Exploring forms of the moist shallow water equations using a new compatible finite element discretisation

TL;DR

This work addresses physics-dynamics coupling in atmospheric modelling by deploying moist shallow water equations discretised with compatible finite elements. A unified, general formulation is developed that can recover four moist SW variants, including a newly explored moist convective pseudo-thermal model, and is coupled to a three-state moist physics scheme. The discretisation uses a two-field de Rham-compatible FE pair with buoyancy and moisture variables, coupled to a semi-implicit, quasi-Newton time stepping scheme. Three standard test cases (steady geostrophic flow, flow over a mountain, and an unstable jet) demonstrate realistic cloud and rain formation and reveal how latent-heat feedback and prognostic buoyancy alter large-scale dynamics and precipitation patterns. The results establish reference solutions across formulations, highlight the role of latent heat in feedbacks, and position compatible FE moist SW as a robust framework for exploring physics-dynamics coupling in modern numerical weather prediction cores.

Abstract

The moist shallow water equations offer a promising route for advancing understanding of the coupling of physical parametrisations and dynamics in numerical atmospheric models, an issue known as 'physics-dynamics coupling'. Without moist physics, the traditional shallow water equations are a simplified form of the atmospheric equations of motion and so are computationally cheap, but retain many relevant dynamical features of the atmosphere. Introducing physics into the shallow water model in the form of moisture provides a tool to experiment with numerical techniques for physics-dynamics coupling in a simple dynamical model. In this paper, we compare some of the different moist shallow water models by writing them in a general formulation. The general formulation encompasses three existing forms of the moist shallow water equations and also a fourth, previously unexplored formulation. The equations are coupled to a three-state moist physics scheme that interacts with the resolved flow through source terms and produces two-way physics-dynamics feedback. We present a new compatible finite element discretisation of the equations and apply it to the different formulations of the moist shallow water equations in three test cases. The results show that the models capture generation of cloud and rain and physics-dynamics interactions, and demonstrate some differences between moist shallow water formulations and the implications of these different modelling choices.

Figures (8)

• Figure 1: Convergence plots for the moist Williamson 2 tests in each framework. (a): the moist convective formulation, (b): the moist convective thermal formulation, (c): the moist thermal formulation, and (d): the moist convective pseudo-thermal formulation. The $L^2$ norm of the error is normalised by dividing it by the $L^2$ norm of the field in each case except for the cloud field, which is not normalised (because the test begins with no cloud). Second-order convergence lines are also shown on the plot for comparison.
• Figure 2: Potential vorticity (defined by $\frac{(\nabla \times \bm{u}) \cdot \bm{\hat{k}} + f}{D}$) at Day 50 in the flow over a mountain test. (a) the moist convective formulation, (b) the moist convective thermal formulation, (c) the moist thermal formulation, and (d) the moist convective pseudo-thermal formulation.
• Figure 3: Accumulated rain at Day 50 in the flow over a mountain test. (a) the moist convective formulation, (b) the moist convective thermal formulation, (c) the moist thermal formulation, and (d) the moist convective pseudo-thermal formulation. We plot $\text{log}_{10}(q_r)$ rather than $q_r$ directly so that we can plot all four configurations on the same scale despite variation in rain quantities, and omit rain values of less than $1 \times 10^{-6}$ from the plot.
• Figure 4: Potential vorticity (defined by $\frac{(\nabla \times \bm{u}) \cdot \bm{\hat{k}} + f}{D}$) at Day 6 in the moist unstable jet test. (a) the moist convective formulation, (b) the moist convective thermal formulation, (c) the moist thermal formulation, and (d) the pseudo-moist convective thermal formulation.
• Figure 5: Accumulated rainfall at Day 6 in the moist unstable jet test. (a) the moist convective formulation, (b) the moist convective thermal formulation, (c) the moist thermal formulation, and (d) the pseudo-moist convective thermal formulation. As in Figure \ref{['fig:W5_rain']}, we plot $\text{log}_{10}(q_r)$ rather than $q_r$ directly so that we can plot all four configurations on the same scale despite variation in rain quantities, and omit rain values of less than $1 \times 10^{-6}$ from the plot.