A stochastic closure for two-moment bulk microphysics of warm clouds: part I, derivations

We propose a mathematical methodology to two-moment parameterization of bulk warm cloud microphysics based on a stochastic representation of the high-order moment terms. Unlike previous bulk parameterizations, the stochastic closure approach proposed here 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 arbitrary collision kernels. Our strategy is a new twofold approach of approximating the kinetic collection equation (KCE). First, by partitioning the droplet spectrum into two large bins representing cloud and rain aggregate particles, we are able to represent droplet densities as the sum of mean and random fluctuation terms. Second, we use a Taylor approximation for the collision kernel around the centres of masses of bulk cloud and rain aggregates which allows the derivation of bulk rate equations for the cloud and rain aggregate droplet numbers and mixing ratios that are independent of the collision kernel. Detailed numerical simulations of the KCE demonstrate that the high-order (quadratic and cubic) stochastic fluctuation terms can be neglected which results in a closed set of equations for the mean bulk cloud and rain aggregates with only five parameters. Considerations of consistency of cloud number concentration and mass conservation constraints further reduce the parameter set to three key entities representing, respectively, the strength of cloud self-collection, the strength of auto-conversion, and the strength of rain self-collection, relative to auto-conversion. In an accompanying paper, bounds on the parameters’ space are derived and the mean bulk two-moment parameterization is validated against direct simulation of the KCE and compared to an existing competitor parameterization.