506 
berg (page 497) predict a similar law of diffusion. It is 
possible that here is the first indication of developments 
of considerable importance for meteorology. 
Further progress in the study of small-scale processes, 
so important for many aspects of civilized life, depends 
on a number of factors. Micrometeorology demands 
measurements whose accuracy approaches that attaimed 
in the laboratory, and for which the development of 
specialized instruments, highly sensitive and yet suffi- 
ciently robust to be used in the open, is imperative. 
On the mathematical side it must be admitted that 
present theories, although often astonishingly success- 
ful for limited classes of problems, are unsatisfactory 
in that they involve a large element of empiricism, and 
rigour has often been sacrificed for simplicity of treat- 
ment. There is, in particular, a need to examine both 
practically and theoretically the basic assumptions of 
the various theories, such as the form of the various 
correlation functions and of the eddy spectrum over a 
wider range of conditions. The characteristic meteoro- 
logical problem, that of diffusion in large density gra- 
dients, is still unsolved and is likely to remain so until 
further progress has been made in the dynamics of 
stratified fluids. Some of the attempts to solve this, 
perhaps the most difficult problem in turbulence, are 
discussed in the next section. 
PROBLEMS ARISING FROM CHANGES IN. THE 
DENSITY GRADIENT 
Much, if not most, of the preceding work could be 
fairly described as a straightforward application of the 
aerodynamical theory of the turbulent boundary layer 
to meteorology. The feature which sharply differen- 
tiates the meteorological problem from those of normal 
aerodynamics is undoubtedly the influence of the den- 
sity gradient on the nature of the flow. In other words, 
the meteorologist must take into account the effects of 
the gravitational field, and there is very little in the 
way of wind-tunnel investigation to guide him here. 
The Richardson Criterion. The basic result in the 
theory of turbulence in a gravitational field is that of 
Richardson [51], who enunciated the criterion that the 
kinetic energy of the eddying motion will increase or 
decrease if the rate at which energy is extracted by the 
Reynolds stresses exceeds or falls below that at which 
work has to be done against gravity by the turbulence. 
From this, it is easy to show that turbulence will in- 
crease or die away according as the Richardson number 
oT 
— 1? 
z & 2 ) 
7) 
0z 
(T = absolute temperature of the environment) is less 
or greater than the ratio of the eddy viscosity to the 
eddy conductivity. If these two coefficients are sup- 
posed identical, we have 
Ri = 
turbulence increases if Rz < 1, 
turbulence decreases if Ri > 1, 
DYNAMICS OF THE ATMOSPHERE 
that is, the critical value (R2.,i2) of the Richardson 
number is unity. 
In his derivation of this criterion Richardson limited 
its application to a system possessing a very small 
amount of turbulence (‘‘ust-no-turbulence”’). The mat- 
ter has since been the subject of intensive investiga- 
tions, chiefly theoretical. Taylor [76] and Goldstein [23] 
showed that in the case of an inviscid incompressible 
fluid with a lmear velocity profile and a continuous 
density distribution, Ri. = 0.25, but the most de- 
tailed and realistic investigation is that of Schlichting 
[57] on the decay of turbulence in the boundary layer 
of a smooth flat plate. Schlichting found that R7crie 
varied between 0.041 and 0.029, depending on the 
inertia effects of the density distribution, a result which 
was later confirmed by Reichardt in the Gottmgen 
“hot-cold” wind tunnel. Recently, a detailed investiga- 
tion of the mathematical aspects of Richardson’s deriva- 
tion has been made by Calder [8], who finds that the 
inclusion of certain terms, neglected by Richardson 
because of his assumption of ‘‘just-no-turbulence,” 
changes the form of the criterion to Rian = 1 — 6, 
6 > 0, but he was unable to give a definite value for 6, 
except that 6 must be small if the initial degree of 
turbulence is small. 
There have been several attempts to ascertain the 
value of Ri., in the atmosphere [63]. Durst agrees 
with Richardson in finding R7,,;, = 1, but Paeschke 
[38] concludes that the Schlichting value, R2.,i: = 0.04, 
is appropriate. Some of the best evidence is that assem- 
bled by Deacon [12] who proposes Ri.,i = 0.15 for 
conditions near the ground, while for the free atmos- 
phere Petterssen and Swinbank [43] suggest Ritcruw = 
0.65. It is evident from these results that at the present 
time the question of the exact value of the critical 
Richardson number (if indeed it exists) is one of the 
most open in meteorology. 
The Heat Flux Equation and the Problem of Con- 
vection. It has been known for some time that the 
early assumption of the identity of the coefficients of 
eddy viscosity (Ky) and eddy conductivity (Kz) is 
open to considerable doubt; thus although in the lower 
layers of the atmosphere the transfer of momentum 
can be explained by taking Ky « 2”, where0 S m S 1, 
the propagation of the diurnal temperature wave in 
warm weather seems to require Ky <2”, where 
1 S m’ S 2. These conclusions are based upon the 
assumption that the eddy flux of heat is proportional 
to the gradient of potential temperature (Taylor) or to 
the difference between the observed gradient and the 
adiabatic lapse rate (Brunt). Ertel [18], in a series of 
papers, questions this assumption because it neglects 
buoyancy effects, and concludes that the flux is more 
likely to be proportional to the gradient of temperature 
itself, a result which in turn has been criticized by 
Prandtl [45] on the grounds that the analysis can only 
be valid for occasions of calm or light winds and large 
lapse rates. 
Ertel’s main conclusion is supported by the later 
work of Priestley and Swinbank [47], who consider that 
the flux of heat has the form 
