CUMULUS CONVECTION AND ENTRAINMENT 
By JAMES M. AUSTIN 
Massachusetts Institute of Technology 
Introduction 
Cumulus clouds are the visual evidence of the over- 
turning of the atmosphere which follows the establish- 
ment of an unstable state of equilibrium. The unstable 
state may be created by such processes as the follow- 
ing: 
1. The heating of the lower atmosphere through its 
contact with a warm surface gives rise to the cumulus 
clouds which appear on a summer afternoon and to the 
cumulus clouds which often develop when a cold air 
mass flows over a warmer surface. 
2. The cooling of the middle troposphere may pro- 
duce an unstable condition in the lower atmosphere. 
The nighttime cumulus which forms over oceans may 
be attributed, in part, to this process. 
3. The saturation of a layer of convectively unstable 
air results in the development of convective cells. A 
good example of this process is the cellular pattern 
of the precipitation im convectively unstable air of 
tropical maritime origin. 
Following the establishment of the unstable condi- 
tion the atmosphere overturns and thereby transports 
heat upward. The manner in which an unstable layer of 
fluid overturns has been the subject of experimental and 
theoretical investigations. 
Bénard has described the convection cells which 
appear in a horizontal layer of fluid which is heated 
from below. The experimental work of Bénard, Idrac, 
Terada, Mal, Walker and Phillipps, and others is re- 
viewed by Brunt [5]. These experiments involved only 
very thin layers of fluid. The evidence of the existence 
of convective cells in the atmosphere, other than cumu- 
lus clouds, includes the observational work of Durst [8] 
and Woodcock and Wyman [16], and the measure- 
ments by radar and aircraft as reported by Wexler [15] 
and Byers and Braham [6]. 
The theoretical problem of the development of con- 
vective cells in an unstable medium of small thickness 
has been analyzed by Rayleigh and others. The theory 
is reviewed by Stommel [14]. However, a theoretical 
analysis of the development of cells of the dimensions 
of cumulus clouds is lacking. Some of the disturbing 
problems are the applicability of the linearized equa- 
tions, the variability of the coefficient of eddy diffusion, 
the compressibility of the atmosphere, and the hori- 
zontal wind shear in the vertical plane. Also, the libera- 
tion of the latent heat of condensation in the cloud 
provides a heat source other than the ground. It would 
appear that the problem of cumulus convection should 
be analyzed from the standpoint of the development of 
cellular circulations. In the absence of a complete solu- 
tion of this problem it has been necessary to make 
various assumptions as to the important aspects of the 
process. 
Parcel and Slice Methods 
Convective currents develop in an atmospheric layer 
which is in an unstable state of equilibrium because a 
perturbation then gives rise to an acceleration of air 
particles away from their original positions. The sim- 
plest approach to the convection problem is to consider 
that a parcel of air can be displaced adiabatically with- 
out any density change taking place in the environment. 
The stability of an atmospheric layer is then tested by 
comparing the density of a parcel which is displaced 
from its position of equilibrium with the density of the 
environment. This technique, which is referred to as the 
parcel method, is reviewed in many meteorological texts. 
The parcel method states that convective cells appear 
in nonsaturated air whenever the actual lapse rate of 
temperature is in excess of the dry-adiabatic rate, and 
in saturated air whenever the actual lapse rate of tem- 
perature is greater than the moist-adiabatic rate. 
HEIGHT ——> 
A 
TEMPERATURE —> 
Fie. 1.—An idealized atmospheric sounding. 
Cumulus clouds represent the condition where the 
rising air is saturated while the environment is non- 
saturated. The parcel-method analysis then gives cumu- 
lus growth provided that (1) the surface layer AC in 
Fig. 1 is heated so that the convective currents in this 
layer give condensation at C (that is, C is the convective 
condensation level), and (2) the lapse rate of tempera- 
ture above C is greater than the moist-adiabatic lapse 
rate. 
One of the principal defects of the parcel method is 
the assumption of an undisturbed environment. The 
slice method eliminates the assumption of an undisturbed 
environment by assuming that there are compensating 
downward currents which are accompanied by adiabatic 
heating. This method gives the same stability criteria 
as the parcel method for the cases of a nonsaturated 
layer and an entirely saturated layer. However, for the 
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