Post on 09-Oct-2019
R-C Modeling, Exoclimes 2011
Radiative-Convective Modeling of Planetary Climate
Raymond T. Pierrehumbert
The University of Chicago
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R-C Modeling, Exoclimes 2011
Outline of the Lecture
• Basic formulation and solution methods
• Some interesting applications
• Beyond 1D: The need for dynamics
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R-C Modeling, Exoclimes 2011
Basic formulation: What is a radiative convective model?
• Represent entire atmosphere by a single vertical column T (p, t), etc.
• Column generally meant to represent global mean climate
• Only vertical energy transport is modeled
– Radiative transport
– Turbulent transport due to convection
– Convection modeled as a 1D mixing process
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R-C Modeling, Exoclimes 2011
Radiating temperature and the greenhouse effect
Tg
T(ps)
Radiation only
TemperatureRadiation
and Turbulence
For many purposes, can assume Tg ≈ T (ps)
For review, see: Pierrehumbert 2010: Infrared Radiation and PlanetaryTemperature, Physics Today 64
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Abisko 2011: Energy Balance and Planetary Temperature
A few observed vertical structures
10
100
1000
150 180 210 240 270 300
March 15 1993, 12Z2.00 S 169.02 W
Pres
sure
(m
illib
ars)
Temperature (Kelvin)
Tropopause
Troposphere
Stratosphere
100 200 300 400 500 600 700 800
0.001
0.01
0.1
1
10
100
Venus
Pioneer Venus Large Probe, 1978Magellan Radio 1990
T
p (b
ars)
60 80 100 120 140 160 180
1
10
100
1000
Titan (Huygens Data)
T
p (m
illib
ars)
100 150 200 250 300 350 400 450
10-5
0.0001
0.001
0.01
0.1
1
10
Jupiter
T
p (b
ars)
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R-C Modeling, Exoclimes 2011
The IR Radiative Transfer model
• Input T (p), composition (e.g. pCO2,q(p))
• → fluxes I+(p), I−(p)
• Heating rate H = g−1d(I+ − I−)/dp
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R-C Modeling, Exoclimes 2011
Pure radiative equilibrium
• H + Hsw = 0 at each p
• Equivalently I+ − I−+ F~ = const.
• Typically ignore scattering for IR, but incorporate it for F~
• Wavelength of incoming stellar radiation�Wavelength of outgoing IR
• ... but this separation can break down for roasters.
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R-C Modeling, Exoclimes 2011
Energy balance for atmosphere transparent to incoming stellarradiation
120 160 200 240 280
2 104
4 104
6 104
8 104
1 105
Radiative-ConvectiveDry AdiabatPure Radiative Equilibrium
T (K)
pres
sure
(Pa)
300ppmv CO2
in Dry Air
Tropopause
-0.004 -0.002 0 0.002
2 104
4 104
6 104
8 104
1 105
Radiative-ConvectiveDry Adiabat
IR Heating Rate (W/kg)
pres
sure
(Pa)
300ppmv CO2
in Dry Air
Warm A
ir Up
Warm A
ir Up
Cold Air Down
Cold Air Down
(∫ pspt
Hdp/g) + Sabs = 0 ; Determines tropopause height
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R-C Modeling, Exoclimes 2011
How to get OLR(Tg)
Time steppingrequired here
IR
OLR
Require zeronet flux
No time steppingin troposphere
StratosphereTroposphere
Tg
ptrop
SW
SW
• Without atmospheric shortwave absorption, stratospheric temperaturedepends on insolation only via Tg.
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• For typical atmospheres stratosphere is optically thin in IR. Can thenuse isothermal stratosphere or ”all troposphere model” and dispensewith time-stepping entirely.
R-C Modeling, Exoclimes 2011
Once you have OLR(Tg) ...
T
OLR
Stellar AbsFlu
x
Plot it, and you’re done!
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R-C Modeling, Exoclimes 2011
When is a 1D model sufficient?
< OLR(Tg, q) >≈ OLR(< Tg >, < q >)
+12(∂TTOLR) < T ′2 > +1
2(∂qqOLR) < q′2 > +(∂TqOLR) < T ′q′ >
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R-C Modeling, Exoclimes 2011
Forms of convective adjustment
• If you are time-stepping only to find an equilibrium, then the convectiveadjustment stage need not conserve energy
• If you are trying to represent the actual time evolution (e.g. the diurnalor seasonal cycle) the convective adjustment stage needs to conserveenergy
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R-C Modeling, Exoclimes 2011
What is conserved during convective adjustment?
