ATMOSPHERIC TURBULENCE AND DIFFUSION 
where v is the kinematic viscosity of the air and n is a 
parameter expressing the degree of turbulence in the 
fluid. The value of n is found from observations of the 
velocity profile. Using mixing-length concepts, an ex- 
plicit power-law form for K, can then be found, which 
when inserted in the equation of diffusion leads to 
results in excellent agreement with observation from 
a smooth saturated surface (Sutton [68], Pasquill [40]). 
For diffusion over rough surfaces, the kinematic vis- 
cosity in (15) must be replaced by the macroviscosity 
N = x2 ; the resulting expressions are then in good 
agreement with observation (Sutton [65)). 
For three-dimensional problems Sutton abandons the 
mixing length cwm differential equation approach and 
proceeds by finding expressions which satisfy Taylor’s 
equation 
cao [ [ Re dé dt, 
the boundary conditions and the equation of continuity. 
For this purpose it is necessary to introduce generalized 
diffusion coefficients C, and C., which are found as 
explicit functions of the mean gustiness, the parameter 
nm occurring in the velocity profile, and either the kine- 
matic viscosity or the macroviscosity, according as the 
underlying surface is smooth or rough. The expression 
for the concentration from a line-source obtained in 
this way 1s not quite as accurate as that derived from 
the differential equation, but is adequate for many 
purposes, and the method has the advantage of pro- 
viding a simple solution for the point-source problem 
where, as yet, the mixing length cwm differential equa- 
tion approach has failed. Recently, Sutton [69] has 
extended this work to cover the case of an elevated 
source such as a factory chimney. 
Detailed Study of Evaporation. The work described 
above has shown that the rate of evaporation from a 
small saturated or free-liquid area can be calculated 
with considerable accuracy, at least in conditions of 
neutral vertical equilibrium and provided that edge 
losses can be disregarded. Since the mean rate of evapo- 
ration decreases with the length of the area downwind, 
and also because it is uncertain if a small area can 
reflect adequately the effect of changing stability (Sut- 
ton [67]), it is extremely doubtful if the conventional 
“evaporimeter” or “evaporation tank” is of any real 
use in assessing the rate of evaporation from areas of 
moderate size, such as reservoirs or small lakes. 
For the estimation of evaporation from natural sur- 
faces without the aid of evaporimeters, two methods 
are possible. One, initiated by Angstrém for lakes and 
later applied by Penman [42] to land, depends upon an 
accurate evaluation of the remaining terms in the equa- 
tion for heat balance. No knowledge of the process of 
turbulent transport is required. The second method, 
used by Sverdrup and others for evaporation from the 
sea and by Thornthwaite and Holzman [77] for land, 
uses the theory of eddy diffusion to evaluate the flux 
of water vapour from the wind and water-vapour pro- 
files. This procedure has been subjected to a critical 
505 
examination by Pasquill [41], who concludes that the 
Thornthwaite-Holzman formula 
ip = pk (q er q2) (up — u) 
(In 2/21)? 
(H = rate of evaporation, @ , @ and uw, uw = specific 
humidities and mean velocities at heights z, and 2. , 
p = air density, k = Karman’s constant = 0.4) is valid 
for conditions of neutral equilibrium and may be used 
in other conditions if errors up to about 20 per cent in 
the rate of evaporation can be tolerated. 
Space does not permit more than a brief mention 
here of the large amount of work which has been done 
on evaporation from the ocean, but some reference 
should be made to the ‘evaporation coefficient” of 
Montgomery [85], defined as 
1 de 
€s — €a A(In 2)’ 
ro = 
where e; = vapour pressure at the surface and e, = 
vapour pressure at a standard height. The rate of 
evaporation from the sea surface is then 
E = pkyV (qs — q)Wa, 
where & is Karman’s constant, ya = Ux/W., where 
W.= velocity at height a, and qg, and q; are the specific 
humidities at heights a and b. Sverdrup [71] has shown 
1/In J 
A) 
where 2p is the roughness length, and has made a critical 
examination of the validity of Montgomery’s concept. 
Richardson’s Diffusion Theory and Its Relation to 
Recent Developments. We conclude this short summary 
of the diffusion problem by mention of some work by 
Richardson which may prove to be of considerable sig- 
nificance. In 1926, Richardson [50], showed in his 
“Distanee-Neighbour Graph” theory, that if / is the 
projection of the separation of a pair of marked par- 
ticles on a fixed direction, and qg is the “number of 
neighbours per unit of J,” 
that with certain assumptions T = 
ie) 
d= | x@)x@ +0 ar, 
where x is the concentration. The equation of diffusion 
then becomes 
9g _ 9 ) mq 99 
at a{FO a 
where F'(l) is a kind of diffusion coefficient. If J) is the 
initial value of /, and J, its value ¢ seconds later, 
(mean of (1, — l)” for all pairs) 
F(t) = = 
Observations by Richardson and Stommel [53] on small 
objects floating in water indicate that 
F() = 0.0704. 
This is a result of considerable importance, because the 
recent statistical theories of Weizsicker and Heisen- 
