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cloud particles which have coalesced and been 
caught up are supercooled or not. If there is no 
glaciation, then we are dealing purely with a 
shower forming as warm rain (by the all-water 
process). If on the other hand there is glacia- 
tion, then the particles of precipitation, which 
have now solidified, can sometimes grow into 
huge hailstones through contiued aggregation 
with supercooled droplets. 
The fact that it makes no difference in this 
mechanism whether the coalescing droplets are 
supercooled or not, has led understandably to 
less importance being attached than formerly to 
the Bergeron-Findeisen process of rain forma- 
tion. There may, however, be grounds for ask- 
ing, whether the disparagement of the Bergeron- 
Findeisen process has not perhaps gone too far. 
At all events we should at least check whether, 
if glaciation were to take place in the growing 
particle later, after an initially pure all-water 
process, this would not bring about an increase 
in the amount of precipitation produced. 
Firstly, here are a few basic considerations. 
The introduction of thermals into the general 
movement of upward currents of air within a 
Cumulus cloud is merely a first approximation 
to suggest the turbulent character of atmos- 
pheric movements. The fundamental works of 
Kolmogorov [1941] and the theoretical reflec- 
tions of von Karman [1948] and Heisenberg 
[1948], which follow on Taylor’s [1922] enquiries 
into statistical-isotropic turbulence, have led to 
an appreciable clarification of the inner struc- 
ture of turbulence. What is above all relevant to 
our present discussion is that a successful dem- 
onstration has been made, on a statistical basis, 
to show that a law of energy distribution exists, 
which in the case of meteorology is valid for an 
extremely wide spectral range of vortex ele- 
ments. It is in accordance with this law that the 
energy contained in the largest vortex elements 
in the atmosphere, which may be several hun- 
dred meters across, is successively transferred 
to others of smaller dimensions until it finally 
disappears in vortices of almost molecular size. 
Two questions are to be raised in this connec- 
tion. 
(1) When Ludlam uses a mathematical de- 
scription of the amount of aggregation to cal- 
culate the growth of hailstones, relying entirely 
on Langmuir’s [1948; Langmuir and Blodgett, 
1948] concept of collection efficiency, which in 
practice is equated to unity, is sufficient allow- 
ance made for the turbulent character of the 
RAYMUND SAN 
GER 
air stream, particularly in view of the fact that 
during ice formation additional heat is liber- 
ated? 
(2) In the mathematical account of how the 
water content of supercooled droplets is trans- 
ported onto neighboring ice particles, is it 
enough to take into consideration only the mo- 
lecular diffusion process? In discussing the prob- 
lem already mentioned of the effect which the 
Bergeron-Findeisen mechanism has on the pro- 
duction of precipitation, we shall have to pay 
particular attention to this second point. With 
regard, finally, to the admissibility of a sta- 
tistical approach to turbulence phenomena, it 
should be pointed out that thereby a variation 
is tacitly included in the product of the growth 
processes, and this may reveal itself in a con- 
siderable multiplicity of size and shape in the 
precipitation particles. 
It is generally acknowledged that, if the ma- 
jor part of the supercooled droplets in a cloud 
were to freeze, this would impair the further 
growth of a solid particle of precipitation 
through coalescence and stop the development 
of hailstones. Ludlam has calculated that for this 
the density of ice-forming nuclei would have to 
be 10/em*, and Weickmann [1953] arrives at a 
similar value by an analogous argument. It is 
a high figure for the density of icing nuclei and 
it could never be realized artificially, for im- 
stance, through seeding the atmosphere, by any 
economically feasible methods. 
Ludlam’s calculation of the nucleus density 
necessary to bring about a sufficient glaciation 
of the supercooled parts of a cloud to prevent 
coalescence, is based, as I have suggested previ- 
ously, on the same theoretical ideas that under- 
lie the Bergeron-Findeisen mechanism. And here 
the transference of the water from the super- 
cooled droplets onto the precipitation particles, 
which have themselves solidified into ice under 
the influence of ice-forming nuclei, occurs ac- 
cording to the laws of molecular diffusion. Since 
the freezing process releases considerable quan- 
tities of heat, however, and thereby initiates pro- 
nounced turbulent intermingling, the following 
experimental enquiry is suggested: May not a 
closer approximation to reality be reached if, m 
the expression for the movement of water from 
the fluid droplet to the ice particle, the constant 
D of molecular diffusion (D = 0.45 em*/see in 
Ludlam’s calculation) is replaced by a constant 
more in keeping with turbulent diffusion? The 
conception of this constant (which cannot, for 
