330 
tion of density and temperature; the coefficients of 
eddy diffusion depend on horizontal and vertical gra- 
dients of pressure and temperature. These elements 
will vary at surfaces of constant height in the upper 
atmosphere while diurnal, interdiurnal, and seasonal 
variations will be different at different latitudes. How- 
ever, apart from the troposphere and substratosphere, 
data are scanty and the effects of meridional variations 
of d and D on the composition of the upper atmosphere 
are hypothetical. 
In the discussion of diffusion in the vertical, under 
steady conditions, large-scale external area sources and 
sinks were considered which covered the entire earth’s 
surface (e.g., helium) or considerable parts thereof (¢.g., 
carbon dioxide). External and internal point or line 
sources can exist (e.g., meteor trails); production and 
destruction of matter may be instantaneous, continu- 
ous, or periodical as exemplified by volcanic eruptions, 
lithospheric helium production, or photosynthesis. As 
demonstrated by the first and second versus the third 
of these examples, the process can or cannot be inde- 
pendent of the state of the atmosphere. It must be 
concluded from the above that diffusion in general is 
not only a three-dimensional, but a time-varying proc- 
ess. Solutions of equations (15), (22), and (63) must be 
found which satisfy appropriate initial and boundary 
conditions. 
13. The Horizontal Coefficients of Eddy Diffusion 
and the Effect of the Representative Wind Field on 
Diffusion Processes. In contrast to molecular diffusion, 
the coefficients of eddy diffusion at a fixed point may be 
different in horizontal and vertical directions; with re- 
gard to equations (12)—(14), D.,, # D. Another impor- 
tant factor is that the eddy-diffusion coefficients depend 
on the averaging process. 
Defant [6] considered the traveling cyclones and anti- 
cyclones of middle latitudes as individual turbulent 
elements in an intensive meridional mixing process 
where ¢?) = 10° em sec — and \%”” > 108 cm are hori- 
zontal values; thus, DS = ¢{ \{? = 10” cm? sec ~ 
(cof. Fig. 1). The magnitude of the virtual time-terms, 
r»Y?/¢S? > 10° sec, manifests the differences of D{?? 
in comparison with D,,, due to normal turbulence, 
since \p/fp = 10 to 10’ sec. Representative with regard 
to DY are monthly means; with regard to D, hourly 
means. The coefficient D{’) effectively equalizes the 
meridional differences of physical properties and com- 
position; however, very little is known about D{?? in 
the upper atmosphere. 
Even in normal turbulence, D, or D, may differ from 
D, denoting nonisotropic turbulence caused by thermal 
stratification [22]. In the surface layer, the isotropy 
coefficient deviates only slightly from unity and it can 
be expected that throughout the entire atmosphere 
D, and D, have the same order of magnitude as D. 
Another important consequence results when the me- 
dium of atmospheric turbulence is not “‘at rest” (cf. 
§4). In reality, k- V equals zero for the atmosphere as 
a whole. The term k-V and the components of V- V 
and V are functions of height, latitude, and longitude, 
since hemispheric circulation cells are superimposed on 
THE UPPER ATMOSPHERE 
the general zonal motions. The most general equation 
of three-dimensional diffusion is equation (63) or 
Ons 
ot 
= —V-lne(Ca > Cp a cr | Cz V)| 
Ge ae 
which can be expressed by the individual time variation 
dn; 
dt 
(96) 
Ons 
== + V-Vnz; = —V-[ne(ca + Cp + cr + C,)] 
— nV-V + gS +48, (97) 
where gS and ¢®™ and the velocity terms as defined 
earlier (see §§2-4) are functions of space and time. 
Lettau [23] found that the term V-(n,C;) depends on 
the wind shear and that the effects of local wind deriva- 
tives increase with time and with the geometrical dimen- 
sions of the air mass under consideration. This explains 
why Fick’s simple equation of diffusion 
dns 
ae V-(KVns) (98) 
or 
Wily aoe, 
Ta K*V-Vns (99) 
results in a ‘diffusion power” K or K* which must 
depend on arbitrary physical parameters, such as time 
of the diffusion process and the geometrical dimensions 
of the distances under consideration. Richardson [82] 
and Stommel [86] maintain the point of view that the 
diffusion coefficient is a function of distance. 
Observations of moisture variations along isentropic 
surfaces in large-scale tropospheric currents to the north 
or to the south led Rossby [33] and Grimminger [13] to 
the concept of lateral diffusion processes where the 
effective coefficients D{") = 10° em” sec" are such that 
DS = 10° D& = 10! Dz, . Lettau [23] considered, as 
a criterion of real coefficients of eddy diffusion, that 
D = pAp when fp and Xp can be statistically related 
with observable properties of irregular motions and 
depend on physical parameters dealing with the energy 
and the heat transfer of the flow. The term D{*) does 
not satisfy the requirements of a real diffusion coeffi- 
cient. Consequently, Lettau [23] termed it an “appar- 
ent” coefficient which may be caused by steady de- 
formation fields superimposed on the mean current. 
This explanation appears to be supported by the find- 
ings of Miller [25] that a strong positive correlation 
exists between the vertical wind velocity and the mean 
speed of the south-to-north components of the air flow, 
and the finding of Grimminger [13] that there is a definite 
indication that the coefficients of lateral mixing increase 
downstream. 
14. Examples of Three-Dimensional Diffusion. All 
terms of equation (96) were accounted for in Diitsch’s 
study of ozone; the only simplification of (96) was that 
Cr = 0. Diitsch [8] systematically investigated the 
Yo,-equilibrium in vertical columns. Direct radiation 
and also direct plus diffuse radiation explained only 
