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

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.

Atmosphere and Ocean Modeling on Grids of Variable Resolution—A 2D Case Study

“…The test was proposed by Williamson et al (Williamson et al 1992, test case 2). The P DG 1 P 2 element was designed carefully to satisfy the primary balances of the linear shallow-water equations without producing strong spurious modes and it was already tested that P DG 1 P 2 is able to represent geostrophic balance properly for the linear shallow-water equations (Cotter et al 2009a;Maddison et al 2011). Here, we want to test if geostrophic balance is still represented properly when grid refinement is strongly changing the aspect ratio between adjacent grid cells, for the nonlinear equations on the sphere.…”