Thermodynamic Consistency and Structure-Preservation in Summation by Parts Methods for the Moist Compressible Euler Equations

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. We present a thermodynamically consistent and structure preserving formulation of the moist compressible Euler equations. When discretized with a summation-by-parts (SBP) method in a single domain, our spatial discretization conserves total mass, total mass of water, thermodynamic entropy, energy, and tracer variance. These properties are achieved by discretizing a skew-symmetric form of the moist compressible Euler equations, using thermodynamic entropy as a prognostic variable, and the SBP property of discrete derivative operators. Additionally, we derive a discontinuous Galerkin spectral element method with recently developed energy and tracer variance stable numerical fluxes, which conserve the invariants when centered fluxes are used and minimally dissipate energy and tracer variance only at element faces when upwind fluxes are applied. Extensive numerical experiments are presented to verify the theoretical results and demonstrate the accuracy and robustness of our numerical method. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/kieranricardo/moist-euler-dg and in the supplementary materials ( moist-euler-dg-main.zip [694KB]). [Formula: see text]