Higher-order approximations of the residual-mean eddy streamfunction and the quasi-Stokes streamfunction

TL;DR

This paper compares higher-order series expansions of residual-mean and quasi-Stokes streamfunctions in idealized ocean models, revealing their accuracy in the interior and limitations near boundaries and complex topography.

Contribution

It introduces a detailed comparison of third-order series expansions and proposes a measure for their convergence behavior in ocean modeling.

Findings

First and second order approximations are accurate in the interior.

