, acting to bring back the original equilibrium situation.

• This sort of low is cause by heating, is called a “thermal” low, warm core. It tends to damp out.

Lets return to our simple problem

Earth’s surface

warmcore

coldcore

PGFHL

LH PGF

Simple Thermal Circulation

• This sort of low is cause by heating, is called a “thermal” low, warm core. It tends to damp out.– Remember the question about the hurricane

being warm core.• What about the divergence and

convergence?

Lets return to our simple problem

Earth’s surface

warmcore

coldcore

PGF HL

LH PGF

DIVERGENCE

CONVERGENCE

CONVERGENCE

DIVERGENCE

Simple Thermal Circulation

• What about the divergence and convergence?– Convergence and Divergence are aligning

over top of each other in the vertical.– Again, in this case there is a tendency for the

circulation to damp out.

Back to the earth again

Still in the atmosphere

Flow over a hill

HILL

Derived a simple form of potential vorticity

vorticitypotential

0)(

Hf

Hf

DtDhorizontal

From scaled equation, with assumption of constant density and temperature.

Flow over a hill(long in the north-south)(can’t go around the hill)

west east

Flow over a hill

HILL

west east

Dep

th, H

Flow over a hill(assume flow is adiabatic)

HILL

west east

Dep

th, H

θ

θ + Δθ

Flow over a hill(far upstream constant zonal flow)

HILL

west east

Dep

th, H

θ

θ + Δθ

ζ=0

Derived a simple form of potential vorticity

vorticitypotential

0)(

Hf

Hf

DtDhorizontal

From scaled equation, with assumption of constant density and temperature.

What happens as air gets to hill?

HILL

west east

Dep

th, H

θ

θ + Δθ

ζ=0

What happens as air gets to hill?

HILL

west east

Dep

th, H

θ

θ + Δθ

ζ=0

Air is lifted. Lifting higher at ground than upper air.(pressure gradient force spreads it out)

What happens as air gets to hill?

HILL

west east

Dep

th, H

+ΔH

θ

θ + Δθ

ζ=0

Air is lifted. Lifting higher at ground than upper air.(pressure gradient force spreads it out)

What happens as air gets to hill?

HILL

west east

Dep

th, H

+ΔH

θ

θ + Δθ

ζ must increase

Air is lifted. Lifting higher at ground than upper air.(pressure gradient force spreads it out)

How does vorticity increase?

What happens in these waves?

Gains cyclonic vorticity

Loses cyclonic vorticitySame as gains

anticyclonic vorticity

Or schematically

Cyclonic Anticyclonic

Rotational

Shear

What happens as air gets to hill?

HILL

west east

Dep

th, H

+ΔH

θ

θ + Δθ

ζ must increase

Air turns cyclonically to increase vorticity.In northern hemisphere turns north.

In the (east-west, north-south) planeD

epth

, H

Dep

th, H

+ΔH

west easts

n

What happens as air goes over hill?

HILL

west east

Dep

th, H

-ΔH

θ

θ + Δθ

Air turns anti-cyclonically to decrease vorticity.In northern hemisphere turns south.

ζ must decrease

In the (east-west, north-south) planeD

epth

, H

Dep

th, H

+ΔH

west easts

n Dep

th, H

-ΔH

What happens as air goes down hill?

HILL

west eastD

epth

, H +ΔH

θ

θ + Δθ

Air turns cyclonically to increase vorticity.In northern hemisphere turns north.

ζ must increase

In the (east-west, north-south) planeD

epth

, H

Dep

th, H

+ΔH

west easts

n Dep

th, H

-ΔH

Dep

th, H

+ΔH

What is happening with planetary vorticity?(In the (east-west, north-south) plane)

Dep

th, H

Dep

th, H

+ΔH

west easts

n Dep

th, H

-ΔH

Dep

th, H

+ΔH

f is greater for deflections to north

f is less for deflections to south

What is happening with planetary vorticity?(In the (east-west, north-south) plane)

Dep

th, H

Dep

th, H

+ΔH

west easts

n Dep

th, H

-ΔH

Dep

th, H

+ΔH

f + ζ is less than earth’s vorticity and wants to turn north.

Arrives here wanting vorticity. “Overshoots”

What is happening with planetary vorticity?(In the (east-west, north-south) plane)

Dep

th, H

Dep

th, H

+ΔH

west easts

n Dep

th, H

-ΔH

Dep

th, H

+ΔH

What happens if wind is from east?

HILL

west east

θ

θ + Δθ

What is happening with planetary vorticity?(In the (east-west, north-south) plane)

Dep

th, H

Dep

th, H

+ΔH

west easts

n Dep

th, H

-ΔH

Dep

th, H

+ΔH

Flow from east planetary and relative vorticity interact together, no overshoot or undershoot.

Excursion into the atmosphere

Middle latitude cyclones

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