ATMOSPHERIC TURBULENCE AND DIFFUSION 
relations between these properties and the mean motion. 
The two approaches have their counterparts in molecu- 
lar theory, one being analogous to the kinetic theory of 
the elastic sphere models while the other has certain 
affinities to the statistical mechanics of an assembly of 
molecules. 
In his initial (1920) contribution Taylor considered 
what statistical properties of a fluctuating field of flow 
are required to determine the diffusion of a group of 
particles suspended in the medium. If the turbulence 
is statistically uniform and steady (w’? mdependent of 
locality and time), the spread of a cluster of particles 
over a plane is given by 
v=o [ [Re dear, 
where X is the distance travelled by a particle in time 
T and R(é) is the correlation coefficient between the 
fluctuating velocities which affect a particle at times 
tand t + £. Taylor’s theorem means, in effect, that a 
knowledge of the mean eddying energy and of the 
correlation R(é) specifies completely the diffusion in 
such a field. The application of this result to meteoro- 
logical problems is considered later. The analysis also 
indicates a length J, , analogous to the mixing length 
but independent of any model, defined by 
he ae { Re dé, 
which is finite if R(E) tends to zero in a suitable manner 
as E— oo. 
Another way of representing such a field utilises the 
correlation R(y) between velocities at points separated 
by a variable distance y. The length Z defined by 
L= [ Rly) dy 
is called the “‘scale of the turbulence” and may be 
regarded as a measure of the average size of the eddies 
which constitute the pattern of flow. Evidence pre- 
sented later suggests that in meteorological problems L 
has no effective upper bound or, in other words, that in 
atmospheric diffusion, account must be taken of eddies 
ranging from the minute convectional whorl to full-size 
disturbances which affect the general circulation of the 
atmosphere. 
In later development of this theory attention has 
been particularly concentrated upon the problem of the 
decay of disturbances (e.g., downstream of a grid in a 
pipe), especially in isotropic turbulence, for which 
Taylor has introduced another important length \ called 
the “microscale of turbulence” and defined by 
ir? = tim {+= 0) 
y0 OF 
The microscale of turbulence indicates the average 
size of the small eddies which are responsible for the 
greater part of the dissipation of energy, and is also 
497 
related to the curvature of the R(y) curve at the 
origin. 
Later Developments; Kolmogoroff’s Theory. The 
most striking advances in the statistical theory of 
turbulence since 1941 are due to Kolmogoroff [34], 
Onsager [387], Weizsicker [79], and Heisenberg [25]. 
These theories have many points of resemblance, and 
the most promising appears to be that advanced by 
Kolmogoroff, whose work has been well summarized 
in English by Batchelor [1]. 
Kolmogoroff’s basic conception is that, at very high 
Reynolds numbers, all turbulent motion has the same 
sort of small-scale structure, although the mean flow 
may differ widely from situation to situation, and that 
the motion caused by the small eddies is isotropic and 
statistically uniform. A turbulent flow may be thought 
of as giving rise to a spectrum of oscillations whose 
wave lengths vary in scale from those characteristic 
of the mean flow to a lower limit below which the 
motion is entirely laminar. Within this spectrum, energy 
is continually bemg transferred from one length scale 
to another by the generation, from any given band of 
oscillations, of a set of smaller oscillations, and so on, 
until it is no longer possible to form smaller eddies. 
Thus the process may be thought of as one m which 
energy from the mean motion is fed into the long-wave 
end of the spectrum by the largest eddies and passed 
down the spectrum in the direction of decreasing wave 
length, until ultimately it is absorbed into the random 
heat motion of the molecules by viscosity. Kolmo- 
goroff then puts forward two similarity hypotheses: 
(1) that the statistical characteristics can depend only 
on the mean energy dissipation per unit mass of fluid 
(ec) and on viscosity; (2) that the statistical charac- 
teristics of the motion due to the larger eddies are 
independent of viscosity and depend only upon the 
energy dissipation «. From these plausible hypotheses 
it is possible, by arguments chiefly of a dimensional 
character, to make certain definite predictions about 
the properties of the mean flow, and in particular, to 
show that as the Reynolds number increases without 
limit, the coefficient of correlation between fluctua- 
tions at points distance y apart tends to the form 1 — 
Ay’'* (A = constant), provided that y issmall compared 
with the scale of the turbulence. Wind-tunnel measure- 
ments so far have provided excellent confirmation of 
deductions made from Kolmogorof{’s hypotheses and 
there is little doubt that the theory constitutes a 
powerful tool for the resolution of many complex prob- 
lems, and one which should especially appeal to 
meteorologists because of the emphasis laid upon the 
passage of energy from disturbances of one size to 
another. An account of this and other theories, together 
with some promising new developments, has recently 
been given by Karman [31], who considers that the 
principal aim of the present period is to find the laws 
which govern the shapes of either the correlation or the 
spectrum functions. 
We now pass from these theoretical and laboratory 
studies to detailed consideration of the meteorological 
problem. 
