Stochastic Differential Equations for the Variability of Atmospheric Convection Fluctuating Around the Equilibrium

Abstract

Most convection parameterization schemes used within current atmospheric models make a convective equilibrium assumption, which breaks at the resolutions currently used by many numerical weather prediction models. To account for fluctuations of the cloud base mass flux about its equilibrium value, stochastic convection schemes have been developed for both deep (Plant & Craig, 2008, https://doi.org/10.1175/2007JAS2263.1) and shallow (Sakradzija et al., 2015, https://doi.org/10.5194/npg-22-65-2015) convection. Due to the need to explicitly track individual clouds in each grid box, these schemes can be computationally expensive. Motivated by the above considerations, the present study demonstrates how the machinery of the above schemes can be reduced to the solution of two ordinary stochastic differential equations for the cloud number and the total cloud base mass flux. The properties of the resulting stochastic processes for the cloud number and the total mass flux are not exactly equivalent to those of the original schemes but recover them to a very good approximation.