Exploring physics-dynamics coupling using moist shallow water equations

One of the key choices for numerical models of geophysical fluids is how parametrisations of physical processes interact with the numerical methods that handle the resolved flow, known in the atmospheric community as the dynamical core. As both the dynamical core and parametrisations of physics processes continue to evolve and improve, the issue of physics-dynamics coupling - how these two different parts of the model interact - becomes ever more important. In this paper we use two variations of the moist shallow water equations to develop a simplified framework that can be used to investigate some of the questions associated with physics-dynamics coupling. The shallow water equations act as a simplified dynamical core that is computationally cheap but still retains pertinent features of the atmosphere, and the introduction of moisture means the addition to the model of a physical parametrisation. This study uses 'split-physics' moist thermal shallow water equations which couple moisture to the shallow water equations via a parametrisation, and also develops a new 'integrated-physics' formulation of the moist thermal shallow water equations which re-formulates the model so that all the moist processes are captured in the dynamics. This integrated-physics model thus requires no physics-dynamics coupling and acts as a ground-truth to compare coupling strategies in the split-physics formulation against. We use both models to examine the effect of varying the approach to how physics is coupled to the dynamics in the semi-implicit quasi-Newton timestepping scheme. The results demonstrate the usefulness of the integrated-physics moist shallow water equations and provide insights into how best to deal with physics in a model's timestep.