WMOD00277

• Contract/Permit #: 14-06-D-6467
• Title: An Operational Adaptation Program for the Colorado River Basin
• Prepared By: Lewis O. Grant, Chappell, Crow, Mielke Jr., Rasmussen, Shobe, Stockwell, Wykstra
• Date: 10/1/1969
• State: CO
• Country: United States
• Weather Modification - Doc Type: Report

appreciable riming. Furman (1967) has also shown
that the tops of clouds during snowfall at Climax are
generally near the 500 mb level.
a. Rate of ice growth by vapor diffusion
If it is assumed that around a growing
ice crystal there is a steady state diffusion of water
vapor and steady state thermal conduction, the rate of
ice crystal growth by vapor diffusion is usually
expressed as
dm/dt=41TCPiSiG'F1F2 (1)
where

G' = D( P / P.)[ 1 + DL,2 P /R T2Kr1 (2)
v 1 1 V v
and m is the mass of an ice crystal; C the electro-
static capacity factor of the crystal; S. the super-
saturation relative to a plane ice surrJ:ce; L. the
latent heat of sublimation; R the gas consti-nt for
saturated air; D the diffusiori coefficient; K the thermal
conductivity of air; P the vapor density; P . the ice
density; T the ambient temperature; F the1ventilation
factor of the crystal in the air flow; F 1 the vapor
factor that corrects the vapor field to 1hat of a super-
cooled cloud.

The ventilation factor may be written as
a function of the Reynold1s Number or
1
Fll+.22Re2 (3)

The Reynold1s Number is defined by
R 2Vr / \) (4)
e e
where V is the velocity, r is the equivalent radius of
a crystal having the sameevolume as a droplet, and \)
is a kinematic viscosity.

The equivalent radius
determined from the relation
1 2/3 1/3
= 2 a c

of a crystal is

(5)

r
e
where a and c refer to the lengths of the crystal axes.
Combining (3), (4), and (5) the ventilation factor may
be expressed

F 1 = 1 + 0.22 (va2/3 cl/3/ \))1/2 (6)

From (6) it is seen that the ventilation
[adUl' ill tllt: gl'UWtll "'y.U<iti0ii is <i [ul1dim of the
crystal habit and size, crystal fall speed, and the
kinematic visco sity. Through the habit and viscosity
terms the ventilation factor is dependent upon pres-
sure and temperature.

For simplification a mean value
of the ventilation factor may be computed for
conditions that frequently occur at Climax. The
settling velocity relative to the environment is taken a
at 60 cps, and a value of 0.18 is given the kinematic
viscosity. This value is applicable for a pressure
of 700 mb and a temperature of -10C.

After substituting these values into (6)
the ventilation factor is found to equal about 1. 7.

Marshall and Langleben (1954) derived
a factor (F?) that corrects the vapor field to that of a
supercooleo. cloud. An expression for this vapor

factor may be written

1
F2 = 1 + r{4 1T 1: r )2
1 c
where r is the radius of an ice
clou d droplet radius.

(7)

crystal and r is the
c

It is seen from (7) that the vapor
factor is a function of the size a. the growing snow
crystal and the sum of all cloud droplet radii per
unit volume.

For simplification a mean vapor factor
may be evaluated for conditions representative of the
snowfall and continental type clouds present in the
Climax area. A mean radius for snow crystals of
about 400 microns is taken, 3and a mean cloud droplet
concentration of 200 per cm having a mean droplet
radius of 6 microns is assumed.

SuLstituting thde v<ilues into (7) gives
a value for the vapor factor of about 1. 05.

The electrostatic capacity factor (C) is
a function of the crystal shape. Based on typical
temperatures found in winter orographic clouds it is
reasonable to consider that the capacity factor may
be approximated by that of a circular disk, or
C = 2r / 1T .

Substituting the values for the ventila-
ting factor (F 1)' vapor factor (F 2) and capacity
factor (C) into (1) the diffusional growth equation
becomes
dm/dt = 14. 4r[SG'p.J (8)
1

In the case of a crystal growing in an
environment at water saturation the bracketed
quantity in (8) may be solved as a function of tempera-
ture (Mason, 1953) and is henceforth denoted F(T).
From (8) it can be noted that the growth by vapor
diffusion of a crystal in an environment at water
saturation has been reduced to a function of crystal
size and cloud system temperatures.
b. Supply rate of condensate
Thc determination of the rate at whi~h
cloud water is supplied within an orograrhic cloud
system is quite complex since it depends upon a
knowledge of the vertical motion fields over the
mountain barrier. This complexity partially stems
from the verticHl motion fi eld which may be com-
posed of three components: orographic, dynamic and
convective.

The orograp,. ic component depends
mainly upon the direction and speed of the wind flow
relative to the orientation of the mountain barrier,
stability, of the air mass crossing the barrier, and
the nature of the vertical wind shear present. The
dynamic component is controlled by the vertical
distributions of mass divergence associated with
traveling disturbances in the westerlies. On
occasion the thermal instability of the air mass
present may result in the development of convective
cells and lines having concentrated and significant
vertical motions.

6