DIFFUSION IN THE UPPER ATMOSPHERE 327 
In steady states without sources or sinks, © = 0; thus, 
c;;k = 0; whenk-V = 0, 
0) = =¢s = D= =". (58) 
Vi 02 
= mexp(—[ Sa (59) 
vi = Vi9 EXP | 5 %)- 
By definition, »; = ni/ny, = ni/n, when 
ie = mye (60) 
if T = To and g = g . The number of particles per cubic 
centimeter therefore is 
N; = Nin EXP (-7/ at ete), 
D. = cH. (62) 
4 
(61) 
where 
Equation (58) shows that the vertical concentration 
gradient is everywhere negative. The term In »; de- 
creases in the ratio of D-/D per 8 km. In layers of high 
turbulence, v; will be practically constant when n; 
decreases proportionally to the density of the atmos- 
phere. When dD/oz < 0, »; must decrease rapidly 
with height. Such effects might be the cause of “dust 
horizons.” With regard to Fig. 2, one would expect 
“dust horizons” at approximately 15 and 70 km, pro- 
vided dust particles exist in the stratosphere. The 
optical phenomena due to scattering of solar radiation 
prove the presence of dust in the stratosphere. With | 
reference to Gutenberg [14], the duration of civil and 
astronomical twilight leads to the assumption of dust 
boundaries at approximately 15 and 60-80 km. 
Difficulties arise with regard to the rate of descent of 
atmospheric dust particles. Stokes’ law, equation (28), 
is true for small spheres when Reynolds’ number— 
defined as Re = ajc;p/u—is smaller than a fixed value. 
In air at NTP, u/p = 0.2 cm’ sec ’; therefore a; must 
be smaller than 10 ” em. More accurate is the formula 
of Oseen, which is valid also for a; > 10 ~ em. Both of 
these aerodynamic formulas fail for nonspherical parti- 
cles and for values of a; of the order of the free path of 
air molecules (10 * em at z = 20 km and 10° cm at 
z = 60 km). Humphreys [18] considered Cunning- 
ham’s corrections of Stokes’ law as applied to the strato- 
spheric conditions. 
As an example, we assume c; = 0.063 em sec | & 
50m day ‘= 20km yr ‘ corresponding to a; = 10 *cm 
when, in equation (28), p; = 2. Such a;- and p;-values 
correspond to volcanic dust particles which can exist 
in varying amounts in the stratosphere (see §14). Equa- 
tion (62) then yields D- = 5 X 10* em’ sec ', which 
equals normal D-values in the troposphere when, in the 
stratosphere, D is alternately larger and smaller than 
5 X 10’. The study of optical phenomena appears to be 
useful for the discussion of turbulence in the strato- 
sphere. 
DIFFUSION IN THE VERTICAL—UNSTEADY 
CONDITIONS 
10. The General Equation. In unsteady states, 
V-(ns¢s) # 0 in equation (15). For nonpermanent 
gases the equation of continuity yields 
Ons 
ae Ne (oG) oP ay? dog&. 
(63) 
For the solution of (15) with regard to (63), we assume 
the gas to be permanent (q§” = 0, qS~ = 0), and 
(a) the No-atmosphere to be steady and at rest when 
ve Kl, 
On _ Onn, _ Ons _ Ov: ee 
ae ae ee ae SN SO, GS) 
(6) fT = T) , andg = m, 
0 In P it il si —2/H 
ee cE? ih = hae (65) 
(c) diffusion to be vertical, 
Ns Cs = vk, We (ns Cs) = oh oy (66) 
Oz 
(d) gravity to be the only external acceleration, 
i, = 0, (67) 
When equations (64)—(67) are considered, the differ- 
entiation of (23) with respect to z yields 
is _ Chin [Or 9 Wal/me@, dia 3) 
ot Q, E 1 dz ( H dz » (|) 
Os => (dsx a Doe a a2 a a : = ie ~ = 
(69) 
Let Q, be constant throughout the layer under con- 
sideration, Q; = Q. Then D is proportional to dws, 
or D varies with height in the same way as does 1/p. 
Consequently, the product pD (usually called the aus- 
tausch coefficient A) is dependent of elevation. We 
introduce a new variable Z, 
e— Ca (70) 
such that 
sw. = (dans)o@ = (dena) (71) 
For steady cases, equation (46) yields 
Py A (72) 
With the aid of equations (68)—(71), 
St ae 
