LONG-WAVE RADIATION 45 
pected as early as 1935 [80] and which was recently 
demonstrated by Hess [23]. 
Radiation cannot be responsible for day-to-day 
changes in the tropopause of the temperate latitudes, 
since its effect is much too slow, as shown by Junge 
[26]; in this case dynamic processes are certainly im- 
portant. On the average, however, radiation of water 
vapor is as responsible for the low temperature at the 
tropopause, as is the decrease of solar radiation from 
the equator to the pole for the meridional temperature 
distribution of the troposphere, although in the latter 
case, day-to-day variations are likewise considerable. 
Numerical and Analytical Radiation Calculations 
All calculations of radiation processes mentioned so 
far are based on the method of the radiation diagrams. 
There is no doubt that these are somewhat cumber- 
some. The determination of the heating and cooling 
processes via the radiation fluxes and their differentia- 
tion with respect to altitude seems especially laborious. 
Bruinenberg [8] has made a valuable contribution here 
toward attaining the objective directly. He puts the 
differentiation with respect to altitude, which can also 
be replaced by differentiation with respect to tempera- 
ture, under the integral of the radiative flux in equation 
(13) or (15). Thus, after calculations of a scope analo- 
gous to those required for the construction of a radia- 
tion diagram, he arrives at expressions which are suit- 
able for numerical integration or summation. However, 
these equations are less suitable for a graphical treat- 
ment. Therefore, Bruinenberg has calculated tables 
which permit very simply the determination and addi- 
tion of the cooling or heating effects exerted on a given 
element by all atmospheric layers above and below it. 
The principal advantage of the individual calcula- 
tion is the fact that the cooling for points closely spaced 
along the vertical can be determined independently and 
with great accuracy. This leads to a result which, though 
not unexpected, has thus far been underestimated in 
its implications. Every break in the vertical tempera- 
ture distribution is manifested as a sharp peak in the 
distribution of the cooling rate. Thus cooling peaks up 
to 38C per day project from the average cooling level 
of 1C per day in the lower troposphere at every break 
of the characteristic temperature curve directed toward 
higher temperatures, and heating peaks up to +1C 
or +2C per day where the characteristic curve breaks 
toward lower temperatures. Heating of +5C per day 
occurs below an inversion, whereas there is a cooling 
of 15C per day at its upper boundary. However all this 
_ holds true only in very thin layers. (Bruinenberg’s 
method is so far applicable only to calculations in the 
lower and middle troposphere; for the tropopause, see 
p. 42.) At these points the radiation simply acts as a 
temperature equalizer in a manner similar to heat con- 
duction. Every break in the characteristic temperature 
curve is equalized at an accelerated rate. From the fore- 
going discussion it follows again that if inversions or 
more or less sharp discontinuities in the temperature 
gradient persist for days, the ordinary radiation proc- 
esses of the water vapor cannot be the cause. Thus the 
question whether long-wave radiation can produce in- 
versions is decided partially in favor of the negative 
[4]. For the maintenance of existing inversions, some 
processes must be constantly active that recreate the 
inversion continuously against the equalizing effects of 
the radiation. Such processes are partly the additional 
radiation from the haze layers below the inversion which 
overcompensate the heating (see p. 42) and partly the 
dynamic processes of shrinking and subsidence. The 
intensity of these dynamic processes can then be esti- 
mated from the calculations of radiation. 
An especially significant level is the earth’s surface. 
For radiation effects, the ground may be conceived as 
replaced by an infinitely extended isothermal layer of 
water vapor or COs. In such a case the corresponding 
characteristic temperature curve extended into the 
ground has a break at the ground surface. If there is a 
temperature decrease with altitude above the ground, 
there must be strong cooling directly at the ground 
surface, whereas there is strong heating at the base of a 
ground inversion. Such radiative processes will scarcely 
become operative, since all observations disprove them. 
However, in every case, they will tend toward an iso- 
thermal state near the ground as an equilibrium con- 
dition. Even if this equilibrium is not reached, its quan- 
titative consideration will possibly lead to a revision of 
the assumptions concerning the magnitude of the aus- 
tausch near the ground as far as has been disclosed by 
measurements of the temperature gradient.* 
Although the graphical and numerical calculation 
methods are carefully worked out, an analytical equa- 
tion can be very advantageous at times because it per- 
mits combination of the radiation process with others 
that can be approached theoretically. Such a possi- 
bility is offered by the quasi-conduction of radiation 
introduced by Brunt [9]. However, this method will not 
include the processes at the ground surface which were 
discussed above. The complicated construction of a 
radiation diagram makes it apparent that a complete 
description by a single convenient equation is impos- 
sible. Simplifications must be made. One such simplifi- 
cation consists of running the temperature lines hori- 
zontally in the Méller diagram (in the water-vapor 
section). This means that we do not change the distribu- 
tion of the absorption coefficients over the wave lengths, 
but that we do assume that the energy curve has the 
same shape for all temperatures, instead of assuming 
Planck’s formula for black-body radiation. This would 
imply, for example, that, if we take the shape of the 
energy distribution at 273K, the radiation for a differ- 
ent temperature would be given by multiplying this 
curve by the factor 74/273 which is not a function of 
the wave length. In the Elsasser chart this assumption 
would mean that the w-lines would be rectilinear and 
convergent to the point where 7 =OK. If, in addition, 
we approximate the abscissa scale X of the Moller chart 
with any convenient function, new problems can be 
solved. 
4. (Note added July, 1950.) In his newest publication, Rob- 
binson [44] also comes to the conclusion that radiation processes 
are very important near the earth’s surface. 
