assume a simple situation in which the atmosphere is transparent to this radiation, and 

 emissions and reflections are isotropic, we may express this upward stream, F, as 



F = E + r (E + r F) (1) 



o o c c 



where: 



E = emitted flux from the surface 

 o 



r = reflectivity of the surface 



E = emitted flux from the cloud base 

 c 



r = reflectivity of the cloud base 



Quantities refer to the radiation 8-13 microns. The second quantity on the right of (1) re- 

 presents the fraction of the downward stream which is reflected upward by the surface. 

 Equation (1) may be solved for F giving 



F = (E + r E ) / (1 - r r ) (2) 



o o c o c 



If we now let: 



4 



Where: 



E = (1 - r ) A0 <r T 

 o o o o 



E = (1 - r ) A0 (tT^ 

 c ceo 



F = A0(rT 



(1 - r ) = emlssivity of the surface 



(1 - r ) = emlssivity of cloud base 



A0. - fractional black body flux between 8-13 microns 



T = surface temperature ( K) 



T = surface temperature of cloud base (°K) 



T = blackbody temperature associated with the upward flux F 



(T = Stefan-Boltzmann constant 



(3) 



We may rewrite (2) as 



4 

 F = A0<rT = 



(1-r ) A0 T^ (r+ r (1-r ) A0 (TT'* 

 o o o o c c c 



-97- 



/ (1-r r ) (4) 



o c 



