LONG-WAVE RADIATION* 
By FRITZ MOLLER 
Gutenberg University at Mainz 
Introduction 
Long-wave radiation occupies a peculiar position in 
the science of meteorology in that its effects in the free 
atmosphere are known only through theoretical cal- 
culations and not through measurements. These calcula- 
tions are, nevertheless, based on experiments in the 
laboratory and in the atmosphere. The atmospheric 
radiation is very seldom measured from balloons [3] 
or from aircraft [17], but quite frequently on the ground 
where the radiation from above is observed. It appears 
that part of the measurements from airplanes are in 
error; and the measurements on the ground have seldom 
found an evaluation that went beyond the formulation 
of empirical equations to approximate their average 
values. The theoretical deductions concerning the radia- 
tion in the atmosphere are much more extensive, and 
it would be most desirable to support them by careful 
measurements, especially since the experimental tools 
are available. 
Initially, much was expected from the investigations 
of long-wave radiation in the free atmosphere. It was 
hoped that they would furnish an explanation for the 
latitudinal differences in the temperature and altitude 
of the tropopause, an interpretation of the variations 
of these values from day to day, and a physical elucida- 
tion of the origin of the inversions in the free atmos- 
phere. In all these problems the solutions at times 
seemed to be near at hand, but then they receded. It 
appears to the author that today the emphasis of 
research is directed more toward the investigation of 
the radiative balance and heat budget in the atmos- 
phere. 
In the following, the techniques of measurement are 
omitted and little will be said concerning the mathe- 
matical-physical basis of the calculations, since two 
detailed treatises which are still up to date are available 
[14, 28]. They cover these subjects very thoroughly. 
On the other hand, we shall discuss in great detail the 
number of instances where long-wave radiation is in- 
volved in the problems of general meteorology. 
Long-wave radiation is a heat radiation; its energy 
is derived from the kinetic energy of the molecules. 
The radiators of this energy are those atmospheric 
gases that have absorption bands in the temperature 
radiation range of 4 to approximately 100 yu, that: is, 
water vapor, carbon dioxide, and ozone. The effect of 
the water vapor is restricted almost exclusively to the 
troposphere, in which carbon dioxide and ozone are 
of little significance. The latter two gases become of 
great importance in the stratosphere between 15 and 
35 km. In addition, heat is radiated by the ground and 
the clouds. 
* Translated from the original German. 
34 
The Absorption Laws 
From the beginning, the theoretical calculations were 
set up in such general terms that it became possible to 
investigate the radiation properties of any atmosphere 
with any temperature distribution and any possible 
arrangement of radiating and absorbing media. This 
was done by the development of special radiation dia- 
grams or radiation charts. This approach was justified 
by two facts: (1) The intensity of the radiation which 
is emitted by an element for a given wave length is 
proportional to the black-body radiation and is thus 
a function only of the absolute temperature; (2) this 
radiation is proportional to the mass of the radiating 
medium, which means, in the troposphere, to the 
amount of water vapor. It is of particular importance 
that the proportionality constant, that is, the absorp- 
tion coefficient k, is really a constant in its first ap- 
proximation and not a function of any other quantities. 
However, in reality such a dependence of & on the air 
pressure and on the temperature does exist and gives 
rise to particular difficulties in advanced investigations. 
It should immediately be pointed out that the con- 
struction and application of radiation diagrams become 
impossible if there exist in the atmosphere two different 
media whose masses in a given volume element vary 
independently of each other from case to case, but 
whose emissive power at the same wave length is of 
the same order of magnitude so that the effect of the 
one medium cannot be neglected as compared to that 
of the other. Such conditions prevail in the stratosphere, 
for instance in the case of ozone and carbon dioxide. 
It is possible to consider two media in the same space 
element of the atmosphere only when one of the radia- 
tors is a gray radiator, that is, when its absorption 
coefficient / is the same for all wave lengths [31]. 
Parallel Radiation. Tf an absorbing medium m is 
penetrated by a beam of parallel rays, absorption takes 
place according to an exponential law. The emergent 
radiation J) is 
Ty = Ip,e*™, 
(1) 
where the subscript \ indicates monochromatic radia- 
tion, and J, is the incident radiation. The absorbed 
portion is 
Al = Gig = y/o (2) 
The length of the path which the radiation follows in 
penetrating m does not appear in this equation. This 
means that the absorption is only a function of the 
penetrated mass m, or that the absorption coefficient 
ky is independent of the density of m throughout the 
penetrated cylinder. If Jp is black-body radiation at a 
temperature 7’, whose value is given by Planck’s law, 
Io SA, i) dw, 
= Gn, 
