Growth by Accretion in the Ice Phase 
R. H. Doveuas 
Meteorological Service of Canada, and McGill University, Montreal, Canada 
Abstract—The growth of spherical ice particles of various densities by sublimation 
and accretion is considered. The less dense the particle, the greater the mass it must 
achieve by sublimation before accretion becomes the dominant growth mechanism. 
Once this stage is achieved, however, growth rates of particles of the same mass are 
relatively insensitive to particle density, the cloud water content exercising the major 
control. With low water content (0.1 gm m™) such as in stratiform clouds and in the 
dilute peripheral regions of cold Cumulus, the precipitation products are essentially 
sublimation elements rather than graupel. In four gm m™* Cumulus, low-density grau- 
pel can grow to millimeter size within six minutes and to centimeter size within ten 
minutes, much denser particles requiring only a few minutes longer to reach the same 
sizes. These times are comparable with the observed elapsed times of about 15 min be- 
tween the detection of the first radar echo and the first appearance of hail at the 
ground. 
Introduction—The accretion process is im- 
portant to the production of graupel and hail; 
there is a need for study of the initial growth 
of the primitive ice particle and its transforma- 
tion from a sublimation to an accretion element. 
The appearance of graupel seems to be related 
to the development of electrical fields in Cumu- 
lus [Fitzgerald and Byers, 1958]. While the 
Thunderstorm Project [Byers and Braham, 1949, 
p. 48] reported very few flight encounters of 
hail in thunderstorms, this may have been due in 
part to the localization of hail within the storm; 
identification of graupel would likely be diffi- 
cult from an aircraft. However Kuettner [1950] 
claims that graupel is the most frequently ob- 
served hydrometeor in thunderstorms occurring 
over the Zugspitze (at 10 kft), beg always as- 
sociated with high electric fields; large hail was 
found to be a rare and by no means a neces- 
sary occurrence in the 125 storms which he 
studied. Thus, while hail may not accompany 
every thunderstorm, it appears that graupel 
does. 
For a small frozen droplet, growth proceeds 
mainly by sublimation until a critical size is 
reached, following which accretion predominates; 
it has usually been particles above this critical 
size whose growth has been studied, Ludlam’s 
[1952] study of the development of ice parti- 
cles and their role in shower initiation being a 
noteworthy exception. Also, the graupel prob- 
lem involves densities which may be as low as 
0.04, according to Magono’s [1954, p. 38] data, 
and relatively few studies have considered den- 
sities even as low as 0.1; the present paper ex- 
amines the growth of spherical particles (such 
as frozen droplets) of densities from 0.05 to 0.9, 
by sublimation and accretion occurring together. 
Growth rates—Sublimational growth at a rate 
(dm/dt), supplies heat to a growing spherical 
particle at a rate 
Q, = L, (dm/dt), = L. 4rSCDAp 
where L, = latent heat of sublimation, S = par- 
ticle radius, C = ventilation factor, D = dif- 
fusivity of water vapor in air, Ap = excess of 
ambient vapor density over that at the crystal 
surface. Growth by accretion at a rate (dm/dt). 
supplies further heat at a rate 
Q: = L, (dm/dt). 
L, being the latent heat of fusion. Heat is lost 
to the environment at the rate 
Q; = 4crSKCAT 
where K = thermal conductivity of the environ- 
ment and AT = excess of temperature at the 
crystal face over that of the environment. In 
the steady state, Q: + Q. = Q;, whence 
Ko Ly ee] 
Apa _ 1 
AT, ~ DE: |) 4aKSCAE 
in which the subscripts a on the left hand side 
indicate the presence of accretion. The physical 
significance of Ap./AT. is shown in Figure 1 in 
an environment which is saturated with respect 
to water, such as would exist in the supercooled 
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