AOSS 401, Fall 2007Lecture 24

November 07, 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 07, 2007

• Homework 6 (Posted this evening)– Due Next Monday

• Important Dates: – November 16: Next Exam (Review on 14th)– November 21: No Class– December 10: Final Exam

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

Rest of Course

• Wrap up quasi-geostrophic theory (Chapter 6)– Potential vorticity– Vertical velocity– Will NOT do Q vectors

• We will have a lecture on the Eckman layer (Chapter 5)– Boundary layer, mix friction with rotation

• We will have a lecture on Kelvin waves (Chapter 11)– A long wave in the tropics

• There will be a joint lecture with 451 on hurricanes (Chapter 11)

• Computer homework (perhaps lecture) on modeling• Special topics?

Material from Chapter 6

• Quasi-geostrophic theory

• Quasi-geostrophic vorticity– Relation between vorticity and geopotential

• Geopotential prognostic equation

• Quasi-geostrophic potential vorticity

Scaled equations in pressure coordinates (The quasi-geostrophic (QG) equations)

Dg

v g

Dt f0

k v a y

k v g

v g

1

f0

k

ua

x va

y p0

tv g

p

J

p

with R

c p

and stability parameter =RdTo

p

d ln0 dp

momentum equation

continuity equation

thermodynamicequation

geostrophic wind

v v g

v a with

v a

v g O(Ro) 0.1

Dv

Dt

Dg

v g

Dt with

Dg

Dtt ug

x vg

y

v g

1

f0

k

Midlatitude - plane approximation :

f f0 fy

0

y f0 y = f0 2cos(0)

ay

with y = a( - 0) (a = radius of the earth)

constant f0 2sin(0)

Ttot (x,y, p, t)T0(p)T(x,y, p, t) with T0

pTp

Approximations in the quasi-geostrophic (QG) theory

Quasi-geostrophic equations cast in terms of geopotential

and omega.

)1

(1

)()()(

2

00

2

0

0000

ffp

ftf

p

J

pf

pf

p

f

ptp

f

p

g

g

V

V

THERMODYNAMIC EQUATION

VORTICITY EQUATION

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

GEOPOTENTIAL TENDENCY EQUATION

Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency

(assume J=0)

))(()1

())((2

02

00

202

p

f

pf

ff

tp

f

p gg

VV

f0 * Vorticity Advection

Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency

(assume J=0)

))(()1

())((2

02

00

202

p

f

pf

ff

tp

f

p gg

VV

Thickness Advection

Advection of 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

Advection of ζ tries to propagate the wave this way

Advection of f tries to propagate the wave this way

Relationship between upper troposphere and surface

vorticity advection

thickness advection

To think about this

• Read and re-read pages 174-176 in the text.

Idealized vertical cross section

Great web page with current maps:

Real baroclinic disturbances

http://www.meteoblue.ch/More-Maps.79+M5fcef4ad590.0.html

Personalize your maps (create a login):http://my.meteoblue.com

Real baroclinic disturbances:850 hPa temperature and geopot. thickness

warm airadvectioneast of thesurface low,enhances the ridge

cold airadvection,enhances trough

Real baroclinic disturbances:500 hPa rel. vorticity and mean SLP

sea level pressure

Positivevorticity, pos. vorticityadvection,increase incyclonicvorticity

Real baroclinic disturbances:500 hPa geopot. height and mean SLP

Upper level systemslags behind (to the west):system stilldevelops

With the benefit of hindsight and foresight let’s look back.

g

t f0

pv g ( g f )

g

t f0

pv g g vg

QG vorticity equation

Advection of relative vorticity

Advection of planetary vorticity

Stretchingterm

Competing

THINKING ABOUT THESE TERMS

g

t f0

pv g ( g f )

g

t f0

pv g g vg

QG vorticity equation

Advection of relative vorticity

Advection of planetary vorticity

Stretchingterm

Competing

WHAT ABOUT THIS

TERM?

Consider our simple form of potential vorticity

vorticitypotential

0)(

H

fH

f

Dt

Dhorizontal

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

There was the assumption that the layer of fluid was shallow.

