502 
trations are to be made to agree with the mathematical 
solution. There are obvious difficulties in ascribing 
physical reality to a quantity whose value depends on 
the distance of the sampling point from an arbitrary 
point in space and any variation in K must be regarded, 
more rationally, as an expression of the fact that (as 
Richardson puts it) in a turbulent fluid the average 
rate of separation of a pair of marked particles is a 
function of their distance apart, a type of diffusion 
quite unlike that contemplated in the kinetic theory of 
gases. In particular, it does not seem possible to assign 
any upper limit to the ‘‘size’’ of eddy which can take 
part in atmospheric diffusion, and there is no definite 
“scale” of atmospheric turbulence. 
The most natural and acceptable form of variation 
of K is to allow the coefficient to increase with the 
depth of the layer under consideration. The mathe- 
matical technique required for the solution of diffusion 
problems is considerably simplified if the interchange 
coefficients are assumed to vary as a simple power of 
the height. The problem of the approach of the wind 
in the friction layer to the geostrophic velocity has been 
dealt with very thoroughly by Kéhler [33] for the case 
Kaz", m = 0. The solution involves Bessel functions 
and the results show a fair measure of agreement with 
observation. Rossby and Montgomery have also dis- 
cussed the same problem by making the reasonable 
assumption that the mixing length (and hence the eddy 
viscosity) first mereases linearly with height in a rela- 
tively shallow surface layer (thickness about 100 m) 
and then decreases slowly so as to allow a small “re- 
sidual turbulence” at the top of the friction layer. 
In the problem of heat transfer Brunt [4], following 
the lines of Taylor’s early work, has shown that if 7 
be the absolute temperature of the air at height z, the 
equation of eddy conduction is 
oT 0 oT 
Pet 2 {xe (4 ali rh, 
where I is the adiabatic lapse rate. This implies that 
the net turbulent flux of heat across a horizontal sur- 
face is proportional to the difference between the exist- 
ing lapse rate and the dry-adiabatic lapse rate, so that 
in a stable atmosphere, the flow of heat due to eddy 
motion is downwards, and the ultimate effect of mixing 
is to produce an atmosphere with constant lapse rate 
equal to the adiabatic value. This equation and the 
conclusions drawn from it have been the subject of 
much discussion in recent years, but this aspect will be 
dealt with at greater length in the section dealing with 
convection. 
From the purely mathematical aspect the equation 
of conduction with K variable has been well explored; 
Haurwitz [24] has given the solution for p = constant, 
T=7+ Bsngt onz = 0, K = a + az(a,u 
constant) in terms of ker and kei functions, and Kéhler 
[33] has published a very thorough treatment of the 
case in which K « z”, m = 0, the solution in this case 
being expressed in terms of Bessel functions of imagi- 
nary argument. The implications of this work are con- 
sidered later. 
DYNAMICS OF THE ATMOSPHERE 
Turbulence in the General Circulation. In 1921, 
Defant [13] introduced a picture of the depressions and 
anticyclones of the synoptic meteorologist as ‘‘eddies”’ 
in the general circulation, a theory which leads to the 
concept of Grossturbulenz (Lettau), the transport of 
heat, momentum and water vapour on a planetary 
scale. A clear account of this work is to be found in 
Chapter XII of Lettau’s Atmosphdrische Turbulenz. 
Lettau adopts the ordinary interchange coefficient the- 
ory of turbulent transport and by consideration of the 
deviations from the geostrophic wind finds, for exam- 
ple, that the meridional component of the Grossaus- 
tausch is of the order of 10° g cm™ sec, with a “mixing 
length” of the order of several degrees of longitude. 
A very different approach, not involying mixing- 
length concepts is that given by Priestley [46] in a 
recent paper in which he develops a method whereby 
the meridional flux of heat, water vapour and momen- 
tum can be evaluated from upper-air soundings. From 
a preliminary survey Priestley concludes that the at- 
mospheric eddy flux of heat is of the magnitude re- 
quired to equalize the heat losses and gains in adjacent 
zones of the earth and atmosphere as a whole, and 
that the zonal stresses arising from deep meridional 
currents can maintain wide zonal circulations against 
the effects of surface friction. 
Richardson’s investigations into large-scale diffusion 
have already been mentioned; in 1932 Sutton [66], by 
means of a semi-empirical theory, showed that in a 
turbulent medium the Einstem law of molecular diffu- 
sion (Brownian motion) should be replaced by the 
equation 
ao = (7(ut)™. (11) 
Here C is a generalized coefficient of diffusion and m 
is a constant whose value is about 1.75. This equation 
is certainly satisfied for diffusion over distances of the 
order of a few hundred metres, and Richardson’s data, 
re-analysed on the basis of this equation, indicate a 
strong probability that the law is valid also for dis- 
tances of the order of hundreds of kilometres. In assess- 
ing the value of this conclusion it should be borne in 
mind that the data on which it is based are extremely 
crude for the greater distances, but the evidence avail- 
able so far is that a law of the type (11) is capable of 
representing atmospheric diffusion, to a first approxi- 
mation at least, over a very wide range. 
It will be evident from the discussion above that al- 
though the broad features of many large-scale atmos- 
pheric processes can be explained reasonably well by 
the simpler forms of turbulence theory, as yet little has 
been attempted in the way of detailed analysis. Many 
opportunities present themselves here. The recent bril- 
liant American work? on the structure of thunder- 
storms has directed the attention of meteorologists to 
the problems of free jets and of the general mechanism 
of the entrainment of air by turbulent mixing on the 
boundary of a thermal current. Much remains to be 
2. Described in The Thunderstorm by H. R. Byers and 
others, Supt. of Documents, Washington, D. C., 1949. 
ee 
