AOSS 401, Fall 2006Lecture 17

October 22, 2007

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

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

dposselt@umich.edu734-936-0502

Class News

• Final exam will be last day of class

Material from Chapter 4

• Vorticity, Vorticity, Vorticity

– Definition of barotropic and baroclinic– Review– Scaling

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

Vorticity, Vorticity, Vorticity

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.

Attributes of important dynamical features

• There is rotation of wind• Which is due to the rotation of the Earth

• Vertical wind requires divergence of the horizontal wind

• Which requires an ageostrophic part of the wind.

Want to formalize the representation of the role of rotation

and divergence

Suppose we have some flow

Imagine at the point flow decomposed into two “components”

A “component” that flows into or away from the point.

Imagine at the point flow decomposed into two “components”

A “component” that flows around the point.

Lets consider a spinning skaterMotion is in the (x,y) plane

Axis of rotation is in the vertical plane

Vorticity

vorticityrelative Earth theof surface the torelative

motion the todueon contributi a is There

Earth. theof

rotation the todueon contributi a is There

hence, system; coordinate inertialin velocity is

fluid in therotation of measure a isvorticity

,definitionby is, vorticity

a

a

U

U

Vertical Component of Vorticity

aUk

vorticityofcomponent

vertical with theconcernedonly generally

In what plane is the motion?In what direction is the vorticity?

Relative Vorticity

y

u

x

v

y

u

x

v

y

u

x

v

x

w

z

u

z

v

y

w

(u,v,w)

vorticityrelative

,,

velocity relative

uk

u

u

Planetary Vorticity

f

fearth

)sin(2vorticityplanetary

)sin(2

uk

Absolute (or total) Vorticity

fy

u

x

veartha

vorticityabsolute

vorticityrelativevorticityplanetary vorticityabsolute

ukukUk

Vorticity and Divergence

• Related to shear of the velocity field. ∂v/∂x-∂u/∂y

• Related to stretching of the velocity field. ∂u/∂x+∂v/∂y

Full equations of motionRemember these?

1 and

)1

(

)()cos(21v

)v()sin(21v)tan(v

)()cos(2)sin(v21)vtan(

222

22

2

RTp

JDt

Dp

Dt

DTc

Dt

D

wΩugz

p

a

u

Dt

Dw

Ωuy

p

a

w

a

u

Dt

D

uΩwΩx

p

a

uw

a

u

Dt

Du

v

u

How would you calculate of the time rate change of the vertical component of vorticity?

The scaled horizontal momentum equation in z coordinates

fuy

p

dt

dv

fvx

p

dt

du

1

1

no viscosity

Relative Vorticity

y

u

x

v

(u,v,w)

vorticityrelative

velocity relative u

Take derivatives

fuy

p

dt

dv

x

fvx

p

dt

du

y

1

1

Take derivatives

)(1

)(1

fuxy

p

xdt

dv

x

fvyx

p

ydt

du

y

Take derivatives

)(1

)(

)(1

)(

fuxy

p

xv

t

v

x

fvyx

p

yu

t

u

y

u

u

Take derivatives

)(1

)(

)(1

)(

fuxy

p

xv

xt

v

x

fvyx

p

yu

yt

u

y

u

u

Pay attention to details of calculus here

Subtract these equationsConservation of vorticity

x

p

yy

p

xy

fv

z

u

y

w

z

v

x

w

y

v

x

uf

t

11)(

))((u

Where do the definitions barotropic and baroclinic make a difference?

What are physical interpretations of these terms?

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

Scale of relative vorticity

1510

sL

U

y

u

x

v

Compare relative vorticity to planetary vorticity

NumberRossby Ro

10

10

10

0

1

0

140

15

Lf

U

f

sf

sL

U

In general planetary vorticity is larger than relative vorticity.

Time rate of change of relative vorticity

211

2102

2

10

10,,

sHL

WU

zw

sL

U

yv

xu

t

Rest of the terms

21122

210

211

290

1011

: termodynamicthat therm

10 :advectionvorticity planetary

10)( : termtilting

10))(( : termdivergence

sL

p

x

p

yy

p

x

sUy

fv

sHL

UW

z

u

y

w

z

v

x

w

sL

Uf

y

v

x

uf

Consider divergence term

290 10))(( : termdivergence

sL

Uf

y

v

x

uf

This term scales larger than all of the other terms. This suggests that the divergence of the horizontal wind must, in actuality, be small; hence, quasi-nondivergent. (∂u/∂x+∂v/∂y)~<10-6s-1

Consider divergence term

)())((y

v

x

uf

y

v

x

uf

We saw that the relative vorticity is less than the planetary vorticity. So

So for quasi-nondivergent flow

21010)(

sy

v

x

uf

Compare relative vorticity to planetary vorticity and

to divergence

100

10

10

0

0

yv

xuf

yv

xu

f

Again we see the importance of the rotation of the Earth.

Assume balance among terms of 10-10s-2

)(

0)(

y

v

x

uf

y

fv

yv

xu

t

y

fv

y

v

x

uf

yv

xu

t

A nuance on vorticity and the scaled equation: potential vorticity

A simple version of potential vorticity

))(()(

y

v

x

uf

Dt

fD

y

fv

Dt

Df

horizontal

returned relative vorticity to equation

A simple version of potential vorticity

)()(

)(

1

y

v

x

u

Dt

fD

fhorizontal

what’s this?

A simple version of potential vorticity

z

w

Dt

fD

fhorizontal

)(

)(

1

Assume constant density and temperature.

A simple version of potential vorticity

H

)()()(

)(

1

)(

)(

1

122

1

2

1

2

1

zwzwdz

Dt

fD

f

dzz

wdz

Dt

fD

f

z

z

horizontal

z

z

z

z

horizontal

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

A simple version of potential vorticity

0)(

)()( 12

H

f

Dt

D

zwzwDt

DH

horizontal

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

Why can we do this?

A simple version of potential vorticity

vorticitypotentialH

f

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

What happens when the vortex meets the mountain?

Surface with a hill.

A less simple version of potential vorticity

vorticitypotential))((

p

gf

This is the isentropic potential vorticity which is conserved for isentropic motion.

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

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.

How are divergence and vorticity related?

• We have gotten to a situation where we have linked the rotational and irrotational components of the wind. divergence and curl

vorticity and divergence

Scaled vorticity equation

))(()(

y

v

x

uf

Dt

fDhorizontal

Pure constant vorticity flow.

Pure divergent flow

Divergence influence on vorticity

Divergence influence on vorticity

Vorticity and divergence

• As a mid-latitude large scale flow diverges (converges) it causes a change in absolute vorticity, primarily, acting on planetary vorticity.

• The relative vorticity has to change in order to conserve the potential vorticity.

• This is the play between relative and planetary vorticity.

Possible development of a surface low.

Earth’s surface

DIVERGENCE

warming

LOW

Changes in depth of vortex

H H CONVERGENCE

And a little more realistically

Concentrates relative vorticity

Dilutes relative vorticity

What if the wind is geostrophic?

Scaled vorticity equation

))(()(

y

v

x

uf

Dt

fDhorizontal

What happens if the wind is geostrophic?

Or returning to potential vorticity

0)( H

f

Dt

D ghorizontal

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 consider the geostrophic potential vorticity

0)()(

such that , function, stream a define

22

ft

xv

yu

g

g

k

Do you recognize this type of equation?

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