Effects of Horizontal Discretization on Triangular and Hexagonal Grids on Linear Baroclinic and Symmetric Instabilities

TL;DR

This paper investigates how different grid discretizations, especially triangular and hexagonal, affect the accuracy and stability of simulating baroclinic instabilities in ocean models, revealing numerical subtleties and the importance of parameter calibration.

Contribution

It extends linear instability analysis to complex grids, highlighting issues with spurious modes and their dependence on flow-grid alignment, and offers guidance on parameter calibration.

Findings

Discretizations on triangular and hexagonal grids are less robust against spurious modes.

Spurious modes do not follow Galilean invariance and depend on flow alignment.

Calibrating viscosity and diffusion suppresses spurious modes effectively.

Abstract

As global ocean general circulation models are run at eddy-permitting resolutions, reproducing accurate growth rates of baroclinic instabilities is a major concern when choosing a discretization of the equations of motion. From this viewpoint, we analyze discretizations on triangular and hexagonal grids with different types of variable staggering used in several ocean circulation models. By extending the linear baroclinic instability analysis in the Eady configuration to discretizations on more complex grids, several numerical subtleties are revealed. In comparison to discretizations on quadrilateral grids, the analyzed discretizations are less robust against unstable spurious modes, partly created by the mesh geometry. Some of the subtleties arise because spurious modes on staggered triangular and hexagonal grids do not adhere to Galilean invariance. As a consequence, their growth rates demonstrate a dependence on the alignment between the background flow and the grid, as well as the strength of a uniform background flow. The interactions with spurious modes become more significant on the axis of symmetric instabilities where the physical and spurious branches of instability are more difficult to separate in wavenumber space. Our analysis shows that in most cases moderate biharmonic viscosity and diffusion suppress spurious branches. However, one needs to carefully calibrate the viscosity and diffusivity parameters for each of the considered discretizations in order to achieve this.