CLOUD DROPLET GROWTH BY CONDENSATION 
ment in time of these moments. We shall however 
find it convenient to use the equivalent set of 
variables No , 7, @’(v > 1), with x defined by (1). 
The time derivative of an average of a function 
of r will depend in part on the rate of growth of 
the individual particles, and in part on any 
changes in the distribution that may occur be- 
cause of turbulent transfer or settling of particles, 
or by coalescence. We shall assume that particles 
are conserved in our air parcel, hence only the 
particle growth by condensation is considered. In 
that case the operations of averaging and differ- 
entiation with respect to time are interchangeable 
in order, so for any differentiable function f(r) 
| (Gh: = ; 
LES TES (3) 
f@) = ai” 
The growth equation and the basic integral con- 
straint—The equation governing the growth of a 
cloud droplet in a cloud of moderate droplet con- 
centration can be stated in the form 
dr A 8} 
#4 (s_ 2) (4) 
The definition of the different symbols is briefly 
discussed in Appendix I. In order to simplify the 
arithmetic operations / is neglected. This step may 
also be defended on the grounds that recent ex- 
perimental work by the author (unpublished) in- 
dicates that its value is less than 2 * 1074 em, 
which limits its effects to the initial stage of cloud 
development. So our basic equation is 
dr A ( 2) 
eee eal (ic eee a 
dt r r 
Our next aim is to see how the supersaturation 
S is coupled to the cooling rate. The total rate of 
production of liquid water in a unit volume of 
cloud is 
(4a) 
q -/ ndnr? (5) 
0 dt 
Substitution for dr /dt from (4a) yields 
q = 4rnNoA (SF — B) (5a) 
The rate of change of the supersaturation is de- 
termined by the combined action of temperature 
change and water consumption. Denoting the 
mixing ratio of vapor to dry air by w, and satura- 
tion conditions by subscript s 
1 dw 
ws dt we dt 
ds d w— ws 
dt dt Ws 7 
w dws (6) 
221 
Microscale 
Mesoscale 
Fic. 1—An example of the logical intereonnec- 
tions between the variables in a model cloud; 
mesoscale variables steer the development on the 
microscale, but they are not influenced by the de- 
tailed behavior of the microscale variables 
Now —pa ~ where pz is the density of the dry 
air, is clearly identical with q in Eq. (5). Further- 
more w, is a function of pressure and temperature, 
and since the system we consider does not ex- 
change heat with its surroundings, it is a function 
of the pressure and of the total amount of con- 
densed water. So (6) may be rewritten as 
dS —4rN,A(ST — B) w 
dt Pas we 
dw,dp — dws dw 
op dt dw dt 
w dws | 4rN oA 
- E dee (St — B) 
Ws OW Pals 
w dws dp 
w? dp dt 
(6a) 
We observe that dw,/dw < 0 anddw,/dp > 0, 
so that (6a) is of the form 
dS/dt = —(1/r)[S — g(p, w)dp/dt]} (7) 
where 7 and g are positive definite functions of p, 
w, and #, and dp/dt is prescribed by the particular 
mesoscale system of which our air parcel is 
thought to be a part. The time constant 7 is seen 
to have the form ¢c/Nor, where the coefficient ¢ is 
a function of p and w. Its evaluation is discussed 
in Appendix II. We find that its magnitude is 
about 1 sec/cm? for fairly typical conditions ob- 
tained in low clouds. Hence our time constant will 
normally be much less than one minute, while the 
convective processes in a cloud generally have 
time scales of several minutes or more. It follows 
