﻿the Insolation of an Atmosphere. 89i 



below t 2 the temperature is given by B c (t) j above t 2 it is 

 given by the function B(t) given by 'formula (28"). At t 2 

 the upward intensity of radiation is that appropriate to the 

 state o£ radiative equilibrium : it is the value I(t ) deducible 

 from (2W) and (27"), 



I(t 2 ) = B(T 2 )+iS(cos« — |?t)r" r - seua . 



Hence t 2 is /Ztc ; roo^ of the equation 



Ic(t 2 ) = I(t 3 ) (40) 



Suppose this equation is solved. i£ />y no means folloivs 

 that 



B e (r 2 ) = B(t 2 ) ; 



i. e. ft fy/ wo means follows that the temperature immediately 

 below the junction is continuous with that immediately above it. 

 Further, even if it happens that these temperatures are equal, 

 it does not follow that the condition for a convective atmo- 

 sphere is satisfied in the region immediately below t 2 . For 

 a physically possible distribution both these conditions must 

 be satisfied. Hence, in general, it is not possible to determine 

 a level t 2 such that a prescribed temperature distribution exists 

 up to t 2 and a radiative one above it. 



The question must therefore be studied in the reverse order: 

 what conditions does the existence of a stratosphere of non- 

 zero optical thickness impose on the temperature distribution 

 in the upper troposphere ? It would make the present paper 

 too long to take up the investigation here. But it appears to 

 be possible to show that if the temperature is continuous at t 2 , 

 then in general (but not necessarily) the temperature gradient 

 is discontinuous there. This is, of course, what is observed. 



In the earth's stratosphere, on the other hand, the ob- 

 served absence of vertical gradients strongly suggests that 

 if it is in strict radiative equilibrium its optical thickness is 

 practically zero. For the particular relation (cosa = ^i or 

 cos \ = nj x /2) which is necessary for an isothermal strato- 

 sphere of non-zero optical thickness cannot be satisfied save 

 in very high latitudes; and even here (as we shall see) this 

 would be prevented by the additional radiation due to 

 world-wide convection. Further, we have seen in § 2 that 

 if the absorption of solar radiation is neglected, an isothermal 

 stratosphere would soon cease to be isothermal and would be 

 disturbed by convection currents. If now the optical thick- 

 ness of the stratosphere is practically zero, a state of radiative 

 equilibrium Will probably extend a little way below thetropo- 

 pause, and the observed suddenness of the demarcation must 



3 M2 



