Energy and entropy conserving compatible finite elements with upwinding for the thermal shallow water equations

TL;DR

This paper introduces a novel compatible finite element method for thermal shallow water equations that conserves energy and entropy, incorporating upwind fluxes to improve stability and accuracy in thermally unstable flow simulations.

Contribution

The work develops a new finite element formulation that combines energy and entropy conservation with upwind fluxes, enabling stable long-term simulations of thermally unstable flows.

Findings

Conserves discrete energy and entropy in semi-discrete and fully discrete schemes.

Upwind fluxes suppress spurious oscillations in thermally unstable flows.

Achieves exact entropy conservation in specific fully discrete cases.

Abstract

In this work, we develop a new compatible finite element formulation of the thermal shallow water equations that conserves energy and mathematical entropies given by buoyancy-related quadratic tracer variances. Our approach relies on restating the governing equations to enable discontinuous approximations of thermodynamic variables and a variational continuous time integration. A key novelty is the inclusion of centred and upwinded fluxes. The proposed semi-discrete system conserves discrete entropy for centred fluxes, monotonically damps entropy for upwinded fluxes, and conserves energy. The fully discrete scheme reflects entropy conservation at the continuous level. The ability of a new linearised Jacobian, which accounts for both centred and upwinded fluxes, to capture large variations in buoyancy and simulate thermally unstable flows for long periods of time is demonstrated for two different transient case studies. The first involves a thermogeostrophic instability where including upwinded fluxes is shown to suppress spurious oscillations while successfully conserving energy and monotonically damping entropy. The second is a double vortex where a constrained fully discrete formulation is shown to achieve exact entropy conservation in time.