298 
The downward fluxes at A and B can be written as 
UA nC 
hy = 05 du, 
0 OU 
(22) 
“3 OSp 
Fz = — du. 
0 OU 
Equations (20) and (22) are consistent with (19). There- 
fore, 
UB Caro Cc uA 
jy oe [ @ a =) du | 2 am, 
0 up OU 
Ou Ou (3) 
Consider a level A’ at a path length w above A and 
a level B’ at the same path length w above B. (The 
linear distances A — A’ and B — B’ are not neces- 
sarily the same.) The flux from B’ through B and the 
flux from A’ through A would be identical except that 
A’ and B’ may be at different temperatures. 
One can write 
OF, OSs oF dT 
= * =. ~ auaT Ge dual du Ce) 
The second integral in (23) represents the additional | 
flux at A because ws > we so that 
UB Cc 
f(a - (2) 
ua \OU Ou) t 
where (0F/du), is the value of (0F/dw) at the top of 
the atmosphere. With these values inserted (23) be- 
comes 
_ fet eEN (all OF 
AF = | (e, =) Au du + (I) Au. (26) 
The cooling is proportional to the divergence of the 
flux. Divide both sides of (26) by Az and let both sides 
approach the limit Az = 0. Then 
oF du ie oF OF 
2 | f — an Sr ea C2 
The divergence of the upward flux can be derived in 
the same manner. However, because the ground acts 
as a black body, the expression for the upward flux has 
no term comparable to the second term in (27). Thus, 
finally, 
OF a) ul = ihe 7 
ot PCy \ 02 0z 
1 oul sr“? (ar F! 
as Le! f aaa 
oF! : =) ] 
7 es) =| — at cee 
Data for Calculating Cooling. According to (28), the 
cooling at any level in the ozone layer can be deter- 
mined by numerical or graphical integration if F is 
known as a function of uw and 7. From (20) 
oF de; 
duoT duoT 
(25) 
(28) 
= 4oT® ss a. ei (29) 
THE UPPER ATMOSPHERE 
The emissivity ey can be determined either from labora- 
tory measurements or from the theoretical transmis- 
sion functions according to (21). 
Elsasser [14] has summarized measured values of the 
emissivities of water vapor and carbon dioxide. Sum- 
merfield® has given data on the 10-1 band of ozone. 
Unfortunately, in none of these cases do the measure- 
ments extend to values of w as small as those met in the 
ozone layer. Some radiation computations have made 
use of extrapolations of these emissivity curves. 
An alternative procedure is to make use of the trans- 
mission functions for the infrared bands. The functions 
can first be tested against and fitted to the observation 
in the range of w where measurements are available. 
In general, it is possible to get a good agreement 
between theory and observations. The transmission 
functions can then be used to extrapolate the available 
information to small values of w. Even this type of ex- 
trapolation, however, is risky. The transmission func- 
tions, as stated above, were derived on the basis of 
idealized bands. That they describe the behavior of the 
actual infrared bands in a certain range is no guarantee 
that they will serve equally well in another range. A 
further difficulty stems from the overlapping of the ozone 
and carbon dioxide bands at 15 yp. Very little is known 
about the former band. How much its presence may 
affect the carbon dioxide emissivities as measured in the 
laboratory is not known. 
Pressure Effects on Infrared Absorption. Still another 
difficult and unsolved problem is the question of pres- 
sure effects on absorption in the infrared. It is certain 
that the half-width of the absorption lines in the infra- 
red varies with pressure. Elsasser [14] gives a rather 
complete discussion of the experimental facts about 
this dependence. 
For water vapor, the half-widths of the absorption 
lines seem to vary as the square root of the pressure. 
Strong [54] has shown a similar square-root dependence 
for the 10—» ozone band. In the ease of carbon dioxide 
evidence is conflicting, and various investigators have 
assumed that the half-width varies as the pressure 
according to laws ranging from square-root dependence 
to direct dependence. This is a question, particularly in 
the case of carbon dioxide, that should be cleared up 
before accurate calculations can be made. 
RESULTS OF CALCULATIONS 
Despite the difficulties discussed above, some investi- 
gators have made calculations based on the available 
knowledge. These calculations correspond qualitatively 
to the observational information available about the 
temperature distribution in the ozone layer. It is the 
purpose of this section to outline the results so far 
achieved. 
Gowan’s Calculations of Equilibrium Temperature. 
E. H. Gowan has pioneered the work in this field. In a 
pair of early papers [23, 24], he made some prelim- 
inary calculations which showed that the suspected 
6. In an unpublished doctoral dissertation at the California 
Institute of Technology in: 1941. 
