Module 3 Climate Modeling Theory - 1

Module 3 Module 3 Climate Modeling Theory - 1Climate Modeling Theory - 1

William J. Gutowski, Jr.William J. Gutowski, Jr.Iowa State UniversityIowa State University

Module 3 Module 3 Climate Modeling Theory - 1Climate Modeling Theory - 1

GOAL:GOAL:Understand basis for modeling Understand basis for modeling climate from (almost) first climate from (almost) first principlesprinciples

Module 3 Module 3 Climate Modeling Theory - 1Climate Modeling Theory - 1

OUTLINE (Part 1):OUTLINE (Part 1): SymbolismSymbolism Conservation LawsConservation Laws

– massmass– thermodynamic energythermodynamic energy– momentummomentum

Equation of StateEquation of State Water in the AtmosphereWater in the Atmosphere

Module 3 Module 3 Climate Modeling Theory - 1Climate Modeling Theory - 1

OUTLINE (Part 1):OUTLINE (Part 1): SymbolismSymbolism

t timex west-east coordinatey south-north coordinatez vertical coordinate

latitude longitude

horizontal windu west-east component of v south-north component of w vertical wind

Some Symbolism

r V

r V

r V

Module 3 Module 3 Climate Modeling Theory - 1Climate Modeling Theory - 1

OUTLINE (Part 1):OUTLINE (Part 1): SymbolismSymbolism Conservation LawsConservation Laws

– massmass– thermodynamic energythermodynamic energy– momentummomentum

Conservation of “M”

dMdt

=?

Conservation of “M”

dMdt

≠0

Source/sink≠0

Conservation of “M”

∂ wM( )∂z

≠0

dMdt

≠0

Conservation of “M”

dMdt

≠0

∇ ⋅

r V M( )≠0

Conservation of “M”

∂ wM( )∂z

≠0

∇ ⋅

r V M( )≠0

Source/sink≠0

dMdt

≠0

Conservation of MassConservation of Mass(Continuity Equation)(Continuity Equation)

∂ρ∂t

=−∇⋅r V ρ( )−

∂ wρ( )∂z

= density [kg/m3]ρ

Source/sink = 0

Conservation of Mass

∂ps

∂t>0

Conservation of Water MassConservation of Water Mass

∂ρq∂t

=−∇⋅r V ρq( )−

∂ wρq( )∂z

+s(q)

q = specific humidity [kg-(H2O)v/kg-air]

s(q) = cond. - evap.

Conservation of Water MassConservation of Water Mass(column integral)(column integral)

∂W∂t

=−∇⋅r Q +E −P

E = sfc. evap.; P = precipitation

W = ρqdz0

∞

∫

r Q =

r V ρqdz

0

∞

∫

Conservation of W

(Precipitable Water)

r Q

r Q

E P

∂W∂t

≠0

Conservation of General Constituent, iConservation of General Constituent, i

∂ρqi∂t

=−∇⋅r V ρqi( )−

∂ wρqi( )∂z

+s(qi)

qi = amount of i [kg-(constituent i)/kg-air]

e.g., CO2, O3, etc.

Conservation of Thermodynamic EnergyConservation of Thermodynamic Energy~ First Law of Thermodynamics ~~ First Law of Thermodynamics ~

Heat input = internal energy) + (work done)

= heating/mass [J-kg-1-s-1]

dt

dp

dt

dTCH p ρ

1+=&

H&

Conservation of Thermodynamic EnergyConservation of Thermodynamic Energy~ First Law of Thermodynamics ~~ First Law of Thermodynamics ~

( ) HT

Cz

wV

t p

&r

θρρθ

ρθρθ

+ƒ

ƒ−?−=

ƒƒ

θ =T poo p( )R /Cp

Conservation of

Thermodynamic Energy

r V ρθ ∂ρθ

∂t≠0

r V ρθ

0?H&

wρθ

wρθ

0?H&

FSH RNET

RNET

Condensation

Conservation of MomentumConservation of Momentum~ Newton’s Second Law ~~ Newton’s Second Law ~

da

r V a,3dt

= (Forces/ mass)∑

Conservation of MomentumConservation of Momentum~ Newton’s Second Law ~~ Newton’s Second Law ~

