326 
result for a Dalton atmosphere; Lettau [21], for the 
general case 0 S @ S 1. 
Paneth and Glueckauf [27] showed that vz. between 
15 and 25 km is slightly larger than at the ground. In 
contrast to the oxygen deficit measured by Regener, the 
helium surplus is not a systematic height function. It is 
relatively small, so that on the scales of Fig. 2, vx.- 
averages for layers of 2.5 km are practically constant. 
The ground concentration is (yze)o = 5.2 X 10 °. The 
assumption that (52) is satisfied yields 
ay, (Vite) ob He diten»/H 
= 3.3 X 10 ’ cm*He cm ~ sec + 
(y0/2) re = 
(54) 
? 
or, integrated over the entire earth’s surface (5.1 X 
105 cm’), 
(yo/n) ze = 17 m* He sec. (55) 
By multiplying (54) with Loschmidt’s number (Table 
I), the units of yo are number of molecules per square 
centimeter per second. Helium is discharged from rocks 
which contain uranium or thorium. Gutenberg [14] 
discussed the total lithospheric helium production and 
estimated its order of magnitude as 10 m* see *, which is 
sufficiently close to (55) in order to justify the assump- 
tion that yo © Yerit. Consequently, helium cannot 
increase with height, even in the Dalton atmosphere 
above 200 km. This would correspond to the absence 
of helium lines in the spectra of the aurora and of the 
night sky. Similar conditions will exist for hydrogen. 
The departure into space at the same steady rate with 
which these lightest gases leave the ground appears 
consistent with Jeans’ theory of escape. 
When 7 ~ Yerit , equations (83) and (84) must be 
studied. Lettau [21] considered a small dvy./dz between 
15 and 25 km and corrected the result (55) slightly. 
When 7 differs considerably from yet, the concentra- 
tion will vary, even in layers where @ is very small. 
Examples of such nonpermanent gases are carbon diox- 
ide (produced at the surface over land), ozone (pro- 
duced in the lower stratosphere), and atomic oxygen 
(produced in the upper stratosphere), when the regions 
of production are left outside the layer under considera- 
tion. The concentration decreases with increasing dis- 
tance from the region of production. The intensity of 
continuous sources can be computed when dy,/dz is 
steady and Q is known. As an example, the CO,-observa- 
tions of Glueckauf [11] in Great Britain, vo = 3.4 X 
10)* and », © 2.5 X 10 * at 2 ~ 7 km, yield 
(10/2) co, % —(vs — v0)D/z 
56) 
= 5 X 10 > cem* CO, cm sec, \ 
for D = 5 X 10* em’ sec’ and dcow, K D. In all 
probability, this rate of CO»-diffusion holds for in- 
dustrialized areas only. For central Europe, Lettau 
[21] derived 10° em* CO, em™ sec from Wigand’s 
observations. A compensating negative flux of carbon 
dioxide will exist in the troposphere over the oceans. 
THE UPPER ATMOSPHERE 
In a similar manner, the average downward-directed 
diffusion of ozone through the troposphere was found 
to be approximately 10 ° em* 0; em ~ sec = 4 X 10° 
molecules em ~ see *; Diitsch [8] estimated (yo)o, as 
7.1 X 10", 2.6 X 10° and 0.4 X 10° molecules em” 
see * at latitudes 0°, 45°, and 80°N, respectively. 
The problem of diffusion equilibrium becomes more 
complex when dy/dz # 0 at certain regions of the layer 
under consideration (cf. equation (27)). Then the main 
difficulties arise from the mathematical analysis of the 
rate of production g‘“” and extinction q‘’ as functions 
of height. These processes will depend on the physical 
state of the atmosphere and on solar radiation. 
With regard to equations (23) and (27), the classical 
model of such investigations will be the distribution of 
ozone. The processes of production and extinction are 
fairly well known and localized in the lower stratosphere 
and at ground level. The requirements that N» can be 
considered the medium of diffusion and that y, 1s small 
are satisfied. Diitsch [8] studied the subject compre- 
hensively and found the effects of molecular diffusion 
small in comparison with those of eddy diffusion. How- 
ever, the vertical ozone distribution is not in equilib- 
rium, that is, the main problem is the explanation of 
time variations at different heights and latitudes due to 
meridional and seasonal differences of g‘” and gq“, 
eddy diffusion, and general circulation. Consequently, 
Diitsch’s results will be dealt with im a discussion of 
three-dimensional diffusion. 
Layers above the stratosphere are characterized by 
the production and destruction of ions due to radiation 
and recombination. According to the theory of ioniza- 
tion, g‘” is a complicated function of height and the 
sun’s zenith distance when g‘° is proportional to the 
square of the number density of ions. It must be men- 
tioned that the assumptions made in Section 4 do not 
hold, especially when Nz is also ionized; the number 
density of ions can become of the same order as that of 
the molecules. Therefore, equation (1) rather than equa- 
tion (15) must be considered in connection with q‘” 
and g~’, and the diffusion coefficient of ionized matter 
must be properly defined. Ferraro [10] found that the 
effect of molecular diffusion is negligible m the E- and 
F,-layers, and small and possibly negligible also in the 
F,-layer. Similarly, Bagge [1] showed that molecular 
diffusion may influence the number density of ions above 
200 km. The effect of eddy diffusion was not considered ; 
when reference is made to Section 7, it follows that eddy 
diffusion should be considered in the E-layer and for the 
distribution of atomic oxygen around 100 km. 
Since the diffusion problems of dissociated and ion- 
ized gases are highly complex and decisively determined 
by forced oscillations due to solar radiation, they will 
not be studied in this article. 
9. Diffusion of Particulate Matter. Let »; be the 
number concentration of particles and let us consider 
horizontal uniform »;-distributions. Then diffusion is 
in the vertical and in (29) 
nec; = Tk. (57) 
