AOSS 401, Fall 2007Lecture 6

September 19, 2007

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

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

dposselt@umich.edu734-936-0502

Class News

• Homework 1 graded

• Homework 2 due today

• Homework 3 posted on ctools between now and Friday

Weather

• NCAR Research Applications Program– http://www.rap.ucar.edu/weather/

• National Weather Service– http://www.nws.noaa.gov/dtx/

• Weather Underground– http://

www.wunderground.com/modelmaps/maps.asp?model=NAM&domain=US

Outline

1. Review from Monday• Continuity Equation• Scale Analysis

2. Conservation of Energy• Thermodynamic energy equation and the

first law of thermodynamics• Potential temperature and adiabatic motions• Adiabatic lapse rate and static stability

From last time

Conservation of Mass

• Conservation of mass leads to another equation; the continuity equation

• Continuity Continuous

• No holes in a fluid

• Another fundamental property of the atmosphere

• Need an equation that describes the time rate of change of mass (density)

Eulerian Form of the Continuity Equation

u

t

x

y

zIn the Eulerian point of view, our parcel is a fixed volume and the fluid flows through it.

Lagrangian Form of the Continuity Equation

The change in mass (density) following the motion is equal to the divergence

Convergence = increase in density (compression)

Divergence = decrease in density (expansion)

uDt

D1

Scale Analysis of theContinuity Equation

• Define a background pressure field

• “Average” pressure and density at each level in the atmosphere

• No variation in x, y, or time• Hydrostatic balance applies

to the background pressure and density

gzdz

zdp

z

zpp

)()(

)(

)(

00

00

00

Scale Analysis of theContinuity Equation

Total pressure and density = sum of background + perturbations (perturbations vary in x, y, z, t)

00 u

Start with the Eulerian form of the continuity equation,do the scale analysis, and arrive at

),,,(')(

),,,(')(

0

0

tzyxz

tzyxpzpp

Scale Analysis of theContinuity Equation

• Expand this equation

wdz

d

yx

u

dz

dw

z

w

yx

u

00

0

0

0

1v

0v

0

u

Remember, ρ0 does not depend on x or y

Scale Analysis of theContinuity Equation

• The vertical motion on large (synoptic) scales is closely related to the divergence of the horizontal wind

wdz

d

wdz

d

yx

u

00

00

1

1v

hu

Scale Analysis of the Horizontal Momentum Equations

Largest Terms

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

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

22

2

Ωuy

p

a

w

a

u

Dt

D

uΩwΩx

p

a

uw

a

u

Dt

Du

U·U/L U·U/aU·W/

aΔP/ρL Uf Wf νU/H2

10-4 10-5 10-8 10-3 10-3 10-6 10-12

Geostrophic Balance

• There is no D( )/Dt term (no acceleration)• No change in direction of the wind (no rotation)• No change in speed of the wind along the

direction of the flow (no divergence)

fuΩuy

p

fvΩx

p

)sin(21

)sin(v2 1

What are the scales of the terms? For “large-scale” mid-latitude

Analysis(Diagnosis)Geostrophic

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

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

22

2

Ωuy

p

a

w

a

u

Dt

D

uΩwΩx

p

a

uw

a

u

Dt

Du

U·U/L U·U/aU·W/

aΔP/ρL Uf Wf νU/H2

10-4 10-5 10-8 10-3 10-3 10-6 10-12

Prediction(Prognosis)

Ageostrophic

• Remember the definition of geostrophic wind

• Our prediction equation for large scale midlatitudes

agg

agg

fuuuffuy

p

Dt

D

fvvvffx

p

Dt

Du

1v

v1

y

p

fu

x

p

fv

g

g

1

1

Ageostrophic Wind and Vertical Motion

• Remember the scaled continuity equation

• Vertical motion related to divergence, but geostrophic wind is nondivergent.

