Transcript METO 621 - UMD | Atmospheric and Oceanic Science

METO 621
Lesson 19
Role of radiation in Climate
• We will focus on the radiative aspects of climate and
climate change
• We will use a globally averaged one dimensional
radiative-convective approximation.
• First we will assume that the atmosphere has
negligible absorption for visible radiation
• Then we will add visible absorption
Radiative Equilibrium with Zero Visible
Opacity
• The surface is assumed to be reflective in the visible and black
in the IR.
• Thus the surface is heated by incoming solar radiation and by
downwelling IR radiation from the atmosphere
• The atmosphere is heated by IR radiation from the surface and
the surrounding atmospheric layers. This will set up a
diffusive-like temperature gradient throughout the optically
thick region.
• At the upper ‘edge’, when the optical depth drops to 1 the
atmosphere radiates to space, at a globally averaged effective
temperature, Te ,determined by the overall energy balance
• Also assume that the optical depth is independent of frequency
– the gray approximation.
Zero Visible Opacity
1/ 4
 S (1   ) 
Te  

4

B


where S is thesolar constant, is thesphericalalbedo
and  B is theStefan - Boltzmannconstant.For theEarth
theeffectivetemperatu
re is 255K.
For a thermalsource we can write
 BT 4 ( )
S ( )  B( )   B d 

and if we assume no scattering, a  0, then
dI ( , u )
u
 I ( , u )  B( )
d
(1)
Zero Visible Opacity
T hehalf - range equations are
dI  ( ,  )

 I  ( ,  )  B( )
d
dI  ( ,  )

 I  ( ,  )  B( )
d
adding and subtracting theseequations




d I  ( ,  )  I  ( ,  )

 I  ( ,  )  I  ( ,  )  2 B( )
d
d I  ( ,  )  I  ( ,  )

 I  ( ,  )  I  ( ,  )
d
Zero Visible Opacity
• In radiative equilibrium the net flux F() is equal to the net
outgoing flux , σBTe4 , which is constant for all .
• If we integrate the equation 1 over solid angles then we get
dF
 4 ( I  B )  0
d
• In radiative equilibrium, the source function is equal to the
mean intensity.
1
1


1
1


B( )   duI( , u )    dI ( ,  )   dI ( ,  )
2 1
20
0

1
Zero Visible Opacity
T hus thegray radiativetransferequation becomes
1
1

dI ( , u )
1


u
 I ( , u )    dI ( ,  )   dI ( ,  )
d
2 0
0

notingthatin radiativeequilibrium
4

T
F  2π  dμ I  ( ,  )  I  ( ,  )  2  d B e  const

0
0
1
1
We can now apply thetwo streamapproximation. We
replaceB( ,  ) with B( )
d (I   I  )

 I   I   2 B( )
d
d (I   I  )

 I   I  ..................2
d
Zero Visible Opacity
which must be solved wit h t heconstraint


F  2 I  ( )  I  ( )  2   BTe4
insertingB( )  (I  I - ) into theleft handside of equat ion 2,
and expressingtheright handside in termsof
F  2  (I ( ) - I - ( ))  2  BTe4 , we get
dB( )  BTe4

 constant
d
2
T hiscan be integrat edto give
 BTe4
B( ) 
 C
2
Zero Visible Opacity
C is a constantof integration. At    * theupward flux at t he
surface is 2  BTe4 . Solving for thedownward flux we get
F  ( *)  2 I  ( *)  2   B (TS4  Te4 )
If we evaluatethesource functionat t hebottomof themedium
 BTe4
1 
B( *) 
 * C  I ( *)  I  ( *)
2
2
(2 BTS4   BTe4 )

2
and
1
C
2

 * 
4
4
2 BTS   BTe 1  
  

Zero Visible Opacity
and thus we get

( *  )  
4
4
2 BTS   BTe 1 


 


Now let us look at   0
1 

I (0)  0 by definition, and B(0)  I (0)
2
1
B( ) 
2
4

T
F (0)  2 I  (0)  2  BTe4 henceB(0)  B e
2
setting  0 in thefirst equation above we get
TS  Te (1   * / 2 )1/ 4  TeG1/ 4
where G is called thesurface greenhousefactor
Greenhouse effect - one atmospheric layer
model
•Є
is the fraction of the IR radiation absorbed by the
atmosphere
Zero Visible Opacity
T wo otherexpressions can be derivedfrom the
equationsdefined before' aftersimplealgebra'.
One for theblackbodysource function
 BTe4

B( ) 
(1  )
2

T heotherfor thetemperatu
re at a given 
1/ 4
1  

Tre ( )  Te  
 2 2 
T hisis known as theradiativeequilibrium expression
for theatmospheric temperatu
re.
Zero Visible Opacity
Finite Visible Opacity
• Any realistic atmosphere absorbs radiation at both IR and
UV/visible wavelengths.
• The procedure to solve this problem is similar to that for the
case of no visible opacity.
• I will give only the solutions, using the two stream
approximation.
S
F
1

F
IR
B( )  I  ( ) 
 I  ( )  a e  / n0
4 
4n
where FaS is theaveragesolar flux over a diurnal cycle,n is the
ratioof k IR / kVIS , and  0 is theaveragesolar zenit hangle over a day.
T 4 ( ) 1
2
 / 
G ( ) 

(
1


)
e
  (1   )
4
Te
2


where  n 0 / 
Finite Visible Opacity
Finite Visible Opacity
  1 or equivalently kIR  kV
• This is the strong greenhouse limit where the solar
radiation penetrates deeply into the atmosphere, In the
deep atmosphere, the greenhouse enhancement “saturates”
to the constant value G(*→∞) = (1+)/2
• The asymptotic temperature is
1/ 4
 1 n 0 

T ( *  )  Te  
 2 2 
1/ 4
 k IR  0 

 Te 
 kV 
• This solution resembles that for Venus, which has a
surface temperature of 800 K. It does not apply to the
Earth or Mars, because of the importance of the surface
in the radiative transfer, and the neglect of convection.
Finite Visible Opacity
• For this case 1
• This represents an isothermal situation where the
solar heating exactly balances the IR escape.
Comparingtheexact value of theratioof thesource
functionsS (  ) / S (  0)   0 3 with thetwo
streamresult  0 /  , one can infer that  1 / 3 is
the best valueto use in opticallythicksituations.
Finite Visible Opacity
• For the case <<1 or kIR<<kV
• This represents the anti-greenhouse case. This is relevant to
numerous phenomena in the solar system
• An inverted temperature structure characterizes the Earth’s
upper atmosphere, where high middle-UV opacity due to
ozone absorption gives rise to a temperature inversion.
• This scenario may have happened in the Earth’s history.
Worldwide cooling causing mass extinction as the result of an
injection of massive quantities of dust (meteoroid impact)
• Stratospheric aerosols ( up to 10), from Mt Toba eruption
some 70,000 years ago may have been responsible for a
subsequent cooling of the Earth for a period of 200 years.