ATMOSPHERIC TURBULENCE AND DIFFUSION 
Bee Lari cer} 
q= rend —wl eG ar r) + wt, 
where w’ is the vertical component of the eddy velocity 
and 7”’ is the temperature anomaly of the eddy at the 
beginning of its upward motion. The first term in the 
brackets is that derived in the classical theory, but the 
second term is independent of the gradient of potential 
temperature and depends essentially on the existence 
of a correlation between the vertical velocity and the 
initial temperature of the eddy, which need not be that 
of the layer from which it came. The second term is 
therefore due entirely to buoyancy, whereas the first 
term could arise if the air were forced upwards by 
purely dynamical means. Unfortunately, the term w’T” 
cannot be isolated from the general heat flux and meas- 
ured separately, and its magnitude can only be inferred 
from arguments of a general nature, but the theory does 
give a fairly satisfactory explanation of certain large- 
scale phenomena, such as the superadiabatic lapse rates 
of the higher regions of the troposphere in clear non- 
subsiding air. The same problem has recently been 
considered by Montgomery [86]. 
The problem of natural convection, that is, of heat 
transport due to the upward motion of the air in calm 
or very light wind conditions has been considered by 
Sutton [64], who, starting from the assumption that the 
vertical flux of heat is proportional to the difference 
between the observed temperature gradient and the 
adiabatic lapse rate, shows that the observations then 
necessarily imply a very rapid increase of Ky with 
height (as 2*"°) in hot, clear weather. On the assump- 
tion that the intensity of the turbulent upward currents 
is determined by the balance between their dissipation 
into smaller eddies and their rate of loss of potential 
energy, Sutton shows that certain simple relations must 
exist between the eddy velocity, the mixing length for 
heat transfer, the temperature gradient, the tempera- 
ture oscillations and the eddy conductivity, provided 
that the upward flux of heat does not vary with height. 
These relations are in harmony with Johnson and Hey- 
wood’s observations of the temperature field during 
the mid-hours of a clear summer day. Sutton also 
showed that a model of convection in which the ground 
is assumed to act as a plane instantaneous source of 
heat at intervals of the order of half-a-minute or so 
could provide an explanation of the temperature field 
found near the ground in warm weather. This implies 
that the typical convectional eddy behaves like a 
“Dubble” which rises because of its buoyancy and 
grows by entraining cold air by turbulent mixing as 
it moves. 
From this brief survey it will be seen that the solu- 
tion of the problem of the transfer of heat from the 
ground to the air, or vice versa, is still far from com- 
plete. At the present time the main need is for accurate 
observations of elements such as the temperature and 
wind oscillations and their correlations (and hence the 
eddy flux) and the radiative flux. On the credit side, 
it appears to be fairly well established that in conditions 
of high lapse rate there is a significant difference be- 
507 
tween the rate of transport of heat and momentum 
near the ground, even when radiation is taken into 
account (Pasquill), but a complete explanation of the 
temperature field in both unstable and stable conditions 
is still to seek. When buoyancy effects are small com- 
pared with purely frictional effects, that is, in the prob- 
lem of forced convection, it seems that there is little 
difference in the transport of heat and momentum by 
eddy motion. 
The Diffusion of Matter and Other Studies in Non- 
adiabatic Gradients. Among the early attempts to 
extend the theory of diffusion to nonadiabatic gradients 
is that of Rossby and Montgomery [55], who, in effect, 
take the mixing length to be given by 
l= kz/V/1 + oRi, 
where o is a “‘constant.”’ This expression was used by 
Sverdrup in a study of turbulence over a snow field; 
his observations indicate that the value of o is about 11. 
In a later contribution Sverdrup examined Best’s ve- 
locity profiles in the light of this relation and found a 
large increase in o as conditions changed from insta- 
bility to stability. This work was later extended by 
Holzman [26], who suggested that 
l= kevV/1 — oRi, 
where o; is another constant so that 1 = 0 when Ri = 
1/o, , when turbulence should be extinguished. 
The Rossby-Montgomery and Holzman formulations 
have been subjected to a critical examination by Deacon 
[12], who finds that the Holzman expression agrees well 
with observations, o; being independent of both rough- 
ness and stability, with a mean value about 7. This 
implies that the turbulent mixing-length becomes negli- 
gibly small if Ri is about 0.14. 
Deacon [12] has applied these studies to an examina- 
tion of two-dimensional diffusion in nonadiabatic gra- 
dients (chiefly superadiabatic lapse rates), using data 
on the travel of gas obtained at the Experimental Sta- 
tion, Alberta, Canada. The results show a certain meas- 
ure of agreement with the theory, except in very larga 
lapse rates, in which the experimental data are rather 
irregular. 
To sum up, it is clear that a satisfactory theory of 
diffusion in large inversions and large lapse rates has 
yet to emerge, and this is particularly true for three- 
dimensional problems. It is unfortunate that the theory 
is most deficient where the need is greatest, that is, in 
conditions of large inversions, when even small sources 
of pollution can produce high concentrations, and the 
problem is one which should receive urgent attention 
from meteorologists if only because of its importance 
in the life of highly industrialized communities. 
REFERENCES 
1. Barcnrtor, G. K., “Kolmogoroff’s Theory of Locally 
Isotropic Turbulence.’ Proc. Camb. phil. Soc., 43:533- 
559 (1947). 
2. Best, A. C., ‘Transfer of Heat and Momentum in the 
Lowest Layers of the Atmosphere.’”? Geophys. Mem., 
No. 65 (1935). 
