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

Kieran Ricardo, David Lee, Kenneth Duru

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.

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. 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.

Paper Structure

This paper contains 31 sections, 6 theorems, 98 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Continuous moist compressible Euler equations
  3. Vector invariant pressure gradient formulation
  4. Thermodynamics and energy conservation
  5. Mathematical entropy and tracer variance
  6. Skew-symmetric split-form and thermodynamic consistent formulation
  7. Thermodynamics
  8. Potentials
  9. Thermodynamic equilibrium
  10. Spatial discretisation
  11. Spaces
  12. Operators
  13. Integrals
  14. DG semi-discrete formulation
  15. Semi-discrete conservation and numerical stability
  16. ...and 16 more sections

Key Result

Theorem 2.1

Consider the the moist compressible Euler equations eq:skew-vel1-eq:skew-water on the smooth spatial domain $\Omega \subset \mathbb{R}^2$ with periodic and wall eq:no-flux-boundary boundary conditions. For continuous solutions $(\vb{v}, \rho, \eta, q_w): \Omega\times\mathbb{R}^{+} \to \mathbb{R}^{5}

Figures (8)

  • Figure 1: Specific entropy of the Bryan-Fritsch test case at $t=0\text{s},800\text{s},1000\text{s},1200\text{s}$.
  • Figure 2: Water vapour mass fraction of the Bryan-Fritsch test case at $t=0\text{s},800\text{s},1000\text{s},1200\text{s}$.
  • Figure 3: Specific entropy perturbation of the three phase bubble test case at $t=0\text{s},200\text{s},400\text{s},600\text{s}$.
  • Figure 4: Ice mass fraction of the three phase bubble test case at $t=0\text{s},200\text{s},400\text{s},600\text{s}$.
  • Figure 5: Conservation errors of the three phase bubble test case.
  • ...and 3 more figures

Theorems & Definitions (12)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 4.1
  • proof
  • Theorem 5.1
  • proof
  • Theorem 5.2
  • proof
  • ...and 2 more