WMOD00434

• Title: The Generation of Secondary Ice Particles in Clouds by Crystal-Crystal Collision
• Date: 11/11/1978
• Weather Modification - Doc Type: Report

2170

JOURNAL OF THE ATMOSPHERIC SCIENCES

then

where vg and v~ have been replaced by the observed
terminal velocities.
If the number of fragments generated can be de- .
termined as a function of the change in momentum,
then the total fragment generation rate can be found.
This "fragment generation function" has been found
for each of five different crystal types and will be
described in Section 3. Suffice it to say the fragment
generation functions take the form

Nijkl=a+(3log.:lMijkl+'Y(log.:lMijkl)2, (8)

where Nijkl is the number of fragments per collision
for crystal type i, size j hit by crystal type k, size l,
and a, (3, 'Yare constants determined for each of the
five crystal types studied.
It should again be reemphasized that Nijkl is an
approximation due to the prohibitive task of obtaining
a true N ijkl.

3) TOTAL FRAGMENT GENERATION

The fragment generation rate for collisions between
crystal type i, size j and crystal type k, size 1 is

(dC) = FijklNijkl,
dt ijkl

where C is the concentration of all crystals in a cloud,
t is time, (dC/dt)ijkl is the contribution to the total
fragment generation rate due to collisions between
crystal type i, size j and crystal type k, size l, Fijkl is
the collision frequency and Nijkl is the fragment gen-
eration function.
The total fragment generation rate for crystal- type
2, size j colliding with all other crystals in a cloud is

(dC) = L: L: FijkzNijkl.
dt ij k I

The total fragment generation rate for all crystal
types and sizes in a cloud is

dC
- = L: L: L: L: F ijklN ijkl.
dt i j k I

From Eq. (1), letting E= 1, dC/dt becomes

dC . .
-= L: L: L: L: C;jCkIAijkllvij-VkdNijkl. (12)
dt i j k I

Now if we define a distribution function for crystal
type and size,

C;j
.Pij=-
C

and

Ckl
Pkl=-,
C

VOLUME 35

dC
-= L: L: L: L: CPijCPklAijkd Vij-Vkd Nijkl, (14)
dt i i k I

where C is a function of time. Since C is not an ex-
plicit function of crystal type or size, it may be
removed from the summations, i.e.,

dC
_=C2 L: L: L: L: PijPkzAijkd Vij-Vkd Niikl. (15)
dt i i k I

"The terms in the summations are functions com-
pletely of crystal type and size distributions. These
may change in time but are independent of the con-
centration C. -

b. Analytic solutions
We define the function

K(t) = L: L: L: L: PiiPkzAijkd Vij-Vkd Niikl (16)
i i k I

and substitute it into Eq. (15)

(9)

dC
-=K(t)C2.
dt

(17)

In general K (t) changes with time because the dis-
tributions change as fragments are produced. The
production of fragments will cause the distributions
to have fewer crystals at large sizes and more crystals
at small sizes. The effect of this should be to cause
K (t) to decrease with time.
As a first approximation K (t) can be assumed to
be constant, i.e.,

K(t) =Ko.

(18)

(10)

Integrating (17) with -the assumption in (18) gIves
the solution

Co

(19)

C

,
1-CoKot

where Co is the initial total concentration of crystals.
(11) This solution is plotted in Fig. 1. It has the following
characteristics:

1) If Ko is zero the concentration does not change
with time and there is no secondary particle generation.
2) If Ko is negative the concentration decreases
with time. However, from Eq. (16) we can see that
K 0 can never be less than zero.
3) If Ko is positive the concentration increases with
time. In fact, it increases at a rate greater than an
exponential increase because it approaches infinity at
some finite time.
4) The greater Ko and Co the faster the concen-
tration increases.

(13)