496 
gomery [55], appears to be more appropriate for such 
cases. Rossby and Montgomery assume that 
l= ke + 20), 
which expresses the intuitive conception that the effects 
of the irregularities are most strongly felt in the immedi- 
ate vicinity of the surface and hardly at all at the 
greater heights. Equation (3), with this form for 1, 
yields 
Papal), (7) 
Ue, Uk 20 
using the boundary condition #@ = 0 on z = 0. Thus 
theoretically, the profile of mean velocity over a rough 
surface is fully determined provided both ux and 2p 
are known (k being regarded as a universal constant). 
These may be combined to form the quantity termed 
by Sutton [63, 65] the macroviscosity, defined by N = 
Uxz9 . The introduction of this quantity enables a single 
profile, applicable to both smooth and rough surfaces, 
to be obtained, thus overcoming the difficulty felt in 
the free use of equations (6) and (7), that neither of 
these tends to the smooth-surface form (5) as 2 — 0. 
The generalized profile is 
=o) Ux2 ) 
Oe le N + 0/9]? 
corresponding to 1 = kz, or 
Sth 
Die, Ue N + 7/9 
if 1 = k(z + 2). Aszo (and hence J) tends to zero these 
equations tend to the smooth-surface form, but in 
most meteorological applications N is of the order of 
10 to 10% cm? sec! and is thus much greater than ». 
Schlichting’s criteria may be written 
smooth flow: NV < 0.13» = 0.02 cm? sec, 
rough flow: N > 2.5» %0.4 cm?sec. 
Power-Law Profiles. A somewhat less satisfactory 
representation of the flow near a boundary is obtained 
by the use of power laws, but this type of profile is 
often much easier to handle mathematically than the 
logarithmic profile. The usual form of the profile near a 
smooth surface is 
a = t(2/z)?, (8) 
where a, is the mean wind speed at height 2 and p 
is a positive number whose value for moderate Reynolds 
numbers is about lv. If r/p = K ou/dz is invariable 
with height, it follows that 
K = K,(z/a)"?, (9) 
where kK, is the value of K at z = 2. Equations (8) 
and (9) constitute the so-called “conjugate power- 
laws” of Schmidt. 
The Vorticity-Transport Hypothesis. The preceding 
analysis is based essentially upon the hypothesis that 
momentum is conserved during the motion of an eddy, 
DYNAMICS OF THE ATMOSPHERE~ 
and for this reason the Prandtl treatment is often 
referred to as the ‘“‘momentum-transport theory.” From 
the aspect of exact hydrodynamical theory this hypoth- 
esis is questionable, since it involves the assumption 
that the pressure fluctuations do not affect the transfer 
of momentum. An alternative hypothesis, in many ways 
more attractive, is that vorticity is conserved and this 
forms the basis of Taylor’s treatment of the problem 
[72]. Space does not allow an adequate discussion of 
the matter here, nor, in the present state of develop- 
ment of the theory of atmospheric turbulence, are the 
differences between the two theories of major impor- 
tance for meteorology, except perhaps in one respect. 
According to the momentum transport theory, the 
rate at which momentum is communicated to unit 
volume of the fluid by turbulence is 
an A aii 
= K 
az oat @) aN 
whereas on the vorticity transport theory, 
These two forms are equivalent if, and only if, K(z) = 
constant. For further details the reader is referred to 
Taylor’s original papers, or to the accounts given by 
Brunt [4] and Goldstein [22]; the latter gives consider- 
able detail as regards tests on the laboratory scale. 
An illuminating discussion, chiefly from the standpomt 
of the experimental worker, of the validity of the 
various hypotheses which have been advanced in the 
last two or three decades in order to develop further the 
Reynolds theory of stresses has been given by Dryden 
[14]. When the predictions of the various theories are 
compared with the results of accurate measurements 
made in turbulent boundary layers, the disquieting con- 
clusion is reached that all such theories fail at some 
point or other. Considerable doubt is now thrown on 
the validity of the mixing-length idea and even on the 
fundamental hypothesis that the eddy velocities and 
the turbulent shear-stresses are directly related to the 
mean velocity and its derivatives at a pomt (equations 
(3) above). These considerations may perhaps indicate 
that one fruitful period of development in the theory 
of turbulence is drawing to its close, and that the time 
is now ripe for advances along quite different lines. 
Statistical Theories of Turbulence. One such funda- 
mentally different approach to the general problem 
is that of the so-called “statistical” theory of turbu- 
lence, initiated by Taylor [73] in 1920 and later ex- 
panded by him in a remarkable series of papers in 
1935 [74]. Subsequent developments were made by 
Karman and Howarth [32] and an account of the posi- 
tion reached by 1943 has been given by Dryden [15]. 
All theories of turbulence are necessarily statistical 
in some sense or other, but whereas the theories de- 
scribed in the first part of this article start with a 
measured distribution of mean velocity and mean pres- 
sure and relate these to the virtual stresses, the methods 
about to be described originate in the statistical prop- 
erties of the fluctuations and seek to establish exact 
