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.

翻译：高效且有效地模拟包含云物理和降水的复杂系统，对于精确的气候建模和预测至关重要。然而，由于微物理过程对水汽和降水的动力学具有关键贡献，模拟这些系统的计算成本高昂。本文针对水的相变动力学，以降水准地转（PQG）模型为原型，建立了合适的随机模型。通过将水汽、相变和潜热释放视为系统的内在组成部分，PQG模型构成了一组偏微分方程（PDEs），其中包含了因水的相变而产生的Heaviside型非线性。尽管系统性地刻画了降水物理过程，但在每个数值积分时间步长上，都需要昂贵的迭代算法来求解PDE反演。作为构建有效随机模型的关键一步，本文设计了一种计算高效的马尔可夫跳跃过程，用于随机模拟饱和与非饱和状态之间的转换，从而避免了使用昂贵的迭代求解器。其转换速率是确定性的，由物理场导出，保证了与自然界的物理和统计一致性。此外，为了保持降水空间格局的一致性，该随机模型引入了一种自适应参数化方案，能够基于空间信息自动调整状态转换。数值测试表明，该随机模型在显著降低计算需求的同时，保留了原始PQG系统的关键特性。它准确地捕捉了观测到的降水模式，包括降雨的空间分布和时间变率，并同时再现了位涡场和纬向平均流等基本动力学特征。