METEOROLOGY — THEORY 199 
applied only to shallow layers and even then m varies 
quite considerably with height, wind velocity, and 
vertical temperature gradient. 
4. In 1936, Sutton,° who in 1982 suggested the 
power law variation, definitely favored the logarithmic 
variation. Furthermore he showed how one could 
handle the problem of varying stability. Sutton. an- 
alyzed different sets of data, ranging to 30 m in 
elevation, to support the logarithmic variation. 
5. In 1936, Sverdrup® criticized Sutton’s logarith- 
mic law and favored a power law in regions of stability. 
As evidence he introduced Rossby and Montgomery’s‘ 
analysis as well as his own data. 
6. In 1937, Sutton’ quite satisfactorily met Sver- 
drup’s criticism and pointed out that all experimental 
evidence suggested the logarithmic variation rather 
than the power law variation. 
This represents only a cross section of opinion and 
perhaps may be summarized as follows: 
1. In an indifferent or unstable atmosphere the 
logarithmic law is generally accepted. 
2. In a stable atmosphere there is more support 
for the logarithmic law than for the power law. 
One writer summarized the situation very aptly 
when he said that all modern mathematical studies 
on atmospheric turbulence are inexact and depend 
on certain wide assumptions. 
Conclusion 
The question now arises, should one assume a 
power law variation, or is the true wind variation 
better represented by a logarithmic law? Certainly 
the experimental evidence tends to favor a logarithmic 
variation. The advantages and disadvantages of either 
assumption may be summarized briefly as follows: 
1. Power law variation. 
a. m varies with stability, wind velocity, rough 
ness, and elevation. 
b. The mathematical analysis is too complicated 
for practical use. 
2. Logarithmic law variation. 
a. Agrees reasonably well with experimental 
data. 
b. Agrees with yon Karman’s logarithmic law. 
von Karman has shown that this law covers 
an exceedingly wide range of turbulence. 
ce. K, like m, varies with stability, wind velocity, 
roughness, and elevation. 
d. If the logarithmic law holds, K is then a 
linear function of height. With this relatively 
simple expression for K it should be much 
easier to handle the diffusion equation than 
in the case of a power law variation. 
e. Provided the integration of the diffusion equa- 
tion is not too complicated, one should be able 
to reconstruct the temperature and vapor 
pressure curves. Consequently the exact shape 
of the M curve can be calculated. 
In conclusion it should be borne in mind that 
theoretical discussion is futile. At best we can only 
make certain assumptions and derive a result. If this 
result agrees with observational data then the original 
assumptions are justified. Furthermore, practical con- 
siderations demand that the final solution be simple 
enough for application in the field. 
It seems certain that over a wide range of elevation, 
say 300 ft, the true wind variation cannot be uniquely 
defined by one specific logarithmic law or one specific 
power law. The most desirable procedure may then 
be an analysis of observations in as simple a manner 
as possible but yet flexible enough to take care of the 
most important changes. Consequently it is suggested 
that experimental data be analyzed on the assumption 
that K varies linearly with elévation, i.c., 
Sot el eens) 
at dzN az dx dz\ dz 
where K = pz-+-q, ana « is the distance measured 
horizontally. If accuracy is not seriously affected it 
is further suggested that approximations be intro- 
duced in order to facilitate the application of the 
results for field use. 
DIFFICULTIES OF LOW-LEVEL 
DIFFUSION PROBLEMS* 
The effect of a temperature inversion is largely a 
secondary one in that by reducing the coefficient of 
diffusion it favors the formation of large humidity 
gradients. The coefficient ot diffusion A is calculated 
from the wind profile which is assumed to satisfy a 
power law of the form U = Az™. 
The difficulty arises because m is fixed once and 
for all before we solve the equation and thus the 
theory cannot take account of changes in the tempera- 
ture gradients of a diurnal character in so far as they. 
affect the humidity distribution. At the same time we 
believe that K is very sensitive to the temperature 
gradient. 
Further, values of m have been used which have no 
meteorological support. The value m = 0.5, for ex- 
ample, implies a wind structure which is absurd if 
extended up to 100 m and it certainly is invalid near 
the ground. Chemical warfare technique measures m 
directly by measuring #, the ratio of the wind at 2 m 
to the wind at'1 m. Even in the very extreme condi- 
tions which prevail over land no value of @ exceeding 
1.35 is observed. This makes m = 0.33. Over the sea, 
even in a low layer, it is very unlikely that a value of 
m differing significantly from 1% would be found. 
The difficulty is that power laws apply only for very 
limited ranges of height and can be extended only by 
using a different power. Their only merit is that they 
enable the equation of diffusion to be solved; the 
By Lt. Comdr. F. L. Westwater, Naval Meteorological 
Service, Royal Navy. 
