GROWTH BY ACCRETION IN THE ICE PHASE 265 
water cloud in which accretional growth would 
occur, 
Now K/DL, = Ap./AT., for sublimation 
alone [Marshall and Langleben, 1954], so that 
a Aps dm 7m 
ease 1 — 2, || eK SCAT) 
AT a AT; diac 
and since AT is positive, the crystal being 
warmer than the air, it follows that Ape/ATa < 
Ap:/AT,, the difference being proportional to 
the accretional growth rate. Thus the effect of 
accretion is to increase the temperature excess 
of the growing particle, and so to reduce its va- 
por density excess which in turn reduces the sub- 
limational growth rate. The total growth rate, 
then, is the sum of the sublimation-only and ac- 
cretion-only components, the former reduced by 
a term which is proportional to the latter. At 
700 mb and —10°C the reduction term is ap- 
proximately three per cent of the accretional 
growth rate; it increases with temperature and 
with pressure to about six per cent at 1000 mb 
and 0°C. Considering all other possible sources 
of error in growth calculations, 1t appears rea- 
sonable to ignore this relatively small reduction, 
and to consider the total growth rate as the sum 
of the two components, sublimational and ac- 
cretional. 
Density of particle and of accreted material— 
The density of the particle is important in de- 
termining its terminal speed, and may of course 
vary during growth. Densities of hailstones are 
usually estimated to be about 0.7-0.8 [see, for 
example, Weickmann, 1953, p. 129]. Densities of 
graupel have been measured by Nakaya [1954, 
p. 115] who reports a value of 0.125, invariant 
with particle diameter, over a range of 1.5 to 6 
mm. Magono [1954], theorizing on the terminal 
speed of graupel, finds agreement with observed 
values when the density is of that order, but 
presents observational data indicating densities 
as low as 0.04. That the density of the accreted 
material is dependent upon the ambient temper- 
ature and liquid water content is suggested by 
riming data of Clark [1948] and by the results 
of Melcher’s [1951] experiments [see Weickmann, 
1953], also by the riming data of Langmuir 
[1944; see Ludlam, 1958, p. 55]. Generally speak- 
ing, the warmer the temperature, the higher the 
cloud water content, and the greater the velocity 
of impact, the denser is the accreted material. 
Thus, for the smallest particles on which accre- 
tion is just beginning, a relatively low density 
Fig. 1—Curves show schematically the equilib- 
rium vapor density with respect to water p, and 
to ice p; as a function of temperature; at the face 
of an ice erystal, growing by sublimation in water- 
saturated air, the temperature exceeds the ambi- 
ent value by A7’, , and the growth rate is propor- 
tional to the vapor density excess Ap;; accretion 
increases the former to A7’, and reduces the latter 
to Apa 
of acereted material is reasonably to be ex- 
pected; with increasing size and fall speed, this 
density should increase, and Magono’s [1954, p. 
38] data do indeed show a roughly linear increase 
in density from 0.04 to 0.17 as graupel diameter 
increases from 1 to 6 mm. Therefore, in consid- 
ering the growth of particles, particularly in the 
early stages, consideration must be given to rela- 
tively low particle densities. 
In the present study, growth has been treated 
arbitrarily at constant density. When a particle 
of initial mass mo, and density oo accretes mate- 
rial of density o., then the particle density be- 
comes g when it has grown to mass m according 
to the equation 
(o — oa)/(oo — oa) = M/m, 
Thus, as the particle mass increases tenfold, the 
density difference between the particle and the 
accreted material decreases to one-tenth its imi- 
tial value. It will be found (Fig. 2) that this 
density change can be accomplished within a few 
minutes. 
Growth calculations—Growth rates were com- 
puted for spherical particles of densities 0.05, 
0.2 and 0.9, representative of the range from 
Magono’s [1954] least dense graupel to dense 
hailstones. Terminal speeds were determined on 
the basis of Langmuir’s [1948] work. Sublima- 
tional growth rates were calculated on the basis 
