AOSS 401, Fall 2007 Lecture 25 November 09 , 2007

AOSS 401, Fall 2007Lecture 25

November 09, 2007Richard B. Rood (Room 2525, SRB)

rbrood@umich.edu734-647-3530

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

734-936-0502

Class News November 09, 2007

• Computing assignment– Second component assigned next week– Due 5 December (or thereabouts)– Involves writing a very simple numerical model…

• Important Dates: – November 12: Homework 6 due—questions?– November 16: Next Exam (Review on 14th)– November 21: No Class– December 10: Final Exam

Today

• Work through Homework 5• Another perspective on planetary vorticity

advection• Finish discussion of QG omega equation• Midlatitude cyclone energetics

5.1) Homework ProblemIn pressure coordinates, the horizontal momentum equation is written as:

pfDtD uku

Derive the equation for the conservation of vorticity. You should pursue the derivation until you have terms analogous to the divergence terms, tilting terms, and the baroclinic or solenoidal term (the term that included the pressure gradient). If there are additional or missing terms, then explain their presence or absence.

Solution to (5.1)1. Recognize that we are in pressure coordinates2. Split the equation into u- and v- components

xfv

DtDu

yfu

DtDv

Solution to (5.1)3. Differentiate the v-equation with respect to x and the u-

equation with respect to y

xfv

DtDu

y

yfu

DtDv

x

Solution to (5.1)4. Subtract the u-equation from the v-equation and

expand the material derivatives

x

fvpu

yuv

xuu

tu

y

yfu

pv

yvv

xvu

tv

x

Solution to (5.1)5. Collect terms, use the definition of divergence and

vorticity, and end up with

0

pu

ypv

xff

DtD V

No solenoidal term, as the equation is defined to be on a constant pressure surface

5.2) Homework ProblemThis is a special homework problem. While we have not formally seen this equation, I think that we have the tools to do this problem.

Given the equation for the conservation of vorticity, ζ, where the prime represents a small quantity and the overbar represents a larger, mean quantity:

constant ,,, and 0 yu

xvvvuuuv

yv

xu

t

With the definition of the velocity field given below, show that the vorticity equation can be written as:

xv

yu

xxu

t

and where,02

What is the criterion for wave solutions to this equation?

Solution to (5.2)1. Plug in the definitions for

2. Then plug in

3. and scale out terms that are products of perturbations

yu

xvvvuuu

,,

xv

yu

and

Solution to (5.2)(Lecture 22, slide 39…)

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.

0))((

)Re(

0

22

)(0

2

klkuk

e

xxu

t

tlykxi

Solution to (5.2)

• Rearrange the dispersion relation to find

• Mean wind must be positive (from the west) for waves to form

)( 22 lkkk

ku

Back to the Quasi-Geostophic System

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

Dgv g

Dt f0

k v a y

k v g

v g

1f0

k

ua

x va

y p0

tv g

p

Jp

with Rc p

and stability parameter = RdTo

pd ln0

dp

momentum equation

continuity equation

thermodynamicequation

geostrophic wind

Application of QG:Prediction of Atmospheric Flow

• Want to know how distribution of geopotential will change in the atmosphere– changes in pressure gradient force (jet

stream, convergence/divergence, cyclogenesis)

• Derived geopotential height tendency equation

Geopotential Tendency Equation

))((

)1(

))((

20

2

00

202

pf

p

ff

f

tpf

p

g

g

V

V

f0 * Vorticity Advection

Thickness Advection

Advection of vorticity

ggggg vf VV )(

Advection of vorticity

Advection of relative vorticity

Advection of planetary vorticity

Advection of vorticity

Φ0 - ΔΦ

Φ0 + ΔΦ

Φ0

ΔΦ > 0

A

B

C

٠

٠

٠x, 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

x, east

y, north

Advection of planetary vorticityStart with straight-line flow

Advection of planetary vorticityIntroduce perturbations and remember conservation of potential vorticity (assume no change in depth h)

0

hf

DtDPV

DtD g

x, east

y, north

Advection of planetary vorticityNorth/south movement change in planetary vorticity Conservation of angular momentum change in ζ

