High order tracer variance stable transport with low order energy conserving dynamics for the thermal shallow water equations

TL;DR

This paper introduces a coupled high order discontinuous Galerkin and low order mixed finite element method for thermal shallow water equations, conserving energy and tracer variance, enhancing turbulence modeling.

Contribution

It presents a novel coupling of high order tracer transport with low order energy conserving dynamics, maintaining stability and turbulence richness.

Findings

High order transport preserves tracer variance and turbulence features.

Coupling maintains energy conservation and stability.

Method verified with standard test cases.

Abstract

A high order discontinuous Galerkin method for the material transport of thermodynamic tracers is coupled to a low order mixed finite element solver in the context of the thermal shallow water equations. The coupling preserves the energy conserving structure of the low order dynamics solver, while the high order material transport scheme is provably tracer variance conserving, or damping with the inclusion of upwinding. The two methods are coupled via a nested hierarchy of meshes, with the low order mesh of the dynamics solver being embedded within the high order transport mesh, for which the basis functions are collocated at the Gauss-Legendre quadrature points. Standard test cases are presented to verify the consistency and conservation properties of the method. While the overall scheme is limited by the formal order of accuracy of the low order dynamics, the use of high order, tracer variance conserving transport is shown to preserve richer turbulent solutions without compromising model stability compared to a purely low order method.