Radiative Equilibrium
• Equilibrium state of atmosphere and surface
in the absence of non-radiative enthalpy
fluxes
• Radiative heating drives actual state toward
state of radiative equilibrium
1
Figure by MIT OpenCourseWare.
2
Extended Layer Models
TOA :  T2 4   Te 4  T2  Te
Middle Layer : 2 T   T2   Ts   Te   Ts
4
1
4
4
4
Surface :  Ts 4   Te 4   T14
1
 Ts  3 Te
4
1
T1  2 4 Te
3
4
Effects of emissivity<1
Surface : 2 A TA 4   A T14   A Ts 4
 2
 TA  5
1
4
Te
321K
 Ts
Stratosphere : 2 t Tt 4   t T2 4
 2
 Tt  1
1
4
Te
214 K
 Te
4
Full calculation of radiative equilibrium
5
Figure by MIT OpenCourseWare.
6
Time scale of approach to equilibrium
7
8
Contributions of various absorbers
Note: All
simulations
have
variable
clouds
interacting
with
radiation
9
10
Problems with radiative equilibrium
solution
• Too hot at and near surface
• Too cold at a near tropopause
• Lapse rate of temperature too large in the
troposphere
• (But stratosphere temperature close to
observed)
11
Missing ingredient: Convection
• As important as radiation in transporting
enthalpy in the vertical
• Also controls distribution of water vapor and
clouds, the two most important constituents
in radiative transfer
12
13
When is a fluid unstable to
convection?
• Pressure and hydrostatic equilibrium
• Buoyancy
• Stability
14
Hydrostatic equilibrium
Weight :  g  x y z
Pressure : p x y   p   p   x y
dw
F  MA :  x y z
  g  x y z   p x y
dt
dw
p
 g  ,
  1   specific volume
dt
z
15
Pressure distribution in atmosphere at
rest
RT
R*
Ideal gas :  
,
R
p
m
1 p
g
Hydrostatic :

p z
RT
Isothermal case :
p  p0 e
z
H
,
RT
H
 " scale height "
g
Earth: H~ 8 Km
16
Buoyancy
Weight :  g b x y z
Pressure : p x y   p   p   x y
dw
F  MA : b x y z
  g b x y z   p x y
dt
dw
p
p
but
  g  b
g
e
dt
z
z

b   e
dw
g
B
dt
e
17
Buoyancy and Entropy
Specific Volume:
Specific Entropy:
  1
s
   ( p, s)
 T 
  

  s
  s  
 s  p
 p  s
Bg
  p
  p g  T 

 T 
    s      s   s
  p s
 z  s
18
The adiabatic lapse rate
First Law of Thermodynamics :
dsrev
dT
d
 cv
p
Q T
dt
dt
dt
dT d  p 
dp
 cv


dt
dt
dt
dT
dp
  cv  R 

dt
dt
dT
dp
 cp

dt
dt
Adiabatic : c p dT   dp  0
Hydrostatic : c p dT  gdz  0


dT
dz 
 g
s
cp
  d
19
g

Earth’s atmosphere:
cp
  1 K 100 m
20
Model Aircraft Measurements
(Renno and Williams, 1995)
This image (published on Monthly Weather Review by the American
Meteorological Society) is copyright © AMS and used with permission.
21
Radiative equilibrium is unstable in the
troposphere
Re-calculate equilibrium assuming that
tropospheric stability is rendered neutral by
convection:
Radiative-Convective Equilibrium
22
Better, but still too hot at surface, too cold at tropopause
23
MIT OpenCourseWare
http://ocw.mit.edu
12.340 Global Warming Science
Spring 2012
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