ζ < 0; anticyclonic

f > f0 ζ > 0; cyclonic

f < f0

ζ > 0; cyclonic

f < f0

f = f0

0

hf

DtDPV

DtD g

x, east

y, north

Advection of planetary vorticityFlow associated with rotation advects adjacent parcels north/south

ζ < 0; anticyclonic

f > f0 ζ > 0; cyclonic

f < f0

ζ > 0; cyclonic

f < f0

0

hf

DtDPV

DtD g

x, east

y, north

x, east

y, north

Advection of planetary vorticityThe wave propagates to the west (actual propagation direction depends on wind speed and wavelength…)

ζ < 0; anticyclonic

f > f0 ζ > 0; cyclonic

f < f0

ζ > 0; cyclonic

f < f0

0

hf

DtDPV

DtD g

VERTICAL VELOCITY

• Important for– Clouds and precipitation– Cyclogenesis (spinup of vorticity through

stretching)• Computed from governing equations

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

Dgv g

Dt f0

k v a y

k v g

v g

1f0

k

ua

x va

y p0

tv g

p

Jp

with Rc p

and stability parameter = RdTo

pd ln0

dp

momentum equation

continuity equation

thermodynamicequation

geostrophic wind

Methods for Estimating Vertical Velocity

1. Kinematic method (continuity equation)2. Adiabatic method (thermodynamic eqn)3. Diabatic method (thermodynamic eqn)4. QG-omega equation (unified equation)

1. Kinematic Method:Link between and the ageostrophic wind

p

gh

ap

yxpppp

pvv

vp

vu)()()(

0 if

)(

2121

Continuity equation (pressure coordinates) and non-divergence of geostrophic wind lead to

which can berewritten as:

and solved for:

0v

pyxu aa

Thermodynamic equation:

Tt u Tx v Ty Sp

Jc p

Assume the diabatic heating term J is small (J=0), and there is no local time change in temperature

2. Adiabatic MethodLink between and temperature advection

p

hp S

TvyT

xTuS

v 1Horizontal temperature advection term

Stability parameter

warm air advection: < 0, w ≈ -/g > 0 (ascending air)cold air advection: > 0, w ≈ -/g < 0 (descending air)

2. Adiabatic MethodLink between and temperature advection

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.

θ - Δθ θ + Δθθ

WARMCOLD

adiabatic = no change in θ

Start from thermodynamic equation in p-coordinates:

pp c

JS 1

Diabatic term

Tt u Tx v Ty Sp

Jc p

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

RadiationCondensationEvaporationMeltingFreezing

3. Diabatic MethodLink between and heating/cooling

mean meridional circulation

1. Kinematic method: divergence vertical motion– 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).

– Therefore: small errors in evaluating the winds <u> and <v> lead to large errors in .

– The kinematic method is inaccurate.

Vertical Velocity: Problems

2. Adiabatic method: temperature advection vertical motion– Assumes steady state (no movement of weather

systems)– Assumes no diabatic heating (no clouds or

precipitation)– What about divergence/convergence?– The adiabatic method has severe limitations.

Vertical Velocity: Problems

3. Diabatic method: Heating/cooling vertical motion– Assumes definition of some average

atmosphere– Assumes vertical motion only due to diabatic

heating– What about divergence/convergence?– The diabatic method has severe limitations.

Vertical Velocity: Problems

4. QG-omega equation Combine all QG equations

• None of the obvious methods work well for for midlatitude waves in general

• Combine information from the full set of QG equations– Geopotential tendency equation

(comes from vorticity equation--combines equation of motion, continuity equation, and geostrophic relationship)

– Thermodynamic energy equation

1.) Apply the horizontal Laplacian operator ( 2 ) to the QG thermodynamic equation

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

3.) Combine 1) and 2)

Jpp

ffp

fp

f

g

g

22

2

0

02

2202

1

1

v

v

4. QG-omega equation Combine all QG equations

4. QG-omega equation Combine all QG equations

• Link between vertical derivative of vorticity advection (divergence/stretching) and vertical motion

Jpp

ffp

fp

f

g

g

22

2

0

02

2202

1

1

v

v

4. QG-omega equation Combine all QG equations

• Link between vertical derivative of vorticity advection (divergence/stretching) and vertical motion

