A Statistical Study of Cloud Droplet Growth by Condensation 
Crags Rootru 
International Meteorological Institute, Stockholm, Sweden 
Abstract—A droplet spectrum is defined in terms of the moments of its frequency 
distribution, and equations are derived for the rate of change of these moments. 
Effects of dissolved salt are not considered; this limits the application of the theory 
to regions well above cloud base. It is shown that the supersaturation adjusts itself 
towards a quasi-steady value within a time given by C(N7)~! where N is the number 
of droplets per cm, 7 is the average droplet radius in em, and C is a coefficient of size 
order one cm-?. The basic parameter used to characterize the cloud development is the 
average growth rate of droplet mass. For very slow rates the droplet size spectrum 
has a tendency to widen, but at higher growth rates a contraction of the spectrum 
occurs. In the limit the linear width of the spectrum is inversely proportional to the 
average radius and the standardized form of the spectrum is invariant. 
Introduction—The development of droplet 
spectra has long been a major concern of people 
engaged in cloud-physics research. Several inves- 
tigators have constructed computation schemes 
for studying in detail how a droplet spectrum is 
formed on a specified nucleus population con- 
tained in a parcel of humid air, as the air is cooled 
past its point of saturation. The causal chain in 
such a model is depicted in Figure 1. One may 
divide the variables of the system into two classes, 
characteristic of the mesoscale and the microscale 
of the development. We shall refer to the meso- 
scale group all those properties of the bulk air, 
which have to do with the hydrodynamical de- 
velopment on the scales of convective elements, 
that is, pressure, temperature, vertical velocity, 
and content of water in liquid and in vapor state. 
To the microscale we refer the details of the drop- 
let size distribution and the chemical properties 
of the aerosol, as well as any turbulence on scales 
that may interfere directly with the coalescence 
process. 
It is evident from the work of previous investi- 
gators that, except at the very onset of the cloud 
formation, the rate of change of the liquid water 
content of the cloud is almost exactly that which 
corresponds to complete utilization of the avail- 
able water vapor. In other words the turnover of 
water is very large compared to the amounts re- 
quired to effect what changes in supersaturation 
that may occur [Howell, 1949; Mason, 1957; 
Mordy, 1959; Squires, 1952]. Since the warming 
of the bulk air by released latent heat and the 
total amount of supported liquid water are the 
only feedback effects from the condensation proc- 
220 
ess on the mesoscale dynamic processes, a de- 
tailed knowledge of the microphysics is not re- 
quired for the study of these scales. On the other 
hand the concept of complete utilization of the 
available water vapor provides an integral con- 
straint on the development of the droplet spec- 
trum, which can be formulated in terms of the 
mesoscale development. 
If a system is governed by integral constraints, 
this forms an ideal basis for an attempt to study 
its development in some suitable statistical terms. 
The present paper is an example of such an ap- 
proach applied to a simple system with some re- 
semblance to what occurs in natural clouds, 
namely, a droplet population with negligible salt 
content, embedded in an air parcel subject to 
steady cooling. 
The basic model—Consider a parcel of humid 
air containing a nearly uniformly distributed pop- 
ulation of cloud droplets. Let the size distribution 
of the droplets be defined by a frequency distri- 
bution n(r), such that Vn(r)6r is the number of 
droplets in the size range r to r + 6r, to be found 
in an air sample of volume V. We shall denote 
averages taken with respect to this distribution 
by a bar (for example, 7). Let us further introduce 
a normalized size variable x, defined by 
TAQ i= ee) 
r 
(1) 
The distribution n(r) is completely defined by its 
moments N,, 
poo 
| r’n(r) dr (v = 0, 1, 2, -:: 
0 
Ny = (2) 
Our goal is to be able to describe the develop- 
