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Module 3 Climate Modeling Theory - 1. William J. Gutowski, Jr. Iowa State University. Module 3 Climate Modeling Theory - 1. GOAL: Understand basis for modeling climate from (almost) first principles. Module 3 Climate Modeling Theory - 1. OUTLINE (Part 1): Symbolism - PowerPoint PPT Presentation
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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 MassConservation of Mass(Continuity Equation)(Continuity Equation)
∂ρ∂t
=−∇⋅r V ρ( )−
∂ wρ( )∂z
= density [kg/m3]ρ
Source/sink = 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 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 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
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
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
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]]
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
= 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?