A Stochastic Precipitating Quasi-Geostrophic Model

Abstract

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.

Figures (8)

• Figure 1: The physical features in the PQG turbulence include the zonal jet (top panel), cloud fraction (middle panel), and snapshots of the PV along with the velocity field $\mathbf{u}_h$ represented by blue arrows (bottom left), and the total water content $q_t$ (bottom right). All spatially 2D plots are fixed-time, $(x,y)$-slices of fields at the intermediate height between levels 1 and 2, where $x$$(y)$ is the zonal (meridional) direction. variables are defined at the intermediate level. Also note that, unless otherwise specified, all plots related to the $q_t$ quantity in this paper have been adjusted by subtracting the saturation mixing ratio $q_{vs}$. Consequently, in the $q_t$ plots, values above 0 (red areas) will indicate rainwater.
• Figure 2: Schematic illustration of the PQG model (left) and the SPQG model (right).
• Figure 3: Comparison of the PV and Heaviside functions. Left column: one snapshot from the PQG model. Middle column: applying the Gaussian kernel to the PQG PV field and the associated Heaviside function computed from the Markov jump process using the SPQG model. Right column: the Heaviside function calculated from the SPQG model without applying the Gaussian kernel smoothing to the PV field.
• Figure 4: Results of the SPQG model: snapshots of the PV, the $q_t$ and the velocity field at different times. All spatially 2D plots are fixed-time, $(x,y)$-slices of fields at the intermediate height between levels 1 and 2, where $x$$(y)$ is the zonal (meridional) direction.
• Figure 5: Results of the PQG model: snapshots of the PV, the $q_t$ and velocity field at different times.