• Suppose the adjustment takes place in a layer from p1 to p2
• The final state is an adiabat. Which adiabat? (e.g. dry adiabat is aone-parameter family defined by T (p) = T (p1)(p/p1)
R/cp)
• First Law: T−1δQ = ds = dcp ln θ
• But during adjustment, δQ 6= 0 at each p. Only have∫ p2p1
δQdp = 0
• Therefore∫ p2p1
T−1δQdp 6= 0; Entropy does not ”mix”
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R-C Modeling, Exoclimes 2011
The answer: Dry or Moist Static Energy
• Define Z(p) from hydrostatic relation
• First law: δQ = d(cpT + gZ) (in dry case)
• Therefore DSE ≡∫ p2p1
(cpT + gZ)dp/g conserved during adjustment
• If q is the mass concentration of the condensible, then in dilute limit(q � 1), MSE density is cpT + gZ + Lq
• Things get interesting (and somewhat unexplored) in the non-dilutelimit, where you need to track the energy carried by the condensate.
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R-C Modeling, Exoclimes 2011
Example: DSE-conserving mixing of a step discontinuity
180 200 220 240 260 280 300 320
20000
40000
60000
80000
100000
Tinit
Tadj
Temperature (K)
Pres
sure
(Pa)
1.4 1051.6 1051.8 105 2 105 2.2 1052.4 1052.6 105
20000
40000
60000
80000
100000
DSEinit
DSEadj
Dry Static Energy (J/Kg)
Pres
sure
(Pa)
Unique adjusted solution for any p1 and p2. Adjust p1 and p2 to make thetemperature continuous at endpoints. (An assumption about how
convection works!)16
R-C Modeling, Exoclimes 2011
Deep atmospheres, optically thick in stellar spectrum
ShortwaveAbsorption layer
No net flux in deep layer → isothermal
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R-C Modeling, Exoclimes 2011
Applications: The conventional habitable zone
• Inner edge: Water vapor runaway (”wet” or ”dry” version).(Runaway = Uninhabitable)
• Outer edge: CO2 runaway.(Runaway = Habitable)
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R-C Modeling, Exoclimes 2011
Water Vapor runaway and inner edge (pure WV atmosphere)
220
240
260
280
300
320
340
360
380
260 280 300 320 340 360 380 400
100 m/s**2, pure water20 m/s**2, pure water10 m/s**2, pure water1 m/s**2, pure water
OLR
(W
/m2 )
Tg
Abs. solar, g = 20 m/s**2 (schematic)
More on this from Colin!19
R-C Modeling, Exoclimes 2011
CO2 runaway and outer edge edge (pure CO2 atmosphere)
0
20
40
60
80
100
120 140 160 180 200 220 240 260
OLR
(W
/m2 )
Surface Temperature (K)
100 m/s2
20 m/s2
10 m/s2
1 m/s23 m/s2
0.01 0.1 1 10 30Surface Pressure (bar)
Abs. solar, g = 20 m/s**2 (schematic)
Ch. 4, Principles ... plus in-prep radiation model intercomparison byPierrehumbert, Abbot and Halevy
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Abisko 2011: Energy Balance and Planetary Temperature
Outer edge: Ice-albedo bifurcation
0
100
200
300
400
500
600
700
800
200 220 240 260 280 300 320 340
OLRL=1517L=1685L=1854L=2865
Flux
(W
/m2 )
Surface Temperature
H
Sn1
Ice covered Ice Free
Sn2
Sn3
AB
A'
B'
1
4(1− α(T ))L~ = OLR(T,CO2)
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Snowbird 2011: Climate sensitivity, feedback and bifurcation
Snowball Earths
Pierrehumbert , Ap. J. L. 2011
Pierrehumbert , Abbot, Voigt & Koll, Ann. Rev. Earth and Plan. Sci. 2011
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Snowbird 2011: Climate sensitivity, feedback and bifurcation
Zero-D Snowball Bifurcation
1 10 100 1000 104 105220
240
260
280
300
320
340
Glo
bal M
ean
Tem
pera
ture
(K)
55% 60% 65%
L
R
CO2 Inventory (Pa)
CO2 Concentration
S1
S2
S3
S4
W1
W2
W3
W4
.000066.0000066 .00066 .0065 .062 .397
Pierrehumbert,Abbot,Voigt & Koll, AREPS 2011
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R-C Modeling, Exoclimes 2011
Applications: H2 worlds
• Conventional outer limit defined by CO2 runaway. Yields Early Marsequivalent orbit
• To make planets in more distant orbits habitable you need a less con-densible greenhouse gas
• So how about H2?
• In a distant orbit, a Super-Earth can hold an H2 atmosphere.
