RADIATIVE TEMPERATURE CHANGES IN THE OZONE LAYER 
integrating this expression over the plane, 
Qa m/2 
soy = dee de | bly eo °" sin 0 dB 
f(u,») t h b 18) 
= QrkIy Hin(kw) du dy, 
where £7, (ku) is an exponential integral. 
The total vertical flux reaching P from all points 
above is thus 
w= | 2nbt Bie(eu) du a. (17) 
0 0 
The integration of (17) solves the computational 
problem. In practice, however, this integration is very 
difficult. In the first place, the black-body radiation 
T, depends on the temperature, which in turn depends 
on the path length wu in an irregular manner that varies 
from time to time in the atmosphere. Secondly, the 
absorption coefficient & depends on the wave length 
din arapid and irregular manner. Indeed, this variation 
in the infrared bands is so rapid as to prevent the direct 
application of methods of numerical integration. 
Fie, 3.—Radiation to P from a volume element above P. 
Elsasser [14] and Schnaidt [52] have introduced a 
simplification that makes the integration over wave 
length practicable. They divide the spectrum into small 
intervals, each interval containing a number of absorp- 
tion lines. Under the assumptions that these lines are 
broadened by pressure effects and that the lines in any 
one spectral interval are all equal in intensity and equi- 
distant, they derive expressions for the average ab- 
sorption in the spectral interval. The absorption coef- 
ficient, which may vary greatly in the interval, can 
then be replaced by a “generalized” absorption coeffi- 
cient which is constant in the interval. 
The transmission function rr (wu) is defined as the 
ratio [/Io of the radiation penetrating a layer of thick- 
ness wu to the radiation incident at the top of the layer. 
For diffuse radiation, the transmission function ry (w) 
is similarly defined in terms of fluxes rather than inten- 
sities. The relation between ry and 77 is 
T(U) = | rr(u sec 6) d sin’ 0. (18) 
297 
Thus for exponential absorption of a beam of mono- 
chromatic radiation, rr = e*” and ty = 2Hi3(ku). 
In these more general terms, (17) may be written in the 
form 
F= -| IN i pl ryllu] dw (19) 
0 0 du , 
where f; is the black-body flux, given by 7J, and 1 is 
the generalized absorption coefficient. 
The transmission functions given by Elsasser [14] 
and Schnaidt [52] apply strictly only to spectral in- 
tervals that contain equal and equidistant lines, each 
line broadened by pressure effects only. Experiment, 
however, has shown that the formulas apply to a good 
approximation to the actual behavior of the infrared 
absorption bands of the atmospheric gases. With the 
aid of these developments one can integrate (19) over 
wave length. The integration over path length must be 
accomplished numerically or graphically in a separate 
calculation for each atmosphere that has a distinctive 
relationship between path length and temperature. The 
cooling at any level in the atmosphere is then propor- 
tional to the vertical divergence of the net flux at that 
level. 
A more direct method of computing cooling in the 
atmosphere has been given by Bruinenberg [5] and 
Brooks [4]. It gives the divergence of the flux, and hence 
the cooling, directly. This method is to be preferred for 
the ozone layer where the fluxes are small in any case. 
Au 
nelly > 
B AZ, Au 
TEMPERATURE ——> 
Fie. 4—Schematie representation of temperature-height 
sounding of the atmosphere. The path length of absorbent, w, 
between A and A’ is the same as that between B and B’. 
Figure 4 gives a schematic representation of the vari- 
ation of temperature with height. Let the total amount 
of absorbent above A be wa, and that above B be wz, 
where ws — Uz = Au. Define the flux from an zsothermal 
column as 
SF (w, 2) = ela, 2) Gl" 20) 
The total emissivity of the isothermal column, ¢;, is 
defined as the ratio of the flux emitted from the column 
to the black-body flux at the temperature of the column. 
Thus, 
1 [-<] 
on | al = ON (21) 
