  Efficient and effective modeling of complex systems, incorporating cloud
physics and precipitation, is essential for accurate climate modeling and
forecasting. However, simulating these systems is computationally demanding
since microphysics has crucial contributions to the dynamics of moisture and
precipitation. In this paper, appropriate stochastic models are developed for
the phase-transition dynamics of water, focusing on the precipitating
quasi-geostrophic (PQG) model as a prototype. By treating the moisture, phase
transitions, and latent heat release as integral components of the system, the
PQG model constitutes a set of partial differential equations (PDEs) that
involve Heaviside nonlinearities due to phase changes of water. Despite
systematically characterizing the precipitation physics, expensive iterative
algorithms are needed to find a PDE inversion at each numerical integration
time step. As a crucial step toward building an effective stochastic model, a
computationally efficient Markov jump process is designed to randomly simulate
transitions between saturated and unsaturated states that avoids using the
expensive iterative solver. The transition rates, which are deterministic, are
derived from the physical fields, guaranteeing physical and statistical
consistency with nature. Furthermore, to maintain the consistent spatial
pattern of precipitation, the stochastic model incorporates an adaptive
parameterization that automatically adjusts the transitions based on spatial
information. Numerical tests show the stochastic model retains critical
properties of the original PQG system while significantly reducing
computational demands. It accurately captures observed precipitation patterns,
including the spatial distribution and temporal variability of rainfall,
alongside reproducing essential dynamic features such as potential vorticity
fields and zonal mean flows.
