322 
ular diffusion, \q and ¢4 correspond to the mean free 
path and the average velocity of molecular motions. 
In eddy diffusion, Ap is a length fixing the scale of 
turbulent displacements (mixing length), and ¢p is a 
velocity fixing the speed of turbulent motions. 
The differences between molecular and eddy diffusion 
are more obvious in terms of \ and ¢ than in terms of 
d and D. The quantity ¢4 corresponds to the internal 
energy or heat of a gas, while the energy for maintaining 
¢> will be taken from external sources, such as the 
potential energy of the horizontal pressure distribution. 
The term Xa is a unique function of the density of the 
atmosphere when Ap depends on geometrical conditions 
like distances from bounding surfaces; the vertical mix- 
ing length is also affected by the vertical thermal 
stratification. 
The value of D can equal that of d when Xa/Apn = 
€p/¢a ; however, this ratio is never unity. Differences 
between the mechanisms of diffusion are emphasized 
when time terms (A/¢) and acceleration terms (¢?/\) are 
studied. Reasonable conditions—such as Aa,p < 2, \v/£p 
< 1 day, and ¢p/X» < g when g = gravity—narrow a 
priori the variability of D-values. The diffusion dia- 
gram, Fig. 1, summarizes our present knowledge of 
X (CM) 
10° 10° 107 10° 102 10! 1 10! 102 103 10% 10° 108 10" 108 102 10!° 3 
\ \ \ | \ \ \ \ WH, 
N y 5 Gris AN [Sas 
ADS NIN| Gy 
2 N S Oz %S, 
\ NI) ZA) SS, 
NIXIN 
ae 4 
De te Ww O% 
\ x |x 
N MM SN hence 
™ ‘ 4 
\, aN S Oy 
AY Ni SK Cy 
BY SS X IN eo 
10 AY Kun loi aS ee eal 7 
Y SEX NINA XIN NINI& 
0,7 7 G2 03 3 G5 Oo > 
D AT HEIGHT z IS DENOTED BY CHARACTERISTIC AREAS ON THE DIFFUSION 
DIAGRAM: 
== |-10 KM FOR ORDINARY TURBULENCE $322 0-1 KM 
25 1-10 KM FOR CUMULUS CONVECTION ++ 25-35 KM 
& 1-10 KM FOR CUMULONIMBUS CONVEGTION -=- 45-80KM 
2 10-25 ,35-45 AND 80-100 KM 
Xx HORIZONTAL GROSS-AUSTAUSCH OF THE GENERAL CIRCULATION 
Fig. 1.—Diffusion diagram. Each point of the X, ¢ plane 
determines a diffusion coefficient (em? sec~!). In molecular 
diffusion, \ = free path and ¢ = mean molecular speed; d = 
AE is fixed by the density and temperature of the atmos- 
phere; consequently, the height variation of dis marked by a 
curve. In eddy diffusion, \ = mixing length and ¢ = mixing 
velocity; owing to the variability of these elements, D = A¢ 
and its variation with height are denoted by characteristic 
areas when the possible variability of D is narrowed by the 
consideration of limiting values of eddy accelerations (¢2/)) 
and time terms (A/¢). 
molecular and vertical eddy-diffusion coefficients in the 
atmosphere. Values of D above 15 km and values of d 
above 100 km are not very reliable. Inasmuch as 
temperature and density are known, d;y, is a unique 
function of height. However, D will vary with height, 
weather, season, and latitude. Inasmuch as D is a 
statistical parameter, it will depend on the averaging 
process when \p/fp is a measure of the minimum time 
THE UPPER ATMOSPHERE 
interval required for the definition of representative 
values of wind and concentration >, . 
4. The Equation of Atmospheric Diffusion. Let us 
define an atmospheric gas as “permanent” when there 
are no sources or sinks, as “neutral”? when gravity is 
the only force acting on the molecules, and “at rest,” 
when the vertical pressure distribution follows from 
the hydrostatic equation, dp,/dz = —n,m.g. Let molecu- 
lar nitrogen, as the medium of atmospheric diffusion, 
be a permanent, neutral gas at rest. Then, in equation 
(1),Fw. = —gk and cy,-k = k-V = 0. However, 
k-V £0. 
Other gases may be nonpermanent and ionized. The 
most important assumption is vy; < 1. Then, the density 
distribution in the N.-atmosphere is not affected by 
the processes of diffusion and turbulence: vy, ~ 1, 
Nn, YN, My, SY Mm, un, & O. Since diffusion equals 
diffusion-velocity times the number of diffusing mole- 
cules per cubic centimeter, the equation of atmospheric 
diffusion follows from (1), (8), and (9), 
Nes = Ns(Ca + Cp + Cr + Can V) 
ath (15) 
= OH ap (81) a> hs 
when Cz is neglected and 
Ca = —dsw.(Vvs)/vs ) (16) 
Cp = ds. MsV In p, (17) 
Cr = —d,y, m:(F; — gk)/kT = —d,y, mf./kT, (18) 
OS Wel. (19) 
Let us define as an auxiliary parameter 
G3 2 (20) 
6 (ds Ti) 
With the aid of (15)—(20), the fundamental equation 
of atmospheric diffusion becomes 
= atte fos 
Ns Cs = ndswa) Q, 's| Ms V(n p) kT (21) 
+ nv.V. 
The equation of continuity is 
Ons 
ot 
ay (domo s ae eon, 
cz V: (n;C.) —— (2 ar oy Se 31) (22) 
Steady states or diffusion equilibriums are defined by 
dn;/dt = 0, unsteady states by On./dt ~ 0. 
In the upper atmosphere, diffusion acts mainly in the 
vertical. Under the assumption that n.c, = yk, 0v;/d% = 
dv./dy = 0, £5 = fak, 0 In p/dz = —mg/kT, andk-V = 
0, when V-k + 0, equation (21) yields a differential 
equation for v, where height is the independent variable 
and the coefficients of the equation are fixed by molecu- 
