ATMOSPHERIC TURBULENCE AND DIFFUSION 
main body of the fluid. A mass of fluid leaving a layer 
z in which it has acquired the mean motion az), and 
moving a vertical distance / while conserving its momen- 
tum, will give rise to the fluctuation 
a(z + 1) — ue) & lL ou/dz 
at the new level. Thus J du%/dz may be regarded as 
representing either an eddy velocity or the friction 
velocity. 
A somewhat different method [22] of introducing a 
characteristic length is as follows: Suppose that H(z) 
is any transferable property, such as momentum, temp- 
erature or concentration of suspended matter, whose 
mean value is constant over any (a, y) plane and which 
is supposed to be conserved during a transfer from the 
plane z = % toz = &. The mean rate at which H(z) 
is transported across a unit area of an (a, y) plane is 
q = —w[E(@) — E@)| © —w'@ — a) dH/éz. 
If a length / be defined such that 
ln/ w? = we — %), 
the rate of transfer is 
gq = —l/w® OE /oz. 
Applying this to the transfer of momentum, in which 
E = pu and g = —7, we have 
t = plv/w® du/dz, 
and so 
K =lV/w?. 
The advantages to be gained by such dissections are 
not immediately obvious, since the analysis gives no 
clue to possible variations of / with any of the variables. 
In practice, the introduction of the mixing length has 
proved very useful, since it has been found that only 
very simple assumptions regarding / are required to 
obtain a satisfactory mathematical representation of 
many complex phenomena. It is doubtful if it is possible 
to assign a definite physical meaning to the mixing 
length, but broadly it is clear that / is a measure of the 
average ‘‘size”’ of the eddies responsible for the mixing, 
or equally, a rough indication of the average depth of 
the layers over which mixing takes place. 
Velocity Profile Near a Boundary. When a turbulent 
stream flows over a smooth surface, three regions of 
motion can be distinguished: 
1. A shallow zone of laminar flow immediately adja- 
cent to the wall in which the shearing stress is chiefly 
due to viscosity (‘laminar sub-layer’’). 
2. The turbulent boundary layer proper, lying above 
the laminar sub-layer, in which the Reynolds stress 
is at least as important as the viscous,stress. 
3. The main body of the turbulent fluid, lying above 
the boundary layers, in which viscosity plays a 
negligible part. 
A surface whose irregularities are large enough to pre- 
vent the formation of a laminar sub-layer is said to be 
“aerodynamically rough.” 
495 
Smooth Surface. We consider again motion in a shallow 
layer in which +r (and hence ux) is invariable with 
height and in which the mean velocity (a) is a function 
of z only. If the velocity profile be assumed to depend 
only upon J, v, ux and the independent variable z, 
it follows that the dimensionless ratio @/us must be 
expressible as a function of J/z and uxz/v, smee these 
are the only nondimensional ratios which can be formed 
from these variables. If, in addition, the scale of the 
mixing is assumed to be proportional to distance from 
the boundary, that is, 1 = kz, where k is a pure number 
(Kaérman’s constant), the definitions of uw, and of 7 lead 
to the first-order differential equation, 
dz l ke? 
whence 
sg We = + const, (4) 
te tb v 
where the ‘“‘constant” is to be determined by a suitable 
boundary condition. The usual requirement that «7 = 0 
on z = 0 cannot be satisfied, but Nikuradse has shown 
that equation (4) agrees closely with observations made 
on flow in smooth pipes in the form, 
tn es) 4 BS AY DE In C2) , ® 
Vv v 
that is, k = 0.4 and the velocity vanishes on the plane 
z = v/9ux , the equation having no meaning for smaller 
values of z. 
Rough Surface. It is a well-established experimental 
fact that in the case of an aerodynamically rough sur- 
face, the influence of viscosity is negligible, in which 
case the solution of equation (3) may be written 
a 
De, Us a 20” (6) 
where 2 is another constant of integration. This form 
of the equation implies that @ = 0 on z = 2 (the equa- 
tion having no meaning for smaller values of 2), so 
that zo , usually termed the roughness length, is a quan- 
tity which may be regarded as specifying, in some 
manner, the effect of the irregularity of the surface on 
the mean flow. This is borne out by the fact that, for 
pipes whose interior surface is uniformly roughened with 
fine grains of sand, it has been found that a definite 
relation exists between the roughness length and the 
size of the irregularities (20 = 10 of the average 
diameter of the grains). Schlichting [56] has gone further 
and has shown that a surface may be regarded as 
smooth if wx2o/v < 0.13 and fully rough if Uxzo/v > 2.5. 
The analysis above appears to be adequate if the 
layer concerned is very shallow, but since it implies 
that the effect of the irregularities is equally felt at all 
distances above the surface it cannot be applied, without 
care, to the deeper layers met with im most meteoro- 
logical applications. An alternative treatment of the 
rough-surface problem, due to Rossby and Mont- 
