Differences in Coalescence Tendencies in Computed 
Condensation Cloud Droplet Spectra‘ 
W. A. Morpy 
International Meteorological Institute, Stockholm, Sweden 
Abstract—Four numerically computed cases of the growth of a population of cloud 
droplets by condensation are analyzed in terms of coalescence tendencies. Following 
the suggestions of Hitschfeld, 1957; Telford, 1955; and Welander, 1959, particular atten- 
tion has been given to the stochastic influence on the coalescence-produced cloud- 
droplet spectrum. The evidence taken from the numerically computed cases strongly 
supports the contention that collisions between small droplets readily produce a wider 
spectrum than the condensation-produced spectrum. The transition time for a conden- 
sation-produced spectrum to change to a predominantly coalescence-produced spectrum 
depends strongly on the maximum supersaturation and resulting cloud droplet concen- 
tration produced at cloud base. 
In a recently published paper [Jordy, 1959] 
the writer described a theoretical investigation 
on the effects of vertical motion and condensation 
nucleus spectra on cloud droplet spectra. The 
purpose of the present paper is to evaluate the 
differences in cloud droplet spectra which resulted 
in these cases in terms of some important factors 
which lead to the subsequent changing of these 
spectra by the coalescence of droplets. 
The coalescence of droplets is mainly depend- 
ent on three factors. These factors are the size 
spectrum, the concentration, and the coalescence 
(collision times collection) efficiencies of the drop- 
lets. An equation which gives the number of con- 
tacts between two droplet sizes is frequently 
given as [for example, Telford, 1955 
n = EN,Nor(ri + 12)? 
E 
(1) 
collision efficiency between 
7, and rz radius droplets 
Ni, and N» are the concentra- 
tions per unit vol of the 7 
and rz size droplets respec- 
tively 
m(r; + 12)? = the area surrounding a drop- 
let in which a contact could 
be made 
v =relative velocity between 
the 7; and r2 droplets 
Customarily the coalescence of droplets has 
been described by following the growth of a larger 
droplet as it sweeps up smaller droplets in a cloud 
with certain assumed characteristics of droplet 
where 
* Contribution No. 1079 from the Woods Hole 
Oceanographic Institution. 
184 
size and vertical velocity [Houghton, 1950; Mason, 
1957; Bowen, 1950; Ludlam, 1951; Langmuir, 
1948]. The growth of the larger droplet in these 
cases is assumed to be continuous, that is 
1 dn 
p at 
lr 4rr8 
= Sane = ExNAn + To)2v = 
(2) 
where 7; and rz are the radii of the collecting and 
collected droplets respectively. In fact of course 
it is discontinuous, each unit of growth being one 
cloud droplet. 
If one adopts the practice of thinking of the 
coalescence of cloud droplets as in (2) above, then 
it is the initial drop size spectrum which largely 
determines the initial and subsequent relative 
velocities between the droplets. Thus droplets 
which are initially slightly larger, will remain 
larger throughout the coalescence process and 
there will be no tendency for smaller droplets to 
overtake them, except that they may follow 
different trajectories in the cloud [Bowen, 1950]. 
Such reasoning has been followed in the work of 
Houghton [1950], Woodcock [1952], Fournier ad’ Albe 
[1955], and others. In this line of thought Wood- 
cock has reported measuring a correspondence 
between the size distributions of sea salt particles 
and raindrops measured in the same localities and 
at the same time. 
As exceptions to the above treatment of this 
problem there have been three very interesting 
investigations to date [Telford, 1955; Hitschfeld, 
1957; Welander, 1959], which study the impor- 
tance of statistical aspects of the coalescence 
process. 
This statistical process in its simplest form may 
