East Asia and Western Pacific METEOROLOGY AND CLIMATE

347
where
V(x) = terminal fallspeed of raindrop with mass x
K(x',x") = collisional kernel between raindrops of mass x 1
mass x"
and
q(x',x") = probability of coalescence when two raindrops of
mass x j and x" collide
S{x|x',x") = function of the number of fragments with mass
between x and x+dx when raindrops of mass x 1
collide and break up
and x"
The first term on the right hand side is the production of
raindrops of mass x due to coalescence. The second term is the
depletion of raindrops of mass x due to coalescence. The third
term is the production of raindrops of mass x due to breakup and
the fourth term is the depletion of raindrops of mass x due to
breakup.
In order to simplify the derivations, the vertical updraft
is taken to be a constant. No and A are assumed to change only
with height and not with time. Other formulations are:
raindrop size distribution: f(x,h)dx = N Q (h) e" D> iD
raindrop mass: x = a D
fallspeed: V = a D
Collfsional Kernel: K(x',x") = \ (D'+D") 2 E|V(D")-V(D J )|
breakup probability: qtx'/.x") - e
2 — A x
fragment number probability: SCxjx'rX 11 )' = {x'+x M )A e
Expressing f(x,h,t)=No{exp(-XD), there are two dependent
variables, i.e. No and A- Thus, two equations are needed in
order to obtain the solutions. The first moment and the second
moment equations of the stochastic collection equation are
derived. Physically, the first moment equation is to keep track
of the total particle mass in a unit volume against height and
the second moment is to keep track of the radar reflectivity
factor against height. The results from this set of two
simultaneous equations give the No and A values at different