ATMOSPHERIC TURBULENCE AND DIFFUSION 
fluid from lower to higher levels must reduce the inten- 
sity of the turbulence and may ultimately cause a transi- 
tion to laminar flow. On the other hand, a fluid in which 
density increases with height is favourable to the forma- 
tion of upward currents and therefore to the mainte- 
nance and the growth of turbulence. The atmosphere is 
a fluid whose lower boundary is subjected to strong 
heating and cooling, especially in clear weather, so that 
near the ground the turbulence of the wind tends to 
rise to a maximum in the mid-hours of a sunny day, 
when there is usually a pronounced superadiabatic 
lapse rate in the lowest layers, and to diminish or even 
die away completely during a clear night, when the 
radiative cooling of the ground creates a marked in- 
version of temperature gradient in the lowest layers. 
The gradient of temperature in the vertical is thus to be 
regarded as a factor exerting a powerful control on the 
turbulence of the wind. A second difficulty peculiar to 
meteorology arises from the variable nature of the 
surface over which the air flows. Provided that the 
obstacles which cover a surface are not too large and are 
evenly distributed, aerodynamic investigations have 
indicated a rational method of allowing for their effects, 
and the same concepts have been applied with consider- 
able success in many problems of atmospheric motion, 
but in meteorology cases frequently arise in which the 
surface irregularities are too large or too unevenly 
distributed to be treated in this way. This is especially 
the case, for example, in considering the diffusion of 
atmospheric pollution in built-up areas. 
Mathematical Treatment. It is generally accepted 
that in any fluid motion the component velocities 
u, v and w, along axes of x, y and 2 respectively, satisfy 
the Navier-Stokes equations, that is, three relations of 
the type 
Ou Ou Ou ou 
» (% ae at” ay +02) 
ou ou ou 
xX, 
Bb & a aye ar =| + pX, 
where p is the density, p is the pressure, » is the dynamic 
viscosity and X the z-component of any external force. 
To these must be added the equation of continuity, 
which expresses the conservation of mass. Exact solu- 
tions of this set of nonlinear equations are known only 
for certain special cases which have little or no interest 
for meteorology. 
The mean velocities which appear so prominently in 
studies of turbulence are usually averages over an 
interval of time (7’), that is, they are defined by the 
relations 
1 pir 1 petit 
== uw dt = =| v dt 
T ine ! T t—1T 4 
1 t+i7 
i — w dt, 
a t—17 
in which case the fluctuations or eddy velocities w’, 
v’ and w’ are defined by 
a 
493 
If the interval of time 7 is sufficiently long, it may be 
asserted that 
w= =w = 0. 
The effect of viscosity is to set up in the fluid a system 
of stresses, namely three normal stresses pzrz, Dyy 
and pz, defined by 
and three tangential stresses, Pz, , Py: , Pzx defined by 
Ou Ov ov Ow 
poe (Sta) mane ta) 
ow Ou 
Pzz = b = ap =) : 
Introducing these stresses into the equations of motion, 
we have three equations of the type 
Ou () to) 
{ recrgies ap (Dex = pu) SF ay (Dey io puv) 
(1) 
oP + (Dex ar puw) ar pX. 
Putting uw = a + w’, etc., and taking means, we have 
@ n_ 2 —- 
ae Ga = al = out?) 
0 = nm ry, S) 
+ a (Day — pud — pu'r') (2) 
Onis _ Ta ‘a 
antes (pr — puw — pu’w') + pX, 
and analogous equations for the y- and zg-components. 
The only formal change which has occurred is that in 
equations (2) certain additional terms, depending upon 
the eddy velocities, have been added to the original 
viscous stresses. These additional terms, called the 
Reynolds stresses, are the mathematical expression of the 
effect of the velocity fluctuations in transporting 
momentum across a surface in the fluid, just as the 
original stresses represent the effect of the molecular 
agitation in transporting momentum by viscosity. In 
general, the Reynolds stresses are considerably greater 
than the corresponding viscous stresses, and it is usually 
possible to ignore the latter in problems of atmospheric 
turbulence. 
No general method has yet been evolved for express- 
ing the Reynolds stresses in terms of the velocity 
components and their spatial derivatives by exact analy- 
sis, and this constitutes the principal difficulty in pro- 
ceeding further along these lines. The study of a special 
case, however, suggests semi-empirical methods by 
which progress can be made despite the formidable, 
difficulties of the complete problem. 
Flow Near a Boundary. Consider a steady flow in 
