Analyzing numerics of bulk microphysics schemes in community models: warm rain processes

Abstract. Implementation of bulk cloud microphysics (BLK) parameterizations in atmospheric models of different scales has gained momentum in the last two decades. Utilization of these parameterizations in cloud-resolving models when timesteps used for the host model integration are a few seconds or less is justified from the point of view of cloud physics. However, mechanistic extrapolation of the applicability of BLK schemes to the regional or global scales and the utilization of timesteps of hundreds up to thousands of seconds affect both physics and numerics.

We focus on the mathematical aspects of BLK schemes, such as stability and positive-definiteness. We provide a strict mathematical definition for the problem of warm rain formation. We also derive a general analytical condition (SM-criterion) that remains valid regardless of parameterizations for warm rain processes in an explicit Eulerian time integration framework used to advanced finite-difference equations, which govern warm rain formation processes in microphysics packages in the Community Atmosphere Model and the Weather Research and Forecasting model. The SM-criterion allows for the existence of a unique positive-definite stable mass-conserving numerical solution, imposes an additional constraint on the timestep permitted due to the microphysics (like the Courant-Friedrichs-Lewy condition for the advection equation), and prohibits use of any additional assumptions not included in the strict mathematical definition of the problem under consideration.

By analyzing the numerics of warm rain processes in source codes of BLK schemes implemented in community models we provide general guidelines regarding the appropriate choice of time steps in these models.