504 
(W. G. L. Sutton [70]), which will be recognized as a 
generalization of the classical equation for the conduc- 
tion of heat in a solid. This is probably the most useful 
way of regarding the equation, for the boundary condi- 
tions which occur in problems of atmospheric diffusion 
are entirely analogous to those familiar in the theory of 
heat flow. Solutions of this equation for various types 
of boundary conditions (chiefly those which arise in 
the problem of evaporation) have been discussed in 
considerable detail by W. G. L. Sutton [70], who based 
his analysis on Goursat’s treatment of the equation of 
conduction of heat. Jaeger [27] has recently given an 
elegant and concise discussion of the diffusion equation 
by the method of the Laplace transform. 
The boundary conditions for the problem of the 
steady continuous infinite cross-wind line-source of 
smoke are easily specified; the concentration tends to 
zero at infinity and increases without limit as the line 
of emission is approached, while the effect of the earth’s 
surface is represented by the condition that there is no 
net flow across the plane z = 0. The solution is then 
easily obtained as 
(Gaara const 
XW, 2 qe eer) 
of a 
(13) 
=) ilar 142 
—const ones p) 2! Pp) 
“exp 7 ; 
in which the various “‘constants” are easily determined 
by the “continuity” condition. 
In the problem of evaporation from a free-liquid or 
saturated area on the plane z = O, of infinite extent 
across wind and of finite length downwind, the bound- 
ary conditions are not as obvious. In his treatment of 
the problem O. G. Sutton [68] introduced the condition 
that the vapour concentration attains the saturation 
value (x;) at all points on the wetted area, so that the 
problem of evaporation becomes that of finding the 
strength of the area source which will maintain this 
constant concentration on the surface despite the re- 
moval of vapour by the turbulent air stream above. 
The solution of the problem constituted by equation 
(12a) and the boundary conditions can then be obtained 
as an incomplete gamma function, namely, 
x(a, Z) = “(1 = const 
f (= Te gre p | 
x * 29+ 1/7 |’ 
from which the total rate of evaporation can be found 
(14) 
0z 
over the area (see W. G. L. Sutton [70], Jaeger [27], 
Frost [21], Pasquill [40]). 
Three-Dimensional Problems. For problems involving 
point or short line sources, or areas finite across wind, 
the term = (% 2x must be retained. So far very 
y y 
by integrating the local rate of evaporation Es =| 
z=0 
little progress has been made with these problems. In 
the first place, the form for K, cannot be settled as 
DYNAMICS OF THE ATMOSPHERE 
easily as that for K, , since there is no resultant shear- 
stress across wind and hence no counterpart of the 
“conjugate power-law” theorem. It is hardly possible 
to consider K, as a function of y, that is, dependent 
upon distance from an arbitrary axis, and AK, must 
therefore be made a function of height (z). This, how- 
ever, introduces considerable mathematical difficulties. 
It is known, for example, that a solution satisfying the 
boundary conditions for a continuous point-source can 
easily be found (in the usual exponential form) if both 
K, and @ are constant or if K, and @ are proportional 
to the same power of z, but these solutions do not agree 
with the experimental data. In the problem of evapora- 
tion, a formal solution for K,, K, and & constant, 
applicable to a rectangular area, can be found in terms 
of Mathieu functions, and Davies [11] has investigated 
the case in which K, and W are proportional to the 
same power of z for an area of parabolic shape, but the 
solution for A, proportional to an arbitrary power of z, 
applicable to a closed area, has not been found. There 
is need for a detailed pure mathematical examination 
of this type of equation, and progress in this branch of 
atmospheric diffusion is bound to be slow until such 
an investigation has been made. 
Formulas for K., Statistical Theory. In applying the 
analysis described above to specific problems, it is 
usually assumed that the velocity profile is known to a 
high degree of accuracy, and that from this, the value 
of the friction velocity ux can be found for any given 
mean velocity. Since 
ee Ue 
* di/dz ~~ du/dz’ 
it follows that K, is also known explicitly at all points 
in the layer in which the shearing stress is constant. 
Thus the two-dimensional problem of diffusion depends 
essentially on the accurate determination of the velocity 
profile near the surface, the underlying assumption 
being that the diffusion of mass by eddy motion is 
identical with that of the diffusion of momentum. 
This method, initiated by Sutton, has been followed 
by Calder [7] in a very exact study of two-dimensional 
diffusion over surfaces covered by both short and long 
grass. Calder uses the logarithmic profile to determine 
the roughness length, friction velocity and zero-plane 
displacement (for the long-grass case), but in the equa- 
tion (12a) he replaces the logarithmic profile by an 
approximate power-law profile and uses solutions (13) 
and (14) above. The theoretical results are in very good 
agreement with the observational data. The method 
cannot be used for three-dimensional problems, and 
for such problems recourse must be had to the statistical 
approach devised by Sutton [67]. 
This theory is based upon Taylor’s “Diffusion by 
Continuous Movements” [73], using as the starting point 
the correlation coefficient R(E) between eddy velocities 
at different times. For flow over a smooth surface, 
dimensional arguments indicate that R(é) may be repre- 
sented by 
0) =| ae (15) 