Forces/mass:Forces/mass: gravitygravity pressure gradientpressure gradient frictionfriction

Conservation of MomentumConservation of Momentum~ Newton’s Second Law ~~ Newton’s Second Law ~

Rotating Frame

r Ω

r R

X

dr V 3dt

= (Forces/ mass)∑

−2r Ω ×

r V 3 +

r Ω

2 r R

Conservation of MomentumConservation of Momentum~ Newton’s Second Law ~~ Newton’s Second Law ~

Rotating Frame

dudt

−uvtanφ

a+

uwa

=−1ρ

∂p∂x

+2Ωvsinφ−2Ωwcosφ+Frx

Conservation of MomentumConservation of Momentum~ Newton’s Second Law ~~ Newton’s Second Law ~

Sphere, Rotating Frame

dvdt

−u2 tanφ

a+

vwa

=−1ρ

∂p∂y

−2Ωusinφ +Fry

dwdt

−u2 +v2

a=−

1ρ

∂p∂z

+2Ωucosφ −g+Frz

rotation of direction

Conservation of MomentumConservation of Momentum~ Newton’s Second Law ~~ Newton’s Second Law ~

Approximation: vertical

dwdt

−u2 +v2

a=−

1ρ

∂p∂z

+2Ωucosφ −g+Frz

Conservation of MomentumConservation of Momentum~ Newton’s Second Law ~~ Newton’s Second Law ~

Approximation: vertical

dwdt

−u2 +v2

a=−

1ρ

∂p∂z

+2Ωucosφ −g+Frz

Conservation of MomentumConservation of Momentum~ Newton’s Second Law ~~ Newton’s Second Law ~

Approximation: vertical

∂p∂z

=−ρg

Hydrostatic ApproximationAccurate to ~ 0.01% for weak vertical acceleration

dudt

−uvtanφ

a+

uwa

=−1ρ

∂p∂x

+2Ωvsinφ−2Ωwcosφ+Frx

Conservation of MomentumConservation of Momentum~ Newton’s Second Law ~~ Newton’s Second Law ~

Approximation: horizontal, extratropical

dvdt

−u2 tanφ

a+

vwa

=−1ρ

∂p∂y

−2Ωusinφ +Fry

dudt

−uvtanφ

a+

uwa

=−1ρ

∂p∂x

+2Ωvsinφ−2Ωwcosφ+Frx

Conservation of MomentumConservation of Momentum~ Newton’s Second Law ~~ Newton’s Second Law ~

Approximation: horizontal, extratropical

dvdt

−u2 tanφ

a+

vwa

=−1ρ

∂p∂y

−2Ωusinφ +Fry

v =+1ρf

∂p∂x

Conservation of MomentumConservation of Momentum~ Newton’s Second Law ~~ Newton’s Second Law ~

Approximation: horizontal, extratropical

u=−1ρf

∂p∂y

Geostrophic ApproximationAccurate to ~ 20 - 30%

Module 3 Module 3 Climate Modeling Theory - 1Climate Modeling Theory - 1

BREAKBREAK

Module 3 Module 3 Climate Modeling Theory - 1Climate Modeling Theory - 1

OUTLINE (Part 1):OUTLINE (Part 1): SymbolismSymbolism Conservation LawsConservation Laws

– massmass– thermodynamic energythermodynamic energy– momentummomentum

Equation of StateEquation of State

Ideal Gas Law

p=ρRTR = gas constantR = R(constituents)

Common practice:R = Rd = 287 J-kg-1-s-1

T = Tv

Module 3 Module 3 Climate Modeling Theory - 1Climate Modeling Theory - 1

OUTLINE (Part 1):OUTLINE (Part 1): SymbolismSymbolism Conservation LawsConservation Laws

– massmass– thermodynamic energythermodynamic energy– momentummomentum

Equation of StateEquation of State Water in the AtmosphereWater in the Atmosphere

q versus latitude & pressureq versus latitude & pressure[g-kg[g-kg-1-1]]

Note: small part of atmosphere, but ...Note: small part of atmosphere, but ...

