where © 
i<) 
4H°Q/(dsw.)0 = const, 
2usQ = const. 
(74) 
(75) 
Equations (68), (69), and (73) prove that the form 
dv,/dt = K*dv,/dz°—usually called Fick’s equation of 
vertical diffusion when K* defines the ‘diffusion power” 
of the medium—is an inadequate expression of atmos- 
pheric diffusion. The differential equation (73) is solved 
by Bessel functions, and the solution expresses the 
concentration as a function of z and (£ — t) when 
allowance is made for the appropriate boundary condi- 
tions of the problem. For example, let Q = Q; att < to, 
and Q = Q;; at t = t ; then, according to (72), 
Vy = (vs0) ae a 
i=) 
Ih 
i 
ll 
att <b, (76) 
when 
lim vs = Op)? 88 (77) 
(t — to) 00 
such that the total amount of s-molecules in vertical 
columns remains constant. 
11. The Time Required for Establishing Equilibrium. 
The “transition period” is the time needed to transform 
state (76) into (77). The special transformation, where 
Q; = 0 (maximum mixture) and Q;; = 1 (Dalton 
atmosphere), was studied by Epstein [9] in terms of 
Bessel functions, and by Maris [24] and Mitra and 
Raksit [26] with the aid of numerical and graphical 
integrations. Lettau [21] derived (68) and (73) in the 
forms given above, which allow us to discuss more 
general cases where 
05 Q: < Qi <1. (78) 
Since the gas is permanent, the total amount of s- 
molecules in vertical columns of infinite height must be 
constant, 
Vv, = [ n, dz = const. (79) 
0 
We define the number integral of s-molecules as 
t= [node limp =%. (80) 
0 Z00 
Since n, = v.n, it follows from (65) and (72) that 
Ns = Nso exp[— (1 sr usQ)z/H], (81) 
and 
es (nso = ns)H ae Noll = 
Uy = SaSaoe : Ws = Tae oe = const. (82) 
Equation (82) relates mo to Q. Therefore, a general 
value of the ground concentration is defined: 
Nsoo = W,/ HZ = veopNo - 
(83) 
Then, 
Nso = Msoo(1 ar usQ); 
Ns00 a ore A 
= ag [—( + usQ)z/H], (84) 
(vs) ; = pO, exp (—p.Q:2/H), 
1 =F MsQi (8) 
THE UPPER ATMOSPHERE 
(W:): = Hneo{1 — exp [—(1 + u.Q.)2/H]}. (86) 
When diffusion (initially characterized by Q;) is Qi; 
at t 2 to, it follows from (86) that the difference of the 
number integrals is a height function: 
(We) = (Ws) e = Ay, 
= Hnswe [exp (—usQ.z/H) > exp (=usQi2/H)). 
: (87) 
The term Ay, is zero at 2 = 0 and at 2 = ©; conse- 
quently, Ay, must have an extreme value at a finite 
level, 2 = z*. The level z* is defined by dAy,/dz = 0. 
The differentiation of (87) with respect to z yields 
H 1+ wsQii 
2° = —— — In =~, 88 
Hs (Qs = Q:) 1 ar MsQ): ( ) 
and 
5 a Bee eer 
lim 2* = lim z* = — In (1 + 4,), 
(@ii—Qi) 1 i) Ms (89) 
Qi 70 
lim z* = ZH. (90) 
(Qii—Qi) 70 
By definition, n, = oy,/dz; therefore, 2* is the level 
where (;); = (ms) ii or (vs); = (vs) xx ; that is, where the 
initial y.-curve intersects the final »,-curve (compare 
equations (76) and (77) and, as an illustration, Fig. 3). 
100 = 
is =(I+H5@)Z/H 
S—=(1+p.a)e 
"soo 
80 | 40 
H-g= ~ 0.86 
S (HELIUM) Ss 
= = 30 
N N 
E = 
© 2 2 
Ww ra) 
ae 35 
10 
H— 
Q=0 
0 (0) i 
0 0.2 0.4 0.6 0.8 1.0 0 0.2 04 06 08 10 1.2 14 
"s/ "soo 
"5/"soo 
Fig. 3.—The height variation of density (relative to the 
ground density) of a light and a heavy gas for the border cases 
of maximum mixture (Q = 0), and of maximum separation or 
Dalton atmosphere (Q = 1). It is assumed that the gases 
have no sources and sinks. 
Our result, equation (90), corresponds to the fact that 
z = H represents the isopyenic level which was dis- 
covered by Wagner and explained by Linke (see Hum- 
phreys [18] and Doporto and Morgan [7]). 
The transition period has no finite value since (5): 
will be approached asymptotically even if Q; changes 
suddenly into Q,; at t = &. The transition period can 
be studied only in relative terms. Epstein and Maris 
measured the transition period by the time necessary 
for accomplishing a certain fraction of the whole trans- 
formation. Lettau [21] Gonsidered the final time interval 
Ay; 
BS as (91) 
