Core Concepts
The author compares variational upwinding schemes for geophysical fluids, highlighting their impact on potential enstrophy conservation.
Abstract
The study analyzes different upwinding schemes for potential vorticity in geophysical fluid dynamics. The anticipated potential vorticity method (APVM) introduces dissipation, while the streamwise upwind Petrov-Galerkin (SUPG) method adds a backscatter term for turbulent dynamics. A new downwinded trial function approach shows promising results in terms of energy conservation and residual errors. The comparison reveals differences in stability, conservation properties, and numerical convergence among the schemes.
Stats
In all cases, the upwinding scheme conserves both potential vorticity and energy.
The APVM leads to a symmetric definite correction to potential enstrophy that is dissipative and inconsistent.
The SUPG scheme introduces a consistent correction to the APVM scheme that acts as a backscatter term ensuring a richer depiction of turbulent dynamics.
Downwinded trial function formulation results in improved energy conservation and smaller residual errors compared to the SUPG scheme.
New temporal formulations allow exact integration of potential enstrophy across each time level.
Results using these formulations are observed to be stable even without dissipation, improving turbulent spectra at grid scale over other schemes with unstable potential enstrophy errors.