A Well-Balanced Central-Upwind Scheme for the Thermal Rotating Shallow Water Equations

TL;DR

This paper introduces a well-balanced central-upwind numerical scheme for the thermal rotating shallow water equations, accurately capturing equilibrium states with temperature and density gradients on a rotating planet.

Contribution

The paper presents a novel flux globalization-based central-upwind scheme that maintains thermo-geostrophic equilibria in the thermal rotating shallow water model.

Findings

Successfully maintains equilibrium states with complex topography and rotation.

Capable of accurately simulating large-scale oceanic and atmospheric motions.

Provides criteria for equilibrium existence, wave-breaking, and instability, confirmed by numerical tests.

Abstract

We develop a well-balanced central-upwind scheme for rotating shallow water model with horizontal temperature and/or density gradients---the thermal rotating shallow water (TRSW). The scheme is designed using the flux globalization approach: first, the source terms are incorporated into the fluxes, which results in a hyperbolic system with global fluxes; second, we apply the Riemann-problem-solver-free central-upwind scheme to the rewritten system. We ensure that the resulting method is well-balanced by switching off the numerical diffusion when the computed solution is near (at) thermo-geostrophic equilibria. The designed scheme is successfully tested on a series of numerical examples. Motivated by future applications to large-scale motions in the ocean and atmosphere, the model is considered on the tangent plane to a rotating planet both in mid-latitudes and at the Equator. The numerical scheme is shown to be capable of quite accurately maintaining the equilibrium states in the presence of nontrivial topography and rotation. Prior to numerical simulations, an analysis of the TRSW model based on the use of Lagrangian variables is presented, allowing one to obtain criteria of existence and uniqueness of the equilibrium state, of the wave-breaking and shock formation, and of instability development out of given initial conditions. The established criteria are confirmed in the conducted numerical experiments.