WMOD00399

• Title: WMO Training Workshop on Weather Modification for Meteorologists - Lecture Notes
• Date: 12/1/1979
• Weather Modification - Doc Type: Report

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2) The continuity equation, which is normally given in the anelasfic formulation
by Ogura and Phillips (1962):

'V . Pe~ = 0

(2)

3) The gas law as applied to dry air and water vapor.

4) The first law of thermodynamics.

5) An assumption of how to deal with the eddy viscosity.

Equations 1, 2 and the first law of thermodynamics are candidates ,in which possible
interactions with the particles like water droplets (< 200 ~m) and drops (> 200 ~m) or ice
particles have to be included. Specifically. it can be stated that the buoyancy term of
equation 1 has to include the changes in total air parcel density due to the presence of
cloud and precipitation particles. Thus. d is defined by accounting for the sum of the
. mixing ratios for cloud water rc. precipitation water rp and ice particle content rs:

d

(rc + rp + rs) ~ = rtot ~,

(3)

and the buoyancy term B reads now:

B = - [~- ~e - rtot 1 ~.

(4)

where rtot is total mixing ratio of all solid or liquid H20 components in any air parcel
consider ed.

The gas law does not need to be changed since the particulate matter occupies
only a fraction of order 10-6 of the total volume.

The first law of thermodynamics needs to be formulated in a fashion so that
latent heats of evaporation and freezing are included as well as heat sources or sinks due
to the presence of particles with temperatures different from that of the surrounding air.

3. Cloud Microphysics

The key equation of microphysics is the so-called kinetic equatio~ which
characterizes the evolution of the size spectra of the water and ice particles in a cloud
environment. Expressing the particle spectra with a number density function f(mi, ~, t)
for water, i = 1, or ice (i = 2) particles of mass m at location x and time It, the
kinetic equation is of the form: -

afi a
ar:- + 'V . f i ~mi + am

[ D iml
fi Dt j

2
E Ci]. + Bi + Fi + Ni 6 (m - mOi)
j=l

(5)

The first term on the left represents the local time change of the particle concentration;
Vmi is the absolute particle velocity, hence the second term represents the:net advection
of particles with mass m; the third term is the net advection of m particles in mass
space. The right-hand side is broken into four terms: the first represent9 the concen-