Fluid of changing depth

What if we have something like this, but the fluid is an ideal gas?

Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency

(assume J=0)

))(()1

())((2

02

00

202

p

f

pf

ff

tp

f

p gg

VV

Still looks a lot like time rate of change of vorticity

Quasi-Geostrophicpotential vorticity (PV) equation

Simplify the last term of the geopotential tendencyequation by applying the chain rule:

v g

p

f02

p

f02

v gp

p

= 0 Why?

stability parameter =RdTo

p

d ln0 dp

Quasi-Geostrophicpotential vorticity (PV) equation

Simplify the last term of the geopotential tendencyequation by applying the chain rule:

v g

p

f02

p

f02

v gp

p

= 0 Why?

stability parameter =RdTo

p

d ln0 dp

THERMAL WIND RELATION

ppf g kv

0

Quasi-Geostrophicpotential vorticity (PV) equation

qtv g q

Dg

Dtq0

Simplify the last term of the geopotential tendencyequation by applying the chain rule:

v g

p

f02

p

f02

v gp

p

q1

f0

2 f p

f0

p

= 0

Leads to the conservation law:

Quasi-geostrophic potential vorticity:

Conserved following the geostrophic motion

Why?

stability parameter =RdTo

p

d ln0 dp

Imagine at the point flow decomposed into two “components”

A “component” that flows around the point.

Vorticity

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

Imagine at the point flow decomposed into two “components”

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

Divergence

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

Potential vorticity (PV): Comparison

q1

f0

2 f p

f0

p

Quasi-geostrophic PV:

THESE ARE LIKE STRETCHING IN THE VERTICAL

Barotropic PV:

PV g f

h

s-1

PV ( f )( gp

)Ertel’s PV:

m-1s-1

Units:

K kg-1 m2 s-1

g

t f0

pv g ( g f )

g

t f0

pv g g vg

QG vorticity equation

Advection of relative vorticity

Advection of planetary vorticity

Stretchingterm

Competing

WHAT ABOUT THIS

TERM?

Fluid of changing depthWhat if we have something like this, but the fluid is

an ideal gas?

Conversion of thermodynamic energy to vorticity, kinetic energy. Again the link between the thermal

field and the motion field.

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 (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 IN 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

• NOW WOULD BE A GOOD TIME FOR A SILLY STORY

VERTICAL VELOCITY

Dp

Dtptv h p w

pz

ptv g p

v a p wg

ptv a p wg

with the help of scale analysis (free troposphere)

wg

Vertical motions: The relationship between w and

= 0 hydrostatic equation

≈ 10 hPa/d≈ 1m/s 1Pa/km≈ 1 hPa/d

≈ 100 hPa/d

p v h

p0

p (v g

v a )

p0

ug

xvg

y

p

p (v a )

p0

x

(1

f

y

)y

(1

f

x

)

p

p (v a )

p0

assume f is approximately constant

p p (

v a )

if v h

v g

p0

Links the horizontal and vertical motions. Since geostrophy is such a good balance, the vertical motion is linked to the divergence of the ageostrophic wind (small).

= 0

Link between and the ageostrophic wind

Vertical pressure velocity

( p1)(p2) (p1 p2) u

x v

y

p

For synoptic-scale (large-scale) motions in midlatitudesthe horizontal velocity is nearly in geostrophic balance.Recall: the geostrophic wind is nondivergent (for constant Coriolis parameter), that is

Horizontal divergence is mainly due to small departures from geostrophic balance (ageostrophic wind).Therefore: small errors in evaluating the winds <u> and <v>

lead to large errors in . The kinematic method is inaccurate.

v g

ug

xvg

y0

Think about this ...

• If I have errors in data, noise.

• What happens if you average that data?

• What happens if you take an integral over the data?

• What happens if you take derivatives of the data?

