ON THE PHYSICS OF CLOUDS AND PRECIPITATION 
location, the concentration of nuclei has a diurnal and 
annual variation, the nature of which is largely de- 
pendent upon the local conditions. Correlations have 
been made of nuclei concentration with visibility, air 
mass, wind direction and force, and so forth. The im- 
portant question of the variation of the concentration 
of nuclei with elevation has not been thoroughly studied 
because of the difficulties in the way of such measure- 
ments. Most of these measurements have been made in 
mountainous regions at various elevations. These show 
a rapid decrease in concentration with elevation. A few 
determinations have been made from free balloons. 
These show a more rapid decrease of concentration of 
nuclei with mereasing elevation than the mountain 
observations. The average vertical distribution of nuclei 
from the balloon ascents shows a count of 22,300 per 
cubic centimeter in the layer from 0 to 500 m decreasing 
to 80 above 5000 m. Of necessity the balloon flights were 
made in anticyclonic weather, and the results cannot 
be considered typical of stormy conditions. In any 
event, the rapid decrease in concentration with in- 
creased elevation indicates that the source of conden- 
sation nuclei is at or near the surface. The nuclei are 
presumably carried aloft by turbulence and convection. 
This reasoning would suggest that the decrease in 
concentration with elevation would be smaller in cy- 
clonic than in anticyclonic conditions. No information 
is available on the change in size of the nuclei with 
elevation. It would be highly desirable to obtain more 
information regarding both the number and the size 
distribution of condensation nuclei in the free atmos- 
phere. 
THE INITIAL PHASE OF CONDENSATION 
Reference has been made above to the factors which 
control condensation on both hygroscopic and non- 
hygroscopic condensation nuclei. Referring again to 
Fig. 1, it is evident that condensation nuclei will not 
attain cloud drop size unless the supersaturation cor- 
responding to the maximum of the curves for hygro- 
scopic particles is exceeded. The magnitude of the 
supersaturation required is dependent on the mass of 
the hygroscopic material in the nucleus. The super- 
saturation required in the case of nonhygroscopic nuclei 
is dependent on the radius of curvature of the nucleus 
as indicated by the upper curve in Fig. 1. As was 
pointed out earlier, the critical supersaturation is prob- 
ably also dependent on how easily the surface of the 
nucleus is wetted. If the nucleus is composed of a 
microporous substance, condensation will occur at a 
lower relative humidity, the exact value depending on 
the size of the pores. Even in this case the nucleus 
cannot grow to cloud drop size unless some initial 
supersaturation occurs. It is therefore generally true 
that cloudy condensation cannot occur without a small 
degree of supersaturation. Such data as are available 
on the size and size distribution of nuclei indicate that 
the supersaturation required is usually less than one 
per cent. This is confirmed by the observation that the 
cloud base corresponds to the saturation level within 
the precision of the measurements. 
169 
As the relative humidity exceeds 100 per cent, con- 
densation occurs first on the largest nuclei, that is, on 
those requiring the smallest amount of supersaturation 
to become active. Nuclei are said to be active when they 
have exceeded their critical supersaturations and are 
free to grow to cloud drop size. If the condensation is 
extremely slow, only the very largest nuclei will become 
active. As the rate of condensation is increased, the 
rate of condensation on the larger nuclei will not be 
sufficient to hold the supersaturation down and ad- 
ditional nuclei will be activated. This shows that the 
concentration of cloud drops is dependent primarily 
on the initial rate of condensation. 
The process which has just been described quali- 
tatively has been investigated theoretically and nu- 
merically by Kohler [29] and by Howell [24]. Kohler 
combined the radius of curvature effect and the effect 
of the hygroscopic solute with the thermodynamic 
equations of the condensation process. He did not 
introduce numerical values and did not consider the 
effect of a distribution of nuclear sizes. Howell assumed 
a broad spectrum of nuclear sizes and several different 
rates of condensation. He was able to show that only a 
very small fraction of the nuclei were activated and that 
under reasonable conditions the initial supersaturation 
was less than one per cent. He places the maximum 
probable supersaturation at about three per cent. By 
using different size distributions of the nuclei, Howell 
found that the concentration of cloud drops was much 
more dependent on the initial rate of condensation than 
on the size distribution of the nuclei. As soon as the 
initial stages of condensation are over, the supersatu- 
ration rapidly declines so that it is extremely unlikely 
that any additional nuclei will be activated. Because 
of the rapid decrease in supersaturation it is possible 
that a few of the smaller nuclei which were activated 
may evaporate. Howell contends that although this 
presumably occurs, the number of drops which start 
to form and then evaporate is very small compared to 
the total. It is reasonably safe to state that the con- 
centration of cloud particles immediately after the 
initial condensation is very nearly equal to the con- 
centration of activated nuclei and that there is little 
likelihood of an merease in the concentration of the 
cloud particles thereafter. 
GROWTH OF CLOUD DROPS 
When the initial phase of the condensation process is 
completed, the dissolved hygroscopic material and the 
radius of curvature have a relatively small effect on 
the further growth of the drop. The steady-state dif- 
fusion equation for the growth of drops as given by 
Houghton [21] is 
A(a’) = Sk (pw = Pow) At, (1) 
where A(a?) is the increment of the square of the drop 
diameter in the time Af, p, is the water vapor density 
at a distance from the drop, and pp, is the density of 
water vapor in equilibrium with the drop. As pointed 
out by Howell [24], this equation is not valid during 
the initial condensation phase nor for very small drops 
