COALESCENCE TENDENCIES IN CLOUD DROPLET SPECTRA 
be illustrated by the following example. Suppose 
a group of 100 large droplets is falling through a 
cloud of randomly spaced, uniform size, small 
cloud droplets. Because of the random spacing 
of the cloud droplets, each of the large droplets 
will have a different collision experience with the 
cloud droplets and after some time will represent 
not a uniform-size group but a spectrum of drop- 
lets located at different levels in the cloud. 
If such reasoning is applied to an initial spec- 
trum of cloud droplet sizes instead of an initially 
uniform-size group, a question arises as to 
whether, after some time has elapsed, the result- 
ing drop size spectrum is more dependent on this 
statistical process than on the initial drop size 
spectrum. In other words one may ask the ques- 
tion; do more of the large droplets present repre- 
sent initially large nucleus droplets or do collisions 
between the smaller sizes of droplets ultimately 
produce more large droplets? 
A complete mathematical investigation of this 
problem has not been made but the evidence 
offered by the three investigators above indicates 
that the statistical nature of the problem is very 
important. 
One way of approaching this question is to 
examine the differences in drop sizes and drop 
concentrations which may result from condensa- 
tion with a view toward assessing their coales- 
cence tendencies, that is, how will n (the number 
of first collisions) change in (1) as r grows by 
condensation? It seems that the assumptions 
which are necessary to make such an appraisal 
are not too gross to prevent an examination of 
important aspects of this process. 
As a model for such investigation, assume a 
constantly rising current of air containing a given 
frequency distribution of salt particles. The con- 
densation conditions attending are then fixed by 
the initial conditions such as temperature, pres- 
sure, rate of rise, and the above assumptions such 
that, until the process of coalescence becomes im- 
portant, we can describe the cloud droplet spec- 
trum at each time and altitude above base 
[Mordy, 1959]. We shall then investigate, as time 
elapses, the characteristics of these changing 
spectra which are conducive to coalescence. 
We shall therefore look first at the way n 
varies with time or height if the growth of the 
particles is controlled only by the condensation 
conditions. 
The fall velocity of the drops we consider is 
satisfactorily described by Stokes law (that is, 
for r < 40y) so that 
2gpr? 
v= 
= kr? (k = constant) 
gn 
If the above expression is substituted in (1) 
and the terms rearranged the equation can be 
written 
n = EnxNiNok(r, + r2)3(r — re) (3) 
Only that part of the condensation process 
that occurs above the zone of maximum super- 
saturation (where NV, and N» are determined in 
this model) need be considered here so that 
wN\N2k may be considered as a constant. The 
variation of # is uncertain and therefore for the 
moment will be included in a new variable 
n = n/#. 
If the In n’ is differentiated with respect to time 
the equation becomes 
1 dn’ 3 
n' dt 
iS d(ry + 12) 
m+ re dt 
(4) 
1 d(r, — 12) 
ri + 12 dt 
The time derivatives of the radii can now be 
obtained from the equation for drop growth by 
condensation. Here we shall for the moment con- 
sider only droplets which have grown until they 
are dilute enough to neglect the hygroscopic ef- 
fect of the nucleus. We shall see in the discussion 
below that this makes very little difference in the 
conclusions we draw from our results. If a@ is the 
ratio of the radius 7; to rz or 7; = are (= ar, see 
(6b) below) the drop growth equations for the 
two drop sizes which appear in equation (4) can 
be written [Jordy, 1959] 
dry a 5 
ina (asF — A) 
' (5) 
arg a 
aileron A 
dt Te e ) 
where 
s = (e — es)/es = supersaturation 
_ eel2DJe 1 | 
jin epRITE eL’DJe 
kR?T3 
= const ~ 8 X 1077 egs at 800mb, 10° C 
A = 2cT/JLp 
= capillarity term coefficient ~1.7 K 10-® cgs 
T = temp of drop 