• Divergence of ageostrophic wind leads to vertical motion on large scales.

wdz

d0

0

1

hu

Closing Our System of Equations

• We have formed equations to predict changes in motion (conservation of momentum) and density (conservation of mass)

• We need one more equation to describe either the time rate of change of pressure or temperature (they are linked through the ideal gas law)

• Conservation of energy is the basic principle

Conservation of Energy: The thermodynamic equation

• First law of thermodynamics:• Change in internal energy is equal to the

difference between the heat added to the system and the work done by the system.

• Internal energy is due to the kinetic energy of the molecules (temperature)

• Total thermodynamic energy is the internal energy plus the energy due to the parcel moving

Thermodynamic Equation For a Moving Parcel

• J represents sources or sinks of energy. – radiation– latent heat release (condensation/evaporation, etc)– thermal conductivity– frictional heating.

• cv = 717 J K-1 kg-1, cvT = a measure of internal energy– specific heat of dry air at constant volume– amount of energy needed to raise one kg air one degree Kelvin

if the volume stays constant.

cvDT

Dt p

DDt

J

Thermodynamic Equation

• Involves specific heat at constant volume• Remember the material derivative

form of the continuity equation• Following the motion, divergence leads

to a change in volume• Reformulate the energy equation in terms of specific

heat at constant pressure

JDt

Dp

Dt

DTcv

uDt

D1

Another form of the Thermodynamic Equation

• Short derivation• Take the material derivative of the equation of

state• Use the chain rule and the fact that R=cp-cv

• Substitute in from the thermodynamic energy equation in Holton

• Leads to a prognostic equation for the material change in temperature at constant pressure

c pDT

Dt Dp

DtJJ

Dt

Dp

Dt

DTcv

Dt

DTc

Dt

DTc

Dt

Dp

Dt

Dp

Dt

DTR

Dt

Dp

Dt

Dp

Dt

DTRp

Dt

D

RTp

vp

)(

(Chain Rule)

Substitute in from the thermodynamic energy equation (Holton, pp. 47-49)

(Use R=cp-cv)

(Ideal gas law)

JDt

Dp

Dt

DTc

JDt

Dp

Dt

DTc

Dt

Dp

Dt

Dp

p

p

J

Dt

Dp

Dt

DTcv

(Cancel terms)

(Material derivative)

Thermodynamic equation

• Prognostic equation that describes the change in temperature with time

• In combination with the ideal gas law (equation of state) the set of predictive equations is complete

JDt

Dp

Dt

DTcp

Atmospheric Predictive Equations

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

Motions in a Dry (Cloud-Free) Atmosphere

• For most large-scale motions, the amount of latent heating in clouds and precipitation is relatively small

• In absence of sources and sinks of energy in a parcel, entropy is conserved following the motion

• Why is this important?– Large scale vertical motion– Atmospheric stability (convection)

Motions in a Dry (Cloud-Free) Atmosphere

• Goal: find a variable that– Is conserved following the motion if there are no

sources and sinks of energy (J)– Describes the change in temperature as a parcel rises

or sinks in the atmosphere• Adiabatic process: “A reversible

thermodynamic process in which no heat is exchanged with the surroundings”

• Situations in which J=0 referred to as– Dry adiabatic– Isentropic

• Why is this useful?

Synoptic Motions

Forced Ascent/Descent

WarmingCooling

Derivation of Potential Temperature

(No sources or sinks of energy)

(Energy Equation divided by temperature)

p

d

c

R

p

d

dp

T

T

p

p

dp

dp

dp

dp

p

pTT

p

p

c

R

T

T

ppRTTc

pDRTDc

pDRTDcT

J

Dt

pDR

Dt

TDc

T

J

Dt

Dp

p

R

Dt

DT

T

c

00

00

00

lnln

)ln(ln)ln(ln

)(ln)(ln

)(ln)(ln

0)(ln)(ln

0 0

(Adiabatic process)

(Integrate between two levels)

(Use the properties of the natural logarithm)

(Take exponential ofboth sides)

Definition of the Potential Temperature

T p0

p

Rdc p

Note: p0 is defined to be a constant reference level p0 = 1000 hPa

Interpretation: the potential temperature is the temperature a parcel has when it is moved from a (higher or lower )pressure level down to the surface.

