COALESCENCE TENDENCIES IN CLOUD DROPLET SPECTRA 
and combining (9) and (10) the following equa- 
tion is obtained. 
1 dn’ —2 dAr L4 3 
ni dt Ar dt 1 (11) 
1+ -} (ay — 1) 
a 
Note: ¢ = dx/dz (see Eq. 7); A is defined in (5). 
Values of y are given in Table 1 for cases from 
the writer’s previous condensation study. Choos- 
ing four representative values of y and computing 
the value of the coefficient in (11) for all radit 
larger than one half the mean radius one can see 
that a good approximation for this quantity is 
y’ = 2 + 6/y for the given values of y. Hence 
the time dependence of a can be rather safely 
neglected in integrating (11). 
Putting 
3 
y = 2) 1+ Tae iNGean as 
( + ‘) (ay — 1) 
a 
when 60 > y > 4, (11) may be integrated to give 
Aro \Y’ 
n! = no! (=) ay) < 275) (12) 
r 
Rq. (12) indicates that if Ar remains <Aro/2, n’ 
will be only from 4 to 16 times greater than 
No. 
It is useful now to examine the variations in 
Ar which oecur as a result of condensation. Eq. 
(10) shows that as long as 1 > 1/ay that Aro/Ar 
varies approximately as ¢-"/®, In this range all 
values of Ar are converging. By the time that 300 
sec have elapsed Ar has a time constant 
of more than 1000 sec. In fact when one includes 
the effect of the salt nuclei in the drops this time 
constant is appreciably lengthened so that Ar 
may be treated as nearly constant in the time 
interval of 1000 see or so which we consider here. 
If variations of two in the values of Aro/Ar are 
allowed for, an error in the estimation of n’ by 
a factor as large as 16 may occur. We shall see, 
however, that this variation is small compared 
to the several orders of magnitude variation in 
n’ computed in the cases of different droplet 
spectra. 
Eq. (12) shows effectively that the most im- 
portant factors (mo, y) which determine n’ are 
determined very near the cloud base and are in- 
187 
TaBLe 1—Values of y for various nucleus 
distributions and vertical velocities 
om rertical No. of cu 
Case velocity) diovs (W) ee Ea 
cm/sec 
I-100 100 2.5 X 107 56.2 
II-15 | 15 Fee 10? 4.57 
III-5 | by 165% 107 4.47 
1 
I1I-100 100 
4X 108 10,2 
TaBLE 2—Case III, 100 cm/sec 
N 7 n’ a 
108 lhc LL. 8: | | 
3 x 107 2a | 6760 | 0.98 
107 12.3 3930 0.96 
3 x 108 12.4 | 1780 0.958 
10° 12.6 835 0.945 
9 X 105 12.9 1167 0.922 
7X 10° 13.3 1495 0.895 
5 xX 105 13.8 | 1581 0.862 
3 xX 10° 14.4 | 1407 | 0.826 
9 xX 10! 15.7 870 | 0.758 
Be 08 | Siser | 943 | 0.636 
9 X 103 | 20.0 | 376 0.595 
2X 103 | 24.0 | 211 | 0.496 
TaBLE 3—Case II, 15 cm/sec 
N ip | n’ | a 
3 xX 107 | ys | 
107 | 17.85 | 3760 | 0.980 
3x 106 | 18.10 2005 | (0.966 
108 ei ged 1000 0.951 
4 X 105 18.7 578 0.935 
2.5 X 105 19.2 563 0.911 
2 xX 105 19.9 698 0.880 
105 20.8 570 0.841 
fluenced by the strength of the vertical current 
and the nucleus distribution on which the drop- 
lets form. 
If Ar is considered a constant then with the aid 
of (7), 7 is known and the spectrum of condensa- 
tion produced droplets can be estimated at each 
altitude (or liquid water content). Once the spec- 
trum is determined an estimate can be made from 
Kq. (3) of the number of ‘potential’ first collisions 
(n’ = n/E) per unit time between any size cate- 
gory and the mean radius. 
Such calculations of n’ are given in Tables 2-5 
for four cases from the condensation study. The 
drop size distributions for these cases are shown 
in Figure 2. The calculations were made for a 
cloud liquid water content of 1g/m* which by (8) 
