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

TL;DR

The paper tackles stable, energy-conserving transport of thermodynamic tracers in the thermal shallow water system by coupling a low-order energy-conserving mixed finite element solver with a high-order tracer transport scheme. It introduces a high-order DG transport for buoyancy in material form, ensuring tracer variance conservation, and links it to the low-order solver via a multigrid hierarchy with a 1:1 mapping of degrees of freedom collocated at Gauss-Legendre points. The results across solid body rotation, thermogeostrophic balance, and thermal-instability test cases show that energy is conserved while tracer variance is controlled, improving stability without excessive diffusion compared to purely low-order approaches. The work extends tracer-variance-conserving formulations from flux form to material transport and suggests applicability to other non-canonical Hamiltonian systems, including moist and compressible frameworks.

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 the multigrid hierarchy of the low order dynamics solver, with the basis functions of the high order transport being collocated at the Gauss-Legendre quadrature points with the low order dynamics on the finest scale multigrid mesh. 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.

Figures (6)

• Figure 1: Mass (left) and tracer variance (right) conservation errors with time for the high order discontinuous Galerkin material advection scheme using upwinded ($\alpha=1$) and centered ($\alpha=0$) fluxes.
• Figure 2: $L^2$ error convergence with grid resolution after a single period (left), and absolute tracer variance conservation error after a single period (right).
• Figure 3: $L^2$ error convergence (left) and normalised conservation errors for the mass, depth weighted buoyancy, relative vorticity and energy (right).
• Figure 4: Normalised conservation error with resolution for the tracer variance (left), and number of iterations to nonlinear solver convergence with resolution (right).
• Figure 5: Buoyancy field for the thermal instability test case using high order upwinded transport at dimensionless times $20.0,40.0,60.0,80.0$.