Thermodynamic consistency and structure-preservation in summation by parts methods for the moist compressible Euler equations

TL;DR

This paper introduces a thermodynamically consistent, structure-preserving discretization of the moist compressible Euler equations, ensuring conservation of key physical quantities in atmospheric modeling.

Contribution

It develops a novel discretization approach that maintains thermodynamic consistency and conserves mass, water, entropy, and energy in moist atmospheric simulations.

Findings

Conservation of mass, water, entropy, and energy demonstrated in simulations.

Discontinuous Galerkin spectral element method with stable fluxes derived.

Numerical experiments verify theoretical conservation properties.

Abstract

Moist thermodynamics is a fundamental driver of atmospheric dynamics across all scales, making accurate modeling of these processes essential for reliable weather forecasts and climate change projections. However, atmospheric models often make a variety of inconsistent approximations in representing moist thermodynamics. These inconsistencies can introduce spurious sources and sinks of energy, potentially compromising the integrity of the models. Here, we present a thermodynamically consistent and structure preserving formulation of the moist compressible Euler equations. When discretised with a summation by parts method, our spatial discretisation conserves: mass, water, entropy, and energy. These properties are achieved by discretising a skew symmetric form of the moist compressible Euler equations, using entropy as a prognostic variable, and the summation-by-parts property of discrete derivative operators. Additionally, we derive a discontinuous Galerkin spectral element method with energy and tracer variance stable numerical fluxes, and experimentally verify our theoretical results through numerical simulations.