﻿890 Mr. E. A. Milne on Radiative Equilibrium : 



given level t = t 2 is in strict radiative equilibrium, that below 

 the given level merely in radiative equilibrium as a whole] 

 below t = t 2 the temperature distribution may be of any form 

 (with or without a lower bounding surface) subject only to 

 the condition that the whole system below t 2 radiates out- 

 wards as much as it absorbs ; in general, convection of heat 

 will be required in the region below t = t 2 in order to maintain 

 a steady state. Then it is easily seen that the temperature 

 distribution above t 2 is exactly the same as if the lower region 

 were in radiative equilibrium in the strict sense ; for the 

 upward intensity at t 2 , namely I(t 2 ), is the same in the two 

 cases. Hence the temperature distribution we have already 

 found applies to the region above t 2 . The importance of 

 this point from the point of view of applications to the 

 earth's troposphere and stratosphere is evident *. 



§ 13. Ilie boundary betiveen troposphere and stratosphere. — 

 It is convenient to denote the regions below and above the 

 level t 2 in our ideal problem by the words " troposphere'" 

 and "stratosphere" respectively, without implying any 

 reference to these actual regions in the earth's atmosphere. 

 Then § 12 shows that under the conditions there stated 

 a stratosphere cannot be isothermal unless its optical thick- 

 ness is zero or cos« = ^n. I£ the optical thickness is not 

 zero and cos a > Jw, the lower parts of the stratosphere must 

 be warmer than the upper. This agrees with § 2, where it 

 was found that Gold's stratosphere is warming up at the 

 base. 



We are now in a position to frame in a precise manner 

 the problem of where the division between troposphere and 

 stratosphere should occur, in the ideal case. Let us suppose 

 that there is a certain distribution of temperature which the 

 processes of convection tend to set up throughout the whole 

 atmosphere. Let the corresponding black body radiation- 

 function be expressed as a function of optical depth, say 

 B c (r). This temperature distribution together with the lower 

 boundary surface implies a definite upward intensity of 

 radiation at any point t, say T c (t), which is determinate and 

 calculable when B c (t) is given. Let t 2 denote the optical 

 depth of the surface of separation between troposphere 

 and stratosphere which it is required to determine. Then 



* The points which are the subject of §§ 10, 11, 12 are substantially 

 made by Emden, in the form appropriate to diffuse radiation. But 

 Emden's analysis is in parts a little complicated by his introducing 

 unnecessarily early into the investigation an empirical expression for 

 the water-vapour in the earth's atmosphere as a function of height. 



