A comparison of variational upwinding schemes for geophysical fluids, and their application to potential enstrophy conserving discretisations

Abstract

Methods for upwinding the potential vorticity in a compatible finite element discretisation of the rotating shallow water equations are studied. These include the well-known anticipated potential vorticity method (APVM), streamwise upwind Petrov-Galerkin (SUPG) method, and a recent approach where the trial functions are evaluated downstream within the reference element. In all cases the upwinding scheme conserves both potential vorticity and energy, since the antisymmetric structure of the equations is preserved. The APVM leads to a symmetric definite correction to the potential enstrophy that is dissipative and inconsistent, resulting in a turbulent state where the potential enstrophy is more strongly damped than for the other schemes. While the SUPG scheme is widely known to be consistent, since it modifies the test functions only, the downwinded trial function formulation results in the advection of downwind corrections. Results of the SUPG and downwinded trial function schemes are very similar in terms of both potential enstrophy conservation and turbulent spectra. The main difference between these schemes is in the energy conservation and residual errors. If just two nonlinear iterations are applied then the energy conservation errors are improved for the downwinded trial function formulation, reflecting a smaller residual error than for the SUPG scheme. We also present formulations by which potential enstrophy is exactly integrated at each time level. Results using these formulations are observed to be stable in the absence of dissipation, despite the uncontrolled aliasing of grid scale turbulence. Using such a formulation and the APVM with a coefficient $\mathcal{O}(100)$ times smaller that its regular value leads to turbulent spectra that are greatly improved at the grid scale over the SUPG and downwinded trial function formulations with unstable potential enstrophy errors.

Figures (16)

• Figure 1: Vorticity field (northern hemisphere) for the Galewsky test case at day 7 using APVM (left), SUPG (center) and downwinded trial functions (right) for the converged solutions.
• Figure 2: Vorticity field (northern hemisphere) for the Galewsky test case at day 20 using APVM (left), SUPG (center) and downwinded trial functions (right) for the converged solutions.
• Figure 3: Energy (left) and potential enstrophy (right) conservation for the converged solutions.
• Figure 4: Mass (left) and un-normalised vorticity (right) conservation for the converged solutions.
• Figure 5: Kinetic energy spectra at day 20 (left), and number of nonlinear iterations to achieve convergence (right).