176 
precipitation for those clouds which extend above the 
freezing level. In the past the extension of a precipitat- 
ing cloud above the freezing level has been taken as 
evidence for the operation of the ice-crystal process. 
Tn view of the existence of nonsupercooled precipitating 
clouds a more rigorous criterion must be adopted. It 
cannot be assumed that the ice-crystal process is operat- 
ing unless it can be established that ice crystals and 
supercooled drops are coexistent. 
The only other precipitation process worthy of serious 
consideration is the coalescence of drops in the gravita- 
tional field. If the drops are of nonuniform size, colli- 
sions will result because of their different terminal 
velocities of fall. The rate of growth by this process is 
dependent on the size and size distribution of the drops 
and on their concentration. Findeisen [14] studied this 
process in a cloud chamber and found that the resultant 
erowth corresponded closely to what would be expected 
if each drop coalesced with all drops in its path. Fim- 
deisen’s measurements were relatively crude and could 
not reveal the collection efficiency of one drop for 
slightly smaller drops. More recently, Langmuir [34] 
has computed the efficiency of collection of small drops 
by larger drops. The details of these computations are 
not presented in the reference but it is believed that the 
results are not completely reliable when the collecting 
and collected drops are of nearly the same size. In 
these computations it was assumed that the drops will 
coalesce if brought into physical contact; the collection 
efficiency is determined by the aerodynamic forces 
which tend to cause the smaller drops to follow the 
air streamlines around the larger drop. 
An intelligent appraisal of the two precipitation pro- 
cesses outlined above must be predicated on a quantita- 
tive analysis. Unfortunately, few such analyses have 
been made, and indeed many of the requisite data are 
lacking. The ice-crystal theory involves a molecular 
diffusion process. An expression similar to equation (1), 
derived for the geometric shape of the ice crystal rather 
than of a sphere, is required to permit a quantitative 
discussion of this process. The solution of the diffusion 
equation for geometric forms approximating ice crystals 
has not been given. In an unpublished study, the writer 
has obtained a solution for a thin circular dise which 
might be a useful approximation to some ice-crystal 
forms. Under the rather severe limitations imposed by 
the lack of a suitable equation, only approximate results 
can be obtained. The time required for an ice crystal of 
mass equivalent to a sphere of 20-» diameter to grow 
to an equivalent sphere diameter of 200 yu is of the 
order of 5 to 10 minutes. It was assumed that the vapor 
was saturated with respect to water at the optimum 
temperature of about —15C. To a fair degree of ap- 
proximation the time required for the growth of the 
crystal increases with the square of the equivalent 
sphere diameter. Thus the time required for a crystal 
to grow to a mass equivalent to that of a raindrop of 1 
mm diameter would be of the order of several hours. 
These numerical values are only approximate and are 
for the most favorable conditions of supersaturation 
with respect to ice. The ice-crystal effect is capable of 
CLOUD PHYSICS 
producing crystals of mass comparable to drizzle ele- 
ments in a few minutes but an excessive time is ap- 
parently required to form crystals of mass comparable 
to raindrops. 
With the aid of Langmuir’s computed collection 
efficiencies of drops by larger drops [84] it is a relatively 
straightforward task to compute the growth of drops 
by accretion in the gravitational field. Langmuir’s paper 
is so recent that no such calculations have yet appeared 
in the literature. The writer has made a few preliminary 
calculations which will have to serve as the basis for 
the present discussion.? The problem was simplified by 
considering the growth of an initially somewhat larger 
drop falling through a homogeneous cloud. It will 
suffice to consider one example in which the diameter of 
the homogeneous cloud drops was assumed to be 20 p, 
the liquid-water content 1 g m~*, and the diameter of the 
larger drop 30 pu. The growth of the drop under these 
conditions 1s presented in Table IV. The relatively long 
Tasie IV. Growrn or A Drop or IniTIAL DIAMETER 30 u 
FALLING THROUGH A CLouD or 20-4 Drops 
ContaInine 1 gm? Liqguip WaTErR 
Time (cumulative) Distance fallen 
Drop diameter (x) 
| (minutes) (cumulative) (meters) 
30 | 0 | 0 
40 45 65 
60 74 163 
100 92 322 
200 | 105 650 
500 116 1475 
1000 | 123 2675 
time required for the drop to grow to 100 », compared 
with the time required for it to grow from 100 to 200 
u, is striking. The larger the falling drop is in relation to 
the smaller cloud drops, the more rapid the accretion 
process. Thus, this process is favored by a broad cloud 
drop-size distribution. The data in Table IV should be 
considered as examples and not as definitive numerical 
values. More refined computations, based on a typical 
drop-size distribution, should be made. 
The two precipitation mechanisms can now be com- 
pared on the basis of the approximate numerical results 
presented above. The ice-crystal effect is more rapid 
than the collision process in the initial stages and is 
independent of the drop-size distribution. In the latter 
stages of growth the collision mechanism is more rapid 
than the ice-crystal effect, and may also initiate pre- 
cipitation in clouds which do not contain ice crystals if 
the drop-size distribution is sufficiently broad. The 
two precipitation mechanisms taken together appear 
to be sufficient to explain the formation of precipita- 
tion. The writer’s concept of the roles of the two 
processes is as follows: In all cases in which ice crystals 
are present, the ice-crystal process is domimant im the 
initiation of precipitation and in causing growth to an 
equivalent sphere diameter of the order of a few hun- 
2. Subsequent to the preparation of this article the writer 
has extended these calculations. They will be found in “‘A Pre- 
liminary Quantitative Analysis of Precipitation Mechanisms,” 
by H. G. Houghton, J. Meteor., 7: 363-369 (1950). 
