[Paper Review] Effects of Horizontal Discretization on Triangular and Hexagonal Grids on Linear Baroclinic and Symmetric Instabilities

Steffen Maaß, Sergey Danilov|arXiv (Cornell University)|Jan 7, 2026

Oceanographic and Atmospheric Processes0 citations

TL;DR

The paper analyzes how horizontal discretization on triangular and hexagonal grids (B and C staggers) affects linear baroclinic and symmetric instabilities, revealing spurious modes tied to grid geometry and the need to tune viscosity/diffusion to mitigate them.

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.

Motivation & Objective

• Assess how triangular and hexagonal grid discretizations (B and C staggering) reproduce linear baroclinic and symmetric instabilities in the Eady configuration.
• Identify numerical subtleties and spurious modes arising from grid geometry and staggering distinctions.
• Evaluate the effectiveness of viscosity and diffusion in suppressing spurious instability branches across discretizations.

Proposed method

• Extend linear baroclinic instability analysis (Eady problem) to triangular and hexagonal grids.
• Develop discrete horizontal operators and Fourier symbols for B and C grids on unstructured meshes.
• Compare advection schemes (FDV, FDCRE, AVI, ASC) and their impact on stability and spurious modes.
• Analyze transport of buoyancy with staggered grid considerations and reconstruction schemes.
• Provide supplementary notebook with Fourier-symbol analysis for hexagonal C grid and triangular A grid for completeness.

Experimental results

Research questions

• RQ1 How do triangular B- and C-grid discretizations reproduce the growth rates of baroclinic and symmetric instabilities in the Eady configuration?
• RQ2 What spurious modes arise due to mesh geometry and staggering on triangular/hexagonal grids, and how do they depend on background flow alignment?
• RQ3 To what extent do viscosity and diffusion suppress spurious instability branches across different discretizations?
• RQ4 How do different horizontal momentum and buoyancy discretizations influence the presence and behavior of spurious modes?

Key findings

• Spurious modes emerge on staggered triangular and hexagonal grids due to disbalance between velocity and scalar degrees of freedom and grid geometry.
• Spurious modes can depend on the alignment between background flow and the grid, and on the strength of a uniform background flow, indicating loss of Galilean invariance in some schemes.
• Moderate biharmonic viscosity and diffusion generally suppress spurious instability branches across discretizations, but calibration is needed per discretization.
• Certain advection schemes (e.g., FDCRE, ASC) improve the physical branch accuracy but may introduce opposite-direction spurious geometric modes.
• The study shows that the interactions with spurious modes are more significant near the axis of symmetric instabilities, where physical and spurious branches overlap in wavenumber space.
• An A-grid (triangular) arrangement is less prone to spurious instabilities, while hexagonal and triangular B/C grids require careful handling.