DIFFUSION IN THE UPPER ATMOSPHERE 
By HEINZ LETTAU 
Geophysics Research Division of the Air Force Cambridge Research Center 
FUNDAMENTALS OF THE THEORY OF 
DIFFUSION 
1. General Remarks. The kinetic theory of gases de- 
fines diffusion as the average motion of selected mole- 
cules relative to other molecules. The physical units of 
diffusion are number per square centimeter per second, 
that is, diffusion velocity times number of selected 
molecules per cubic centimeter. Diffusion depends on 
the composition of the gaseous mixture. A classical 
model considers two sets of molecules of different mass, 
effective diameter, and velocity distribution. The theory 
of diffusion results in complicated equations, even in 
the simple case of binary mixtures in a closed system 
when the physical state is well defined. 
Pure, dry air is more complex than a binary mixture. 
Water (vapor, liquid, solid) and particulate matter 
(nuclei, smoke, dust) complicate the composition of the 
atmosphere. The atmosphere is not a closed system and 
the physical state beyond the scope of soundings is 
uncertain in many respects [35]. Sources! and sinks? of 
constituents must be considered. Large- and small-scale 
atmospheric motions produce eddy diffusion. 
Diffusion equilibrium results when molecular and 
eddy diffusion balance the effects of forces acting on 
the molecules or the effects of sources and sinks so that 
the composition is a steady function of height or of 
height and the horizontal coordmates. Time variations 
of composition result from variations of the physical 
state—especially of motion and turbulence—and from 
intensity variations of sources and sinks of certain 
constituents. 
Because of the scarcity of direct observations, the 
problems of diffusion in the upper atmosphere were 
developed theoretically and on the basis of conjecture. 
In this article, the author has tried to point out the 
inadequacies of the field by thoroughly outlining the 
assumptions necessary for a mathematical analysis of 
atmospheric diffusion. In general, applications of the 
theory must be confined to the lowest 200 km. Future 
work will be more promising when theoretical research 
can be supported by more and better observations 
from the stratosphere and ionosphere. 
2. Molecular Diffusion. The study of nonuniform 
gases by Chapman and Cowling [4] resulted in the 
following general equation of diffusion velocity in a 
binary gas mixture: 
V 
C — & = —dp 2 — (1 — p)V In p 
142 x (1) 
By RE een) erp 
mkT Vi V2 
1. Gas and particle production of the lithosphere and hydro- 
sphere, volcanic activity, photochemical reactions, industrial 
processes, etc. 
2. Outflow of light gases into space, condensation, sedimen- 
tation, coagulation, recombination. etc. 
The symbols in equation (1) are explained in Table I. 
Tas_eE I. List or SymBous 
The subscript s denotes one constituent of the mixture; 
for a binary mixture, s = 1 and 2. The rectangular coordinates 
Z, y, and z are oriented so that z is positive towards the zenith. 
The unit vectors i, j, and k are in the directions z, y, and z. 
Constituent gases 
Cs Ceri + Csyj + c.2k = vector of mean molecular motion 
number of molecules per cm? 
mass of the molecule 
IS) 
Hv ww wa 
; hydrostatic partial pressure 
F, external acceleration acting on the molecules 
dp coefiicient of thermal diffusion 
dss = coefficient of self-diffusion 
Gaseous mixture 
dy, = dx, = coefficient of mutual diffusion 
n = Dn; = total number of molecules per em? 
(Losehmidt’s number) 
= 2.705 X 10" at NTP 
NTP = normal temperature and pressure 
hydrostatic pressure; normal pressure = 1013 mb 
n./n = number concentration 
Dv.m; = mass of a fictitious average molecule 
Vs 
QS 
Hoi il 
absolute temperature; normal temperature = 273K 
v 
p = nm = density of the mixture 
Ls = (m, — m)/m = relative weight factor 
k = 1.372 X 10 ergs per degree = Boltzmann’s constant 
Fundamental equations are Dalton’s law: 
= 905 5 (2) 
and the equation of state: 
ps = nk or p = nkT. ; (8) 
Equation (1) shows that the vector of diffusion veloc- 
ity Cc: — Cc. has four constituents, 
C1 — Co = Ca + Cp + Cr + Cr, 
where cz is ordinary diffusion velocity, 
C, is pressure diffusion velocity, 
Cr is forced diffusion velocity, 
Cr is thermal diffusion velocity. 
Experiments are usually based on ordinary diffusion 
due to initial nonuniform composition in a closed sys- 
tem, Vv. ~ 0, when c,, cr, and cr are neglected. In 
the atmosphere, pressure diffusion (an indirect effect 
of gravity since the direct effect of gravitational acceler- 
ation on all molecules is the same) must be considered; 
c, is due to the gradient of In p set up by gravity and/or 
rotation in compressible media. The most important 
example of forced diffusion is the effect of electric 
fields on the motion of ionized gases. Thermal diffusion 
results when different parts of the mixture are main- 
tained at different temperatures; however, the degree 
of separation produced by cr is small; experience has 
proven that dr/di. 0.1 and decreases when m2/7% 
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