• Gravitational lensing has detected Super-Earths in distant orbits.
• Pierrehumbert and Gaidos, ApJL 2011
• Also relevant to Steppenwolf planets
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R-C Modeling, Exoclimes 2011
Pure H2 atmosphere
0 50 100 150 200 250 300
0.001
0.01
0.1
1
10
Temperature Profiles, Absorbed Stellar Radiation 5 W/m2
AdiabatT (M star)
T (G star)
Temperature (K)
Pres
sure
(bar
)
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R-C Modeling, Exoclimes 2011
Top-of-Atmosphere Energy Balance
1
10
100
1000
1 10
Flux
per
uni
t sur
face
are
a (W
/m2 )
Surface pressure (bar)
280K300K
320K340K
360K
L = 40 W/m2L = 80 W/m2
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R-C Modeling, Exoclimes 2011
Beyond 1D: The role of large scale dynamics
• Horizontal heat transport
• Lapse rate
• Subsaturation (Relative Humidity)
• Clouds
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R-C Modeling, Exoclimes 2011
Intro to General Circulation: The Hadley Cell
90S 60S 30S 0 90N30N 60N1000
800
600
400
200
100
July
Excess EnergyOut
Excess EnergyOut
Excess Energy In
90S 60S 30S 0 90N30N 60N1000
800
600
400
200
100
Excess EnergyOut
Excess EnergyOut
Excess Energy In
January
Gets more global for slow rotation, small planet (Venus,Titan); moreequatorially confined for rapid rotation, large planet (Jupiter, Saturn).
Earth is ”Mr. In-Between.”28
R-C Modeling, Exoclimes 2011
Intro to General Circulation: Extratropical synoptic eddies
Transport hot air poleward/upward, cold air equatorward/downward.
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R-C Modeling, Exoclimes 2011
Beyond 1D: Lapse Rate
• Hadley cell sets the entire tropics to the moist adiabat, even thoughconvection is active in only a small proportion of the tropics
• In midlatitudes, there may be little convection, and lapse rate is deter-mined by large scale transports of heat by transient baroclinic eddies.Lapse rate can have large geographic and seasonal variation.
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R-C Modeling, Exoclimes 2011
Example: Siberian lapse rate
200 220 240 260 280 300
200
300
400
500
600
700
800900
1000
Temperature (K)
Pres
sure
(m
b)
55N Asian Continental Interior
Synoptic eddies increase winter stability by sliding in cold polar air at lowaltitudes
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R-C Modeling, Exoclimes 2011
Beyond 1D: Subsaturation
• Atmospheres with a condensible component need not be saturated
• Subsaturation determines concentration of the condensible (e.g. watervapor)
• Subsaturation is a dynamical phenomenon
• Subsaturation affects runaway at both the inner and outer edge of theHZ.
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R-C Modeling, Exoclimes 2011
Where does subsaturation come from?
T2<T1
T1 , q1
q2<q1
T3>T1
q3=q2<q1
Saturated Subsaturated
Lift
and
cool Subside and heat
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R-C Modeling, Exoclimes 2011
Rapid subsidence creates large T (and hence p) gradients
100 200 300 400 500 600 700 800
10000
100000
Temperature (K)
Pres
sure
(Pa)
Subsidence from 10000 Pa to surface
Moist A
scentDry Descent
Radiative Cooling
”Rapid” = ”Rapid compared to radiative cooling”
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R-C Modeling, Exoclimes 2011
RH feedback leads to metastable non-runaway states
240
260
280
300
320
340
280 300 320 340 360 380 400
RHmin = 25%RHmin = 50%RHmin = 75%RHmin = 100%
OLR
(W/m
2 )
Surface temperature (K)
N2/H
2O Atmosphere
RH = RHmin + (100%−RHmin)(T − T0)/(T1 − T0)
for T0 < T < T1
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R-C Modeling, Exoclimes 2011
Subsaturation in FMS GCM dynamic simulations
• 3D dynamic general circulation model
• Tide-locked, various orbital periods
• Idealized moist thermodynamics (includes latent heat release)
• Grey gas radiation; no effect of condensible on IR optical depth
• Carried out by Feng Ding
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R-C Modeling, Exoclimes 2011
A few take-home points
• Radiative-Convective modeling is still a valuable tool.
• Ideal for exploratory work on new problems, testing convection andradiation schemes, etc.
• Energy-conserving convective adjustment for atmospheres with con-densation of a major constituent still has some wrinkles to be workedout.
• Even for planets with quite uniform surface temperature, dynamics hasimportant zero-order climate effects via lapse rate, subsaturation andclouds.
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