Geostrophic balance preserving interpolation in mesh adaptive shallow-water ocean modelling

TL;DR

This paper introduces a novel interpolation method based on Helmholtz decomposition to preserve geostrophic balance in mesh-adaptive shallow-water ocean models, ensuring steady states remain balanced after mesh changes.

Contribution

The authors develop a balance-preserving interpolation technique for dynamic mesh adaptivity in shallow-water models, maintaining geostrophic balance with machine precision.

Findings

Guarantees balanced and steady states are preserved after interpolation.

Prevents imbalanced perturbations from polluting solutions.

Effective for states close to geostrophic balance.

Abstract

The accurate representation of geostrophic balance is an essential requirement for numerical modelling of geophysical flows. Significant effort is often put into the selection of accurate or optimal balance representation by the discretisation of the fundamental equations. The issue of accurate balance representation is particularly challenging when applying dynamic mesh adaptivity, where there is potential for additional imbalance injection when interpolating to new, optimised meshes. In the context of shallow-water modelling, we present a new method for preservation of geostrophic balance when applying dynamic mesh adaptivity. This approach is based upon interpolation of the Helmholtz decomposition of the Coriolis acceleration. We apply this in combination with a discretisation for which states in geostrophic balance are exactly steady solutions of the linearised equations on an f-plane; this method guarantees that a balanced and steady flow on a donor mesh remains balanced and steady after interpolation onto an arbitrary target mesh, to within machine precision. We further demonstrate the utility of this interpolant for states close to geostrophic balance, and show that it prevents pollution of the resulting solutions by imbalanced perturbations introduced by the interpolation.