DIFFUSION IN THE UPPER ATMOSPHERE 
where (n;); refers to the initial state when (c,){; is the 
vertical diffusion velocity as fixed by the final diffusion 
conditions (Q;:) and the initial concentration-distribu- 
tion (y;);. The changes of both », and c; with the prog- 
ress of the transition are not considered; therefore, 
At, in equation (91) represents a minimum value of the 
transition period. 
Expressions more accurate than equation (91) could 
be obtained; however, the gain in accuracy does not 
justify the added complexity of computation in view 
of the crude assumptions that @ is independent of 
height and changes its value suddenly and simulta- 
neously throughout the entire atmosphere. 
With the aid of (87) and (91), 
H = = 
te = a1 fu.) Ose ms @us — Q)z/H) — 1}..(92) 
Let us investigate the three special transformations: 
A 
(a) complete mixture (Q: = 0) > as 
Dalton atmosphere (Q;; = 1) 
(b) Dalton atmosphere (Q; = 1) > 
partial separation (Q;; = «) We) 
(c) partial separation (Qi = 4) >  _ 
partial separation (Q;; = e), 
when 0 < e < « << 1. The diffusion velocity follows 
from (15)—(19) and from the special assumptions above, 
(a) (Giz = 
(0) (cs) is  Ca/€ 
(c) (or FZ Ca €/ € 4 
As an abbreviation, we define Ar, = H’/us (dswo)o; 
then equation (92) yields 
— exp (~4.2/H)] 
<= Gas ye" = o 
(94) 
(a) At, = Ar,e 7" [1 
(WAN 
TLE in: 
ele: = €2) Me® 21a 
a Jel 
e*Texp (uz/H) — 1] (95) 
(c) Ati; & At; 
The transition period is zero at the lower and upper 
boundary of the atmosphere, thus attaining a maximum 
value at the level defined above, that is, at 2 = 2. 
Figure 4 illustrates the conditions for argon when in 
(0), €1 = 1% X 10; that is, Dy) = 0.54 X 10° em’ sec 
or A = 65g cm ‘sec ; and in (c), e remains as before, 
& = 0.30 X 107°; that is, Do = 0.60 X 10° em* sec or 
A = 72¢cem ‘sec. 
For argon, 2* = 6.7 km in (a) and (0). At this level, 
the transformation of a complete argon-nitrogen mix- 
ture into a Dalton atmosphere takes at least 34,000 
years. The reverse transformation of a Dalton atmos- 
phere into a steady state of partial separation charac- 
terized by a constant austausch coefficient of 65 g cm | 
sec takes at least 1.5 months. The changes owing to 
329 
a 10 per cent variation of austausch of the order given 
above are accomplished in not less than 5 days. 
In ease (a) the transition period above 200 km is less 
than one hour whereas it is measured by years below 
100 km. Evidently, very slight turbulence will prevent 
At, 
SECONDS HOURS DAYS YEARS 
oy] 1000 100,000 
1 10 100| 1 Of © HO) Tf eo | 10,000 
200 A —_— — 
ARGON, [Lg 70.42 
150 t 
Q;.=1 
100 oie 
= 
x“ 50 
N 
25 ial 
HEIGHT 
6 
T 
b OI 
wt 
qm o 
@) ¢ 
Oo. 
Sato 
SS 
32 £9) 
=p 10 
=) @ 
KS) 
oon 
a 
ey 
o) 
OOONE i 
(e) 2 8B 
I 
i 
4 5 6 7 8 9 fi 2 Is 
LOG At, / SEC 
Fic. 4.—The height variation of the minimum values of the 
transition period A¢, in three different cases of initial and final 
diffusion conditions. The computation was carried out for 
argon; other gases will have different values for At, and for 
z*, the height of maximum transition period (see Fig. 3), but 
the order of At, will not differ from the above. The height scale 
is proportional to 21/3. 
the establishment of a Dalton atmosphere below 100 km 
when the effects of occasionally enlarged turbulence 
will be extinguished after less than one minute above 
200 km. This result supports the findings concerning 
the height of the datum level (see §7). 
When similar computations are based on more accu- 
rate assumptions (dQ/dz # 0), the results might be 
interesting with regard to seasonal and long-time varia- 
tions of turbulence, or with regard to the intensity 
of the general circulation and its effect on the composi- 
tion of the upper atmosphere. Time variations due to 
variable rates of production of gases should also be 
investigated with the aid of equations (63) and (73). 
However, the effect of natural or artificial external 
sources on the composition of the atmosphere will be 
very small and measured only within geological epochs. 
THREE-DIMENSIONAL DIFFUSION 
12. The General Causes of Three-Dimensional Dif- 
fusion. As a result of the earth’s shape, the use of 
spherical coordinates (longitude, latitude, height) is 
natural. Atmospheric diffusion is a three-dimensional 
problem when the physical state of the atmosphere 
and/or the strength of sources and sinks depends on 
height, latitude, and longitude. As indicated by equa- 
tion (4), the coefficient of molecular diffusion is a fune- 
