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

TL;DR

This paper introduces a novel stochastic parameterization method for warm cloud microphysics that does not rely on specific droplet size distributions and is adaptable to various collision kernels, improving modeling flexibility.

Contribution

The authors develop a new two-fold modeling approach using stochastic processes and Taylor approximations to derive a flexible, kernel-independent bulk microphysics parameterization.

Findings

Parameterization does not assume specific droplet size distributions.

Model equations are simplified to coupled ODEs without ad-hoc parameters.

Method accommodates any collision kernel and captures key microphysical processes.

Abstract

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.