ATMOSPHERIC TURBULENCE AND DIFFUSION 
explained concerning the details of large cellular con- 
vective motion, first studied in the laboratory by 
Bénard and investigated mathematically by Rayleigh, 
Jeffreys and others. The whole problem of turbulence 
in the upper atmosphere has recently arisen in an acute 
form in relation to the design of large aircraft. Finally, 
the applicability of the Reynolds process of averaging 
in dealing with major atmospheric motions is still un- 
certain and it is here, perhaps more than anywhere else, 
that the statistical theories are likely to make the most 
decisive contributions. 
SMALL-SCALE DIFFUSION PROCESSES 
Because of their importance in military operations 
and in studies of atmospheric pollution, specialized 
problems relating to the spread of suspended matter 
(smoke and gas) over distances of the order of a few 
kilometres have been the subject of intensive research, 
both practical and theoretical. In experimental investi- 
gations a measure of control is possible, and the data 
on diffusion thus obtained are among the most reliable 
and accurate in micrometeorology. A summary of the 
properties of continuously generated smoke clouds, 
based on extensive trials conducted at Porton, Eng- 
land, has been given by Sutton [67]; these refer exclu- 
sively to conditions of small temperature gradient and, 
as yet, no corresponding set has been published for in- 
versions or for conditions of a superadiabatic lapse rate. 
Concurrently with the experimental work, there has 
been a great deal of activity on the theoretical side, 
with the result that a workable but semi-empirical 
theory of turbulent diffusion has been built up and 
verified for conditions of neutral equilibrium (adiabatic 
lapse rate). This work will now be summarized briefly. 
Properties of Continuously Generated Clouds in Con- 
ditions of Small Temperature Gradient. The smoke 
cloud from a continuous point-source stretches down- 
wind in a long cone, and measurements of concentration 
taken at fixed points are to some extent dependent 
upon the period of sampling. This is because the natural 
wind is made up of fluctuations of all periods, and the 
“instantaneous” aspect of the cloud, being mainly influ- 
enced by the small-scale eddies, differs considerably 
from the “‘time-mean” aspect. The narrow ‘‘instanta- 
neous cone” swings slowly over a wider front and is 
contained within an enveloping ‘“‘time-mean cone.” The 
properties which have been measured almost invariably 
refer to the “‘time-mean”’ aspect, that is, the concentra- 
tions at fixed points are averages over periods of time 
of not less than three minutes. 
For a continuously generated cloud, the concentra- 
tion of smoke at any point is directly proportional to 
the strength of the source and approximately inversely 
proportional to the mean wind speed, while the cross- 
wind and vertical distributions of concentrations are 
approximately of the “normal law of error” type. From 
the point of view of the mathematician, the results of 
greatest importance are: The central or peak concen- 
tration in the cloud from a continuous point-source 
decays as 1/x'76 (¢ = distance downwind) and as 
1/x°® in the cloud from an infinite cross-wind con- 
503 
tinuous source, and the maximum concentrations at 
z = 100 m in a mean wind of 5 m sec are 
point-source of 1 g sec! .............. 2mg m=“ 
infinite line-source of 1 g see! m.... 35 mg m=. 
The fundamental problem for the mathematical 
physicist is to find means whereby these properties can 
be derived from measurements of the relevant meteoro- 
logical factors, such as the profile of mean velocity, 
gustiness and temperature gradient. 
Theoretical Aspects. The general equation of dif- 
fusion in a turbulent medium is 
(12) 
@ fae ON @ a> OR @ fa OR 
eee I IK 
Ox (« =| M oy ( 5 =) ¢ dz ( ; ) 
where x is the mean concentration of smoke, the density 
of the air being supposed constant. It is convenient to 
take axes in which z is measured in the direction of the 
mean wind, y across wind and ¢ vertically, so that 
>. = ® = O. For these localized problems the mean 
wind may be supposed to vary only with height. 
Two-Dimensional Problems. In the case of a long line 
source across wind, or an area source in which lateral 
edge effects are negligible, equation (12), neglecting 
downwind diffusion in comparison with vertical dif- 
fusion, reduces to 
apn OX @) ff Oh 
me) Ox 02 (x. zy 
for the steady state. To deal further with this equation 
means that a(z) and K, must be known explicitly. We 
shall consider here only power-law formulations; so 
far no solutions have been published in which @ is 
expressed in terms of In z. Most investigations assume 
the Schmidt ‘conjugate power-law” relation, that is, 
if @ = &(z/a)?, then K = K,(z/a)'-?, but since this 
is equivalent to the assumption that the eddy shearing- 
stress t = Kp di/dz is invariable with height, the 
resulting equation can hold only in a relatively shallow 
layer (¢ < 25 m) near the surface. This limitation is 
of importance in questions dealing with the variation 
of humidity profile in air passing from sea to land 
(v. Booker), and results obtained in investigations which 
assume the conjugate power-law relation to hold in 
deeper layers must be regarded with considerable doubt, 
and any close agreement with observation to be a 
matter of chance rather than of sound reasoning. 
With the conjugate power-law relation, the two- 
dimensional steady-state equation becomes 
U1 1—2p Ox —p 0 ( 1—p a) 
2 i — ea 
Ky Ox dz\ Oz 
By a suitable change of variables this equation may be 
converted to the form 
(12a) 
p = p/p + 1), 
