  We propose a mathematical methodology to derive a stochastic parameterization
of bulk warm cloud micro-physics properties. Unlike previous bulk
parameterizations, the stochastic parameterization does not assume any
particular droplet size distribution, all parameters have physical meanings
which are recoverable from data, and the resultant parameterization has the
flexibility to utilize a variety of collision kernels. Our strategy is a new
two-fold approach to modelling the kinetic collection equation. Partitioning
the droplet spectrum into cloud and rain aggregates, we represent droplet
densities as the sum of a mean and a random fluctuation. Moreover, we use a
Taylor approximation for the collision kernel which allows the resulting
parameterization to be independent of the collision kernel.
  To address the two-moment closure for bulk microphysical equations, we
represent the higher (third) order terms as points in an
Ornstein-Uhlenbeck-like stochastic process. These higher order terms are
aggregate number concentration, and aggregate mixing ratio, fluctuations and
product fluctuations on regions of a 2D space of pre-collision droplets. This
2D space is naturally partitioned into four domains, each associated with a
collision process: cloud and rain self-collection, auto-conversion, and
accretion. The stochastic processes in the solution to the kinetic collection
equation are defined as the sum of a mean and the product of a standard
deviation and a random fluctuation. An order of magnitude argument on the
temporal fluctuations of the evolving cloud properties eliminates the terms
containing a standard deviation and a random fluctuation from the mean
evolution equations. The remaining terms form a coupled set of ODE's without
any ad-hoc parameters or any assumed droplet size distributions and flexible
enough to accept any collision kernel without further derivations.