• Link between temperature advection and vertical motion

Jpp

ffp

fp

f

g

g

22

2

0

02

2202

1

1

v

v

4. QG-omega equation Combine all QG equations

• Link between vertical derivative of vorticity advection (divergence/stretching) and vertical motion

• Link between temperature advection and vertical motion• Link between diabatic heating and vertical motion

Jpp

ffp

fp

f

g

g

22

2

0

02

2202

1

1

v

v

4. QG-omega equation Combine all QG equations

• This is still a complicated equation to analyze directly—simplify it

• Assume small diabatic heating (scale out last term)• Use the chain rule on the first and second terms on the

right hand side and combine the remaining terms

Jpp

ffp

fp

f

g

g

22

2

0

02

2202

1

1

v

v

ffp

fp

f g 2

0

02

2202 12 v

Advection of absolute vorticityby the thermal wind

4. QG-omega equation (simplified)

Simple, right?

“Advection” by thermal wind?

• How to analyze this on a map?

pp

gT k

vv

ThermalW

ind

is

Perpendicular to

Thickness

Look at contours of constant thickness

Gradient of

What about that Laplacian?• QG omega equation relates vorticity advection by the

thermal wind with the laplacian of omega• Assume omega has a wave-like form

• This leads to

• which means

lykxppW sinsinsin

00

2

0

0222

2202 1

pflk

pf

2

2202

pf

2

2202

pf

Vertical Motion on Weather Maps

• Laplacian of omega is proportional to -ω• Omega can be analyzed as:

• Remember, from definition of omega and scale analysis

• Positive vorticity advection by the thermal wind indicates rising motion

f

fpg 2

0

1v

wg

Vertical Motion on Weather Maps

• Positive vorticity advection by the thermal wind indicates rising motion

pp

pp

p +

Lines of constant thickness

Descent

Ascent

Vertical Motion on Weather MapsSurface 500 mb

700 mb

Vertical Motion on Weather MapsSurface 500 mb

700 mb

Vertical Motion on Weather MapsSurface 500 mb

700 mb

Vertical Motion on Weather MapsSurface 500 mb

700 mb

Vertical Motion on Weather MapsSurface 500 mb

700 mb

Energetics of Cyclone Development

http://www.aos.wisc.edu/weather/wx_models/gblav_104_12UTC.shtmlhttp://tempest.aos.wisc.edu/wxp_images/gfs104_06UTC/gblav_c300_h120.gifhttp://tempest.aos.wisc.edu/wxp_images/gfs104_06UTC/gblav_c850_h120.gif

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: p

p + pp + 2p

pp + pp + 2p

T+2TT+TT

T

T+2TT+T

Baroclinic:

ENERGY HERE THAT IS CONVERTED TO MOTION

Barotropic/baroclinic atmosphere Barotropic: p

p + pp + 2p

pp + pp + 2p

T+2TT+TT

T

T+2TT+T

Baroclinic:

DIABATIC HEATING KEEPS BUILDING THIS UP

Barotropic/baroclinic atmosphere

• Energetics:– Baroclinic = temperature contrast = density

contrast = available potential energy– Extratropical cyclones intensify through

conversion of available potential energy to kinetic energy

Available Potential Energy

• Defined as the difference in potential energy after an adiabatic redistribution of mass

COLD WARM

Available Potential Energy

• Defined as the difference in potential energy after an adiabatic redistribution of mass

COLD

WARM

Energetics in the atmosphere

• Diabatic heating (primarily radiation) maintains the equator to pole temperature contrast

• Strength of temperature contrast referred to as “baroclinicity”

• Cyclones at midlatitudes reduce this temperature contrast—adjust baroclinic atmosphere toward barotropic

Energetics in the atmosphereAbility to convert potential energy to kinetic energy directly related to tilt with height (offset) of low/high pressure

Back to this weekend’s East Coast cyclone

(animations of 500 mb and surface)

Back to this weekend’s East Coast cyclone

• Occlusion describes the transition of a cyclone from baroclinic (west-ward tilt with height) to barotropic (“vertically stacked”)

• Once there is no more westward vertical tilt with height, no further development can occur

• We will look at this schematically next time

Next time

• Brief wrap-up of midlatidude cyclone dynamics

• Excursion into the boundary layer• Introduction to the next computing

assignment