A Stochastic Closure for Two-Moment Bulk Microphysics of Warm Clouds: Part I, Derivations

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 ODEs without any ad-hoc parameters or any assumed droplet size distributions and flexible enough to accept any collision kernel without further derivations.