222 
that the actual supersaturation should be very 
well approximated by 
S = g dp/dt (8) 
Since the supersaturation is so sensitive to the 
actual rate of condensation, we may conclude that 
the water vapor will be condensed as fast as it is 
being made available by the cooling. The total 
rate of condensation is thus a function of the ex- 
ternal influences on our air parcel, and essentially 
independent of the details of the condensation 
process. If we denote this rate by qo, then from 
(5a) 
SF — B = qo/4rN 0 (9) 
This relation holds as long as the implied varia- 
tions in S are slow in a time scale defined by the 
time constant T. 
The development of the droplet spectrum—The 
growth equation for a single drop as stated in (4a) 
can be transformed into a set of equations for the 
statistical parameters of the droplet spectrum by 
the following procedure. First we introduce the 
variables 7 and 2, as defined above, 
cet ks ene: d B 
qe aa aN sea) 
We shall limit ourselves to cases where r < 27, 
that is, the upper limit for the droplet size is 
twice the mean radius. In that case the fractions 
in (10) can be developed in geometric series 
(10) 
dee 
ee dt ; dt 
M (10a) 
AS AB 
= en) se 22) Ol See) 
Tr p=0 T p=0 
After rearrangement of terms 
(1 + 2) dr r3 dz 
3A dt A dt 
* (10b) 
= > [Sr — (1+ »)B](—2)# 
u=0 
Averaging all terms in (10) yields 
dr? 
Ss SS SB 
SA dha 
s (11) 
+ (esr = @ tw Ble# 
p=2 
This equation is used to eliminate d7*/dt from 
(10b). If now the equation is multiplied through 
with an arbitrary power x” of 2, and averages 
CLAES ROOTH 
are taken again, then the following expression is 
found for the rate of change of any a” 
et dx = 2SF 3B)x™ 
ates aii ae 24 
+ > (-pelsr — 1 + Bl 2 
iz 
= [gmt aa ze(am + zmt)| 
It was shown above that the expression Sr — B 
is a function of the enforced average rate of mass 
growth. We shall introduce the notation 
Si —B=oB 
in (11) and (12), and so, with some transforma- 
tion of the derivatives, our system of equations 
is 
re diln7 £2 
a =o+ » (—1)'e — px 
AB dt ro 
Fara ee 
ae is In (z")1m = — (26 = 1) (13) 
8 
, gmth — gh(gm + amt) 
> Stes ) ———_ 
SF fzS ( ) ( K = 
This is an infinite set of equations, each con- 
taining an infinite number of terms. The possi- 
bility of deriving solutions to the system depends 
upon the rate of convergence of the series. Some 
conclusions may be drawn however, without ac- 
tually going to the problem of solving the equa- 
tions. It is first of all clear that the condensation 
process is a powerful agent for changing droplet 
spectra only when the droplets are small. All rates 
are inversely proportional to 7*. It is further evi- 
dent that spectra of the same shape, in the sense 
that the set of a” is identical between them, will 
develop in the same way if subject to the same 
forcing in terms of the specific growth rate o. A 
doubling, say, of the mean radius, will carry with 
it the same change in spectral shape irrespective 
of whether it means growth from 2 to 4, or from 
20 to 40 microns. The time scale would differ by a 
factor 1000, however. 
We shall now have to consider the actual values 
of the parameter o. Figure 2 shows how it is re- 
lated to vertical velocity and total number of ac- 
tive particles, for the pressure 900 mb and tem- 
perature 10°C. The relation is rather insensitive 
to variations in p and 7 within the range of me- 
teorological interest. It is obvious that values suf- 
ficiently low to change the sign of the leading 
terms in (13) would hardly occur. Points corre- 
