224 
size distributions exist that are stable in a sta- 
tistical sense, like the spectral-energy distribution 
in statistical turbulence theory. 
Acknowledgments—The approach used in this 
paper was inspired by the lectures on statistical 
methods in hydrodynamics given by Phil D. 
Thompson at the International Meteorological 
Institute in Stockholm in 1958-59. Numerous dis- 
cussions about problems in cloud physics with 
Wendell A. Mordy, of the same Institute, pro- 
vided the stimulus necessary to complete it. 
APPENDIX I 
The Growth Equation 
The equation for droplet growth by condensa- 
tion has been discussed extensively by several au- 
thors. We shall only briefly state the results, a 
more thorough discussion with references is found 
in the paper by Neiburger and Chien in this vol- 
ume. 
One considers the spherically symmetric case of 
diffusion of water vapor onto a droplet of radius 
r. The steady state provides a satisfactory ap- 
proximation for natural cloud conditions. The 
boundary conditions are given by the vapor den- 
sity in the surrounding air, taken to be valid at 
infinite distance from the drop, and the equi- 
librium value at the droplet surface, with due ac- 
count taken of the capillary effect, and of the 
heating of the drop by the condensation. In ad- 
dition one has to consider the vapor-pressure 
jump at the phase boundary. In the case where 
the exponential influences of heating and surface 
tension can be linearized equation (4) becomes 
(nuclei neglected) 
dr _ Dps 1 27M 
dt i MI2Dp; r + I RTr 
KRT? 
where the symbols used are 
D diffusion coefficient for water vapor in air 
K heat conductivity of air 
L latent heat of condensation for water 
M molecular weight of water 
R universal gas constant 
DISCUSSION 
degree of supersaturation 
absolute temperature 
vapor pressure jump parameter 
droplet radius 
surface tension of water 
ps vapor density at saturation 
SL ya ttc) 
APPENDIX II 
Evaluation of the Time Constant 
in Eq. (7) 
The time constant 7, that occurs in (7) for the 
rate of adjustment of the supersaturation in a 
cloud, is defined by 
1 Od In ws | 4rN AF 
-=/1 w 
T ow PaWs 
If we substitute for A from Appendix I, and 
transform the partial derivative inside the brack- 
ets, then 
1 | d In ws =a 4rN FD 
-= 7p. == SS > 
T 
oT ow ML?Dp; 
KRT? 
But (T/dw), = —L/c, and (0 In w,/OT), = 
ML/RT*, so 
il MLw 4arN TD 
-= 1 +t —— ee eee 
T Cale 1 ML?Dps 
KRT? 
At p = 900 mb and 7 = 283°K 
a 0.5/N oF 
REFERENCES 
Howe tu, W. E., The growth of cloud drops in uni- 
formly cooled air, J. Met. 6, 134-149, 1949. 
Mason, B. J., The Physics of Clouds, Oxford Univ. 
Press, 470 pp., 1957. 
Morpy, W. A., Computations of the growth by 
condensation of a population of cloud droplets, 
Tellus, 11, 16-44, 1959. 
SaqurreEs, P., The growth of cloud drops by conden- 
sation, I, General characteristics, Austr. J. Sct. 
Res., 5, 59-86, 1952. 
Woopcock, A. H., Salt nuclei in marine air as a 
function of altitude and wind force, J. Met., 5, 
362-371, 1953. 
Discussion 
Dr. M. Neiburger—I do not want to pretend 
that I followed the details of the computation, 
but I think this approach looks like the sort of 
thing to which we will have to turn. Now, I am 
not sure I see how the variations with time of the 
shape of the spectrum are taken into account, or 
how one goes back to the variations of shape from 
the time spectrum if the spectrum has to be char- 
