CLOUD DROPLET GROWTH BY CONDENSATION 
sponding to cases studied by Howell [1949] and 
Mordy (1959) have been entered in the diagram. 
It is interesting to note that Howell’s Case 3 gives 
a value of o = 1.5, which brings it near to the 
critical value. This case, somewhat inexplicably, 
according to Howell, produced a very wide 
droplet spectrum. The condensation nuclei were 
considered in his computation, and they would 
obviously tend to widen the resulting droplet 
distribution, but it seems obvious that the param- 
eter o should be of comparable importance also 
in case the dissolved salt in the droplets is 
considered. 
The fact that the parameter o normally is ex- 
pected to be much larger than unity lends support 
to the concept that (13) might be approximated 
by retaining only the first term on the right-hand 
side in each equation. The equations can then be 
solved, to give the relations 
7 = F(t = 0) + 3ABat 
z(t) \t/n F(0) \ee-/¢ 
a(0)) ra) 
The variable x was normalized with respect to 
F. If we replace it by another variable £, defined by 
g= t// x2 = (r — 7) /+/ (r — 72 
then (14) gives 
em (ft) \1lm 2(0)\12 fet) \ tie 
se ea = =1 (16) 
en(0) x°(t) x(0) 
that is the standardized distribution is invariant 
within this approximation, and the linear width 
of the spectrum, as represented by the standard 
deviation of r from its mean, is proportional to 
Fate. 
(14) 
(15) 
It may seem surprising that the capillarity term 
in the growth equation should play the same role 
irrespective of the size of 7, but this is explained 
by the fact that the supersaturation is inversely 
proportional to 7, and so is the capillary term. 
If the ambient supersaturation is high, then the 
difference in growth rates between various mem- 
bers of the droplet population is controlled by 
the factor r in the growth equation. But if the 
supersaturation is low, then the capillary term 
may overcompensate the effect of this factor. The 
influence of the parameter o on the individual 
droplets is seen clearly if we write the growth 
equation (4a) in the form 
rar: GSE 1 1 (17) 
ABdt fr r 
99° 
ase 
] 
| Howell | x 
| Mordy | | a 
100 Mardy |» 
| Mordy III] © 
w, cm/sec 
Fic. 2—The growth forcing functions as a func- 
tion of particle number and vertical velocity; the 
points marked represent conditions in computa- 
tions made by Howell [1949] and Mordy [1959]; 
lines connect points representing identical nucleus 
spectra 
Differentiation of the right-hand side with re- 
spect to ¢ gives the result that the growth rate is 
largest for the droplet size r = 27/(¢ + 1). It is 
interesting to note that the condition for contrac- 
tion or expansion of the linear width of a fairly 
narrow spectrum is identical with the condition 
that the radius of maximum growth rate be 
smaller or larger than the mean radius. 
Possibilities for further development—The model 
considered in this paper is very simple. The re- 
sults are equally simple and straightforward. It 
seems possible, therefore, that the present formu- 
lation of the problem of droplet growth by con- 
densation could be used as a starting point for 
more sophisticated studies of cloud development. 
It is my opinion, that such extended models 
should not attempt to treat this particular prob- 
lem in greater detail, but that one should build 
up the more complicated system by blocks repre- 
senting its different aspects, each of them being 
simplified in similar extent as the present study. 
The cloud base problem, that is, the question of 
finding the most economic treatment of the initial 
cloud formation and the question of which char- 
acteristics of the nucleus spectrum one should use, 
in order to describe its behavior in clouds in the 
most pregnant way, seems to be one that should 
be attacked along these lines. In the coalescence 
problem the obvious approach is a generalization 
of the ordinary ‘basic-cloud-versus-growing-rain- 
drop’ approach by consideration of a basic cloud 
droplet population, described essentially with the 
method advanced here, and a population of drop- 
lets growing by coalescence alone. One would also 
like to approach the problem of whether drop 