Estimating the vertical velocity: Adiabatic Method

Start from thermodynamic equation in p-coordinates:

Sp 1 Tt uTx vTy

- (Horizontal temperature advection term)

Sp:Stability parameter

Tt uTx vTy Sp

J

c p

Assume that the diabatic heating term J is small (J=0), re-arrange the equation

Estimating the vertical velocity: Adiabatic Method

If T/t = 0 (steady state), J=0 (adiabatic) and Sp > 0 (stable):• then warm air advection: < 0, w ≈ -/g > 0 (ascending air)• then cold air advection: > 0, w ≈ -/g < 0 (descending air)

If local time tendency Tt0 (steady state)

Sp 1 u

Tx vTy

v h T

Sp

Horizontal temperature advection term

Stability parameter

Adiabatic Method

• Based on temperature advection, which is dominated by the geostrophic wind, which is large. Hence this is a reasonable way to estimate local vertical velocity when advection is strong.

Estimating the vertical velocity: Diabatic Method

Start from thermodynamic equation in p-coordinates:

pp c

JS 1

Diabatic term

Tt uTx vTy Sp

J

c p

If you take an average over space and time, then the advection and time derivatives tend to cancel out.

mean meridional circulation

Conceptual/Heuristic Model

Plumb, R. A. J. Meteor. Soc. Japan, 80, 2002

•Observed characteristic behavior•Theoretical constructs•“Conservation”

•Spatial Average or Scaling•Temporal Average or Scaling

YieldsRelationship between parameters if observations and theory are correct

One more way for vertical velocity

Quasi-geostrophic equations cast in terms of geopotential

and omega.

)1

(1

)()()(

2

00

2

0

0000

ffp

ftf

p

J

pf

pf

p

f

ptp

f

p

g

g

V

V

THERMODYNAMIC EQUATION

VORTICITY EQUATION

ELIMINATE THE GEOPOTENTIAL AND GET AN EQUATION FOR OMEGA

Quasi-Geostrophic Omega Equation

1.) Apply the horizontal Laplacian operator to the QG thermodynamic equation

2.) Differentiate the geopotential height tendency equation with respect to p

3.) Combine 1) and 2) and employ the chain rule of differentiation (chapter 6.4.1 in Holton, note factor ‘2’ is missing in Holton Eq. (6.36), typo)

2 f0

2

2

2 p

2 f0

v gp

1

f0

2 f

Advection of absolute vorticityby the thermal wind

Vertical Velocity Summary

• Though small, vertical velocity is in some ways the key to weather and climate. It’s important to waves growing and decaying. It is how far away from “balance” the atmosphere is.

• It is astoundingly difficult to calculate. If you use all of these methods, they should be equal. But using observations, they are NOT!

• In fact, if you are not careful, you will not even to get them to balance in models, because of errors in the numerical approximation.

One more summary of the mid-latitude wave

Idealized (QG) evolution of a baroclinic disturbance(Read and re-read pages 174-176 in the text.)

L

H

+ warm airadvection

- cold airadvection

- neg. vorticityadvection

+ pos. vorticityadvection

500 hPageopotential

p at the surface

p at the surface

Waves• The equations of motion contain many forms

of wave-like solutions, true for the atmosphere and ocean

• Some are of interest depending on the problem: Rossby waves, internal gravity (buoyancy) waves, inertial waves, inertial-gravity waves, topographic waves, shallow water gravity waves

• Some are not of interest to meteorologists, e.g. sound waves

• Waves transport energy, mix the air (especially when breaking)

Waves

• Large-scale mid-latitude waves, are critical for weather forecasting and transport.

• Large-scale waves in the tropics (Kelvin waves, mixed Rossby-gravity waves) are also important, but of very different character.

• This is true for both ocean and atmosphere. • Waves can be unstable. That is they start to

grow, rather than just bounce back and forth.

• And, with that, Chapter 6, of Jim Holton’s book rested comfortably in the mind of the students.

Below

• Basic Background Material

Couple of Links you should know about

• http://www.lib.umich.edu/ejournals/– Library electronic journals

• http://portal.isiknowledge.com/portal.cgi?Init=Yes&SID=4Ajed7dbJbeGB3KcpBh– Web o’ Science

A nice schematic

• http://atschool.eduweb.co.uk/kingworc/departments/geography/nottingham/atmosphere/pages/depressionsalevel.html

Mid-latitude cyclones: Norwegian Cyclone Model

• http://www.srh.weather.gov/jetstream/synoptic/cyclone.htm

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

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