MASS (pgm) 
10° 
--- 00:05 
o-2 
1o'b 14 0-9 
Coalescence 
8 Ob 2 #4 
TIME (min.) 
6 18 20 
Fic. 2—Mass of a spherical ice particle as a 
function of time; the coalescence curve shows the 
growth of a liquid droplet in 4 gm m-*Cumulus; 
cross-marks on the 4 gm m“$ curves indicate parti- 
cle diameters of 1 cm and (on the coalescence 
curve) of droplet radius 50u; open circles indicate 
masses at which the sublimational and accretional 
growth rates are equal (in the As case, growth is 
predominantly sublimational within the range of 
the figure) 
of a water-saturated environment at 700 mb 
and —15°C. Accretional growth rates were cal- 
culated by means of the growth equation 
am IP (S + r)UEW, d 
—= 2 uh 
dt i" us 7 T 
where S = particle radius, 7 = droplet radius, 
U = velocity of particle with respect to the 
droplet, Z = collection efficiency of particle with 
respect to the droplet, W,dr = liquid water con- 
tent contained by droplets of radius 7 to (r + 
dr). Droplet growth by condensation and coa- 
leseence has been treated at length by Hast 
[1957], whose derived cloud droplet distributions 
appropriate to 1 and 4 gm m™* Cumulus were 
used in the present work; for stratiform cloud 
of low water content, Diem’s [1948, p. 261] dis- 
tribution for Altostratus was used. 
For convenience in comparing the collection 
R. H. DOUGLAS 
characteristics of various particles in various 
environments, it is useful to consider an ‘effec- 
tive collection efficiency’ H’, the fraction of the 
liquid water content, contained within the swept 
volume, which is accreted. Thus H is defined by 
the equation 
dm/dt = 2S°v,.E’W 
where v, = terminal speed of the collecting par- 
ticle, and W = liquid water content. In this way 
EH’ incorporates the effects of droplet size dis- 
tribution and of relative speeds. Figure 3 shows 
EY’ as a function of mass, for particle densities 
of 0.05, 0.2, and 0.9, in 1 and 4 gm m™® Cumulus 
and in Altostratus. The improvement in collec- 
tion as particle density increases, and as one 
proceeds from Altostratus to increasingly dense 
Cumulus, is clearly dicated, as also is the ap- 
proach, for large particles, toward a maximum 
value. However an extension of these curves to 
even larger masses tends to show a decrease in 
EY’; such a decrease at large radii is evident in 
Langmuir’s [1948] Table 4 and in Gunn and 
Hitschfeld’s [1951] Figure 1. 
Figure 4 shows the growth rate as a function 
of particle mass, obtained by the addition of the 
sublimational and accretional rates. The parti- 
cle masses at which these two rates are Just 
equal, when the particle density is 0.9, are indi- 
cated by the arrows; for densities of 0.2 and 
0.05 the critical masses are displaced to higher 
values by factors of about 3 and 8, respectively. 
Thus the denser the particle and the higher the 
cloud content, the smaller the mass, and the ra- 
dius, to which growth must proceed before ac- 
cretion becomes dominant. Considering cloud of 
low liquid water content, it is seen that particles 
of mass greater than 10° to 10° »gm may be ex- 
pected to exhibit riming. This is in general agree- 
ment with observations of (unrimed) spatial 
dendrites and of crystals with droplets attached; 
of these erystal types, the largest observed by 
Nakaya [1954] had masses in the range 300-500 
pgm. Particles of masses greater than these were 
in the form of graupel [see Nakaya, 1954, Fig. 
224). 
Also shown in Figure 4 is the growth curve, 
by accretion only, of a 0.9-density particle in 
4 em m™ cloud. This particle density is nearly 
enough equal to that of a liquid droplet that this 
curve may be used to describe droplet growth by 
coalescence, and to compare droplet growth with 
graupel growth. 
