494 
which the mean motion has the same direction at all 
points and in which the turbulence, specified by the 
mean squares of the oscillations, is uniform in all 
directions. Such conditions are approached fairly closely 
in motion very near a plane solid boundary, such as the 
surface of the earth. In the notation of the previous 
section, if x be in the direction of mean flow and z 
distance normal to the surface (2 = 0), 
a= az), .=W=0. 
If we introduce the eddy shearing-stress 7 = —pu’w’, 
the equations of motion reduce to the single equation 
in which molecular terms have been neglected. If the 
pressure gradient dp/d% is mvariable throughout the 
shallow layer concerned, 
where 7» is the value of 7 as z — 0. 
In many aerodynamical problems the pressure gradi- 
ent is effectively zero, and in micrometeorological appli- 
cations 0p/dx is usually small compared with 70/2. 
For moderate values of z the shearing stress 7 may thus 
be considered invariable with height and equal to the 
value at the surface, 7). This assumption is almost 
always made in problems of turbulence near the ground. 
For a more detailed examination of the meteorologi- 
cal problem the reader should consult papers by Ertel 
[17] and Calder [6], who conclude that in the atmos- 
phere the constancy of 7 with height may usually 
be safely assumed for values of z not exceeding about 
25 m. In this special case the transport of momentum 
across any horizontal plane is effectively measured by 
the eddy shearing-stress — pw’w’, that is, by a quantity 
involving the correlation between the eddy velocities 
w’ and w’. The existence of this correlation expresses the 
fact that gusts or positive values of wu’ are more fre- 
quently associated with downward-moving air, and 
lulls (megative uw’) with upward-moving air, than vice 
versa. 
The form of the profile of mean flow, however, cannot 
be deduced unless some further hypothesis is introduced. 
The earliest attempt to frame such a hypothesis appears 
to have been that of Boussinesq [3] in 1877. There is an 
obvious analogy between the action of the velocity 
fluctuations in a turbulent fluid and the motion of the 
molecules in a gas, which suggests that the effects pro- 
duced by the turbulence may be ascribed to the move- 
ments of discrete masses of fluid, called eddies, from 
one level to another. In the corresponding problem in 
purely laminar flow the shearing stress is ndu/dz, which 
suggests that, on this analogy, the Reynolds stress may 
be expressed as the product of the gradient of mean 
velocity and a virtual viscosity, that is, 
—pu'w’ = A ot/dz = Kp dt/az. (8) 
T= 
DYNAMICS OF THE ATMOSPHERE 
The quantity A, called an interchange coefficient 
(Austauschkoefizient), corresponds to the dynamic vis- 
cosity uw, and the quantity K, called the eddy viscosity, 
corresponds to the kinematic viscosity v. It is obvious 
that the same concept can be applied equally well to 
the transport by turbulence of heat or suspended 
matter, by defining in a similar fashion the eddy con- 
ductivity and the eddy diffusivity. This simple but 
powerful concept has had a much greater influence on 
dynamical meteorology than on other branches of fluid 
motion. As an initial assumption it is natural to suppose 
that A and K behave exactly like their molecular coun- 
terparts, that is, are true constants and thus independ- 
ent of position in the field. If this were strictly true, 
turbulent motion would be simply an enlarged copy of 
laminar flow, which is far from being the case. This 
hypothesis is now abandoned, but it should be recog- 
nized that in meteorology it has played an invaluable 
part in bringing to prominence the enormous difference 
in magnitude between molecular and turbulent trans- 
port. The magnitudes usually quoted in meteorological 
literature for AK are 10%, 104 or 10° cm? sec, which 
should be contrasted with the kinematic viscosity of 
air, which is of the order of 10—! cm? sec“. 
The Mixing-Length Hypothesis. It was soon recog- 
nized, especially by workers in aerodynamics, that the 
eddy viscosity itself was likely to prove too complicated 
as a starting poimt for the analysis of turbulent flow. 
An attempt to improve the treatment without entirely 
abandoning the analogy with molecular theory was 
made by Prandtl [44] in 1925. This has proved extremely 
fruitful although, like many other theories of turbulence, 
it is semi-empirical, and intuitive rather than analytical. 
Measurements of turbulent flow indicate that the 
virtual stresses set up by the turbulence are approxi- 
mately proportional to the square of the mean velocity. 
It is convenient to define an auxiliary reference velocity 
ux , known as the friction velocity (Schubspannungs- 
geschwindigkeit) for which this relation is exact, that is, 
Us = 7/p = ww’. 
Thus ux is a quantity of the same order of magnitude 
as the eddy velocities (that is, m most meteorological 
applications wu; is about 49 of the mean speed, the 
exact value depending on the nature of the terrain). 
A list of typical values of uw, has been given by Sutton 
[63]. Prandtl then introduces a characteristic quantity 
1, called the mixing length (Mischungsweg), defined by 
Uy = 1 du/dz, 
so that 
K = u,l = P 0u/oz. 
Thus / is a length which resembles, but is very much 
greater than, the free path of the kinetic theory of 
gases.! It may be interpreted in a general way as the 
distance which an eddy moves from its point of depar- 
ture from the mean motion until it mixes again with the 
1. Compare » = 4cd, where c = molecular velocity, \ = 
free path. 
