696 
downward motion, the environment is continually 
changing while the cloud grows upward and, therefore, 
it might be expected that the entramment and mixing 
at a particular cloud level are changing with time. When 
this factor is added to the other aspects of mixing it is 
apparent that entrainment is a complex problem. At 
this point the question may be raised as to whether the 
problem of cumulus convection could be more readily 
solved by an approach which recognized the cloud 
and its surroundings as a cell rather than one which 
attempted to treat the cloud and environment sepa- 
rately. 
Effect of Mixing on the Cloud Properties 
Even though it is not yet possible to determine the 
precise details of the mixing of environmental air with 
the ascending cumulus cloud, an insight into the growth 
of cumulus cells may be obtaimed by a theoretical 
analysis of the effect of the mixing on such aspects of 
the cloud as its temperature and liquid-water content. 
A graphical technique for the determination of the 
lapse rate of temperature within the cloud is described 
below and will be followed by a thermodynamic analysis 
of mixing. 
—— PRESSURE 
B D c UA 
(Te»We) (T,w) (T*,w*) 
TEMPERATURE —> 
Fie. 2.—Graphical computation of the effect of mixing on 
the lapse rate of temperature. AF is a moist-adiabatic, BH is 
the environmental lapse rate, and DZ is the lapse rate which 
arises from mixing. 
Graphical Procedure. In the graphical procedure the 
following assumptions are made: 
1. The clouds are formed as the result of heating at 
the ground. 
2. Vertically moving columns of air are subjected to 
adiabatic temperature changes. 
3. The rising air mixes with the air of the environ- 
ment. The physical properties of the environmental air 
can be obtained from the radiosonde data. 
4. When mixing occurs, the nonsaturated air of the 
environment becomes saturated at constant pressure by 
the evaporation of some of the liquid water of the cloud. 
This is saturation by the wet-bulb process. 
5. The lapse rate of temperature within the cloud is 
obtained from the computed values of the temperature 
of the cloud top as the cloud grows upward from the 
condensation level, This assumption ignores the fact 
LOCAL CIRCULATIONS 
that the condition of the cloud is probably changing 
continually as a result of the influence of the descending 
currents upon the temperature and humidity of the 
environment. 
The graphical procedure for the determination of 
the lapse rate of temperature within the cloud is illus- 
trated in Fig. 2. Suppose that at some point in its ascent 
along the moist-adiabatic a cloud parcel (A in Fig. 2) 
is permitted to mix with its environment. If the mixing 
takes place at constant pressure and if some value K 
is assigned for the proportion of the environment en- 
trained by the cloud, the temperature 7, and the mix- 
ing ratio w, of the cloud air are given by 
T,, = (T* + KT.)/( + K), (2) 
Wy = (we + Kae)/(1 + BK), 
where (7*, w*) and (7, w.) are, respectively, the 
cloud and environmental variables before the mixing 
occurs. As a result of the mixing, the cloud point on the 
thermodynamic diagram has moved from A(Z’*, w*) 
I 
mb mb 
50077 500 
600 600) 
« 
2 700 700 
wo 
Ww 
& g00 800 
900 900 
1000 1000 
-20 -1I0 O TO 20F9 S040 
TEMPERATURE °C 
-20 -I0 (e) 10) 20) 30° 40 
TEMPERATURE °G 
mb mb 
500 500 
600 600 
: | 
c 
a 500 700 
wo 
é | 
= 800 800) 
900} 900 
1000 1000 
-20 -I0 0 10 20 30 40 -20 -I0 0 10 20 30 40 
TEMPERATURE °G TEMPERATURE °C 
Fria. 3.—Cloud lapse rates of temperature which could arise 
through mixing. The solid lines, dashed lines, and dotted lines 
are respectively the environmental, moist-adiabatic, and cloud 
lapse rates of temperature. The numbers represent the humid- 
ity of the environment in per cent. In the two upper diagrams 
three parts of cloud air mix with one part of environmental air 
at every 50-mb step; in the two lower diagrams five parts of 
cloud air mix with one part of environmental air at every 50- 
mb step. 
to C(I, w.), where the cloud parcel is no longer 
saturated. Let the parcel become saturated at constant 
pressure by the evaporation of the liquid water of the 
cloud. The final state of the cloud parcel is then given 
by the wet-bulb temperature 7 and the saturation 
mixing ratio at this temperature w. 
Permit the cloud to continue its ascent but now 
along the moist-adiabatic through D, and at a pressure 
p’ repeat the process of mixing and evaporation. The 
condition of the air in the cloud at a pressure of p’ is 
then given by the pomt H(7’, w’) on the thermo- 
dynamic diagram. The lapse rate of temperature within 
the cloud, between p and v’, is given by the line DE. 