0

20

40

60

80

100

120

-40 -20 0 20 40

esat(T)

Temperature [oC]

… … waterwater• saturatessaturates• changes phasechanges phase

q specific humidity [kg-kg-1] mass (H2O)v/mass air

e vapor pressure [Pa]partial pressure by water molecules

m mixing ratio [kg-kg-1]

mass (H2O)v/mass dry air

RH relative humidity [%]ratio: m/msat

Some Further Symbolism

EP P

Q Q

R

Water CycleWater Cycle

E

E

Water CycleWater Cycle

Heat absorbedHeat absorbed

Heat releasedHeat released

Water is thus a primaryWater is thus a primary form of heat transportform of heat transport

heat absorbed when evaporates heat absorbed when evaporates

released when water condensesreleased when water condenses

largest individual source of energy largest individual source of energy

for the atmospherefor the atmosphere

Water CycleWater Cycle

Radiation absorbed by water & re-emittedRadiation absorbed by water & re-emitted

Water is thus a primaryWater is thus a primary form of heat transportform of heat transport

heat absorbed when evaporates heat absorbed when evaporates

released when water condensesreleased when water condenses

largest individual source of energy largest individual source of energy

for the atmospherefor the atmosphere

andand greenhouse gas greenhouse gas

~ transparent to solar~ transparent to solar

absorbs/emits infraredabsorbs/emits infrared

RH vs. latitude & pressureRH vs. latitude & pressure[%][%]

RH

70 70

precipitation vs. latitude & longitudeprecipitation vs. latitude & longitude[dm-yr[dm-yr-1-1]]

[dm-yr[dm-yr-1-1] = [100 mm-yr] = [100 mm-yr-1-1] =[0.27 mm-d] =[0.27 mm-d-1-1]]

Lift Moist ParcelLift Moist Parcel

T

z

9.8 K/km

RH

T

z

z

Lift Moist ParcelLift Moist Parcel

RH

T

z

z

100 %

Lift Moist ParcelLift Moist Parcel

LCL

Lifting Condensation Level

Stable PrecipitationStable Precipitation

condensationcollision

coalescence

Stable PrecipitationStable Precipitation

condensationcollision

coalescence

T

z

Lift FurtherLift Further

LCL

T

z

Lift FurtherLift Further

Environment’s T(z)

T

z

Lift FurtherLift Further

Environment’s T(z)

T

z

Lift FurtherLift Further

Level of free convection

T

z

ConvectionConvection

Level of free convection

Module 3 Module 3 Climate Modeling Theory - 1Climate Modeling Theory - 1

Final Question:Final Question:How much heating by How much heating by condensation?condensation?

Use 1Use 1stst Law of Thermodynamics Law of Thermodynamics

= heating/mass [J-kg-1-s-1]

dt

dTCH p=&

H&

Assume: no work doneAssume: no work done

Apply to precipitating column

Heat released Mass condensed Mass falling out

P [m/s]

Apply to precipitating column

Heat released Mass condensed Mass falling out

Pρw [kgw-m-2-s-1]

Apply to precipitating column

Heat released LPρw [J-m-2-s-1]

Pρw [kgw-m-2-s-1]

Apply to precipitating column

Heat released LPρw [J-m-2-s-1]

Apply to precipitating column

Mass heated ps/g [kgair-m-2]

= Heating/mass = LPρwg/ps [J -(kgair)-1-s-1]

Heat released LPρw [J -m-3-s-1]

Apply to precipitating column

Mass heated ps/g [kgair-m-2]

H&

P = 1000 mm-yr-1 =3.2.10-8 m-s-1

Ps = 1000 hPa =10+5 Pa

[ ]spw

p

pCgLP

sCHdt

dT

ρ=

−= −1deg&

dT/dt = 7.7.10-6 deg-s-1 = 0.67 deg-day-1 (Radiation ~ -1 deg-day-1)

How much heating by condensation?How much heating by condensation?

Coming: Module 5 Coming: Module 5 Climate Modeling Theory - 2Climate Modeling Theory - 2

OUTLINE (Part 2):OUTLINE (Part 2):

RadiationRadiation

Surface ProcessesSurface Processes

Earth SystemEarth System