  Analytical solutions are derived for the steady-state size distributions of
precipitating rain and snow particles assuming growth via collection of
suspended cloud particles. Application of the Liouville equation to the
transfer of precipitating mass through size bins in a cascade yields a
characteristic gamma distribution with a "Marshall-Palmer" exponential tail
with respect to particle diameter. For rain, the principle parameters
controlling size distribution shape are cloud droplet liquid water path and
cloud updraft speed. Stronger updrafts lead to greater concentrations of large
precipitating drops and a peak in the size distribution. The solutions provide
a means for relating size distributions measured in the air or on the ground to
cloud bulk microphysical and dynamic properties.