• The temperature at the top of the continental divide is -10 degrees celsius (about 263 K)

• The pressure is 600 hPa, R=287 J/kg/K, cp=1004 J/kg/K• Compute

1. potential temperature at the continental divide

2. The temperature the air would have if it sinks to the plains (pressure level of 850 hPa) with no change in potential temperature

p

d

c

R

p

pT

0

304 K

F 64C 17 K290 oo T

• For a dry adiabatic, hydrostatic atmosphere the potential temperature does not vary in the vertical direction:

• In a dry adiabatic, hydrostatic atmosphere the temperature T must decrease with height. How quickly does the temperature decrease?

Dry Adiabatic Lapse RateChange in Temperature with Height

z

0

p

d

p

p

d

p

d

p

d

p

dc

R

c

g

z

T

z

T

p

TR

c

g

z

T

z

T

p

g

z

p

pp

z

pzc

TR

z

T

z

T

ppzc

R

z

T

Tz

p

p

c

R

zT

zz

p

p

c

RT

p

pT

p

d

1ln

ln

lnln11

lnlnln

lnln lnln

0

0

00

(take the vertical derivative)

(logarithm of potential temperature)

(Definition of d lnx and derivative of a constant)

(Multiply through by T)

(Hydrostatic balance)

(Equation of State)

Dry adiabatic lapse rateThe adiabatic change in temperature with height is

For dry adiabatic, hydrostatic atmosphere

Tz

g

c pd

d: dry adiabatic lapse rate (approx. 9.8 K/km)

pc

g

z

T

z

T

Atmospheric Static Stabilityand Potential Temperature

• Static: considering an atmosphere at rest (no u, v, w)

• Consider what will happen if an air parcel is forced to rise (or sink)

• Stable: parcel returns to the initial position

• Neutral: parcel only rises/sinks if forcing continues, otherwise remains at current level

• Unstable: parcel accelerates away from its current position 0

0

0

z

z

z

Static Stability

• Displace an air parcel up or down• Assume the pressure adjusts instantaneously; the parcel

immediately assumes the pressure of the altitude to which it is displaced.

• Temperature changes according to the adiabatic lapse rate

envenvparcelparcel

envdenvparceldparcel

envparcel

d

TT

TRTR

pp

TRp

Static Stability

• Adiabatic: parcel potential temperature constant with height

• For instability, the temperature of the atmosphere has to decrease at greater than 9.8 K/km

• This is extremely rare…• Convection (deep and shallow) is

common• How to reconcile lack of instability

with presence of convection?

0

0

0

z

z

z

Static Stability and Moisture

• The atmosphere is not dry—motion is not dry adiabatic

• If air reaches saturation (and the conditions are right for cloud formation), vapor will condense to liquid or solid and release energy (J≠0)

• Average lapse rate in the troposphere: -6.5 oC/km• Moist (saturated) adiabatic lapse rate: -5 oC/km

Consider the Upper Atmosphere

Atmosphere in Balance

• Hydrostatic balance (no vertical acceleration)• Geostrophic balance (no rotation or divergence)• Adiabatic lapse rate (no clouds or precipitation)

• What we are really interested in is the difference from balance.

• This balance is like a strong spring, always pulling back.

• It is easy to know the approximate state. Difficult to know and predict the actual state.

Next time• Ricky will be lecturing Friday, Monday, and

Wednesday• We have essentially completed chapters

1-2 in Holton• We have derived a set of governing

equations for the atmosphere• Chapter 3 will introduce simple

applications of these equations• First exam covers chapters 1-3—three

weeks from today!