Implementation of bulk cloud microphysics (BLK) parameterizationsin atmospheric models ofdifferent scales has gained momentum in the last two decades.Utilization of these parameterizations in cloud-resolving models when timesteps used for the hostmodel integration are a few seconds or less is justified from the point of view of cloud physics.However, mechanistic extrapolationof the applicability of BLK schemesto the regional or global scales and the utilizationof timesteps of hundreds up to thousands of seconds affect both physics andnumerics.We focus on the mathematical aspects of BLK schemes, such asstability and positive-definiteness. We provide a strict mathematical definition for theproblem of warm rain formation. We also derive a general analytical condition(SM-criterion) that remains valid regardless of parameterizations for warm rainprocesses in an explicit Euleriantime integration framework used to advanced finite-difference equations, whichgovern warm rain formation processes in microphysics packages inthe Community Atmosphere Model and the Weather Research and Forecasting model.The SM-criterionallows for theexistence of a unique positive-definite stable mass-conserving numerical solution, imposesan additional constraint on the timestep permitted due to the microphysics(like the Courant-Friedrichs-Lewy condition for the advection equation), and prohibits use of anyadditional assumptions not included in the strict mathematical definition of theproblem under consideration.By analyzing the numerics of warm rain processes in source codesof BLK schemes implementedin community models we provide general guidelines regardingthe appropriate choiceof time steps in these models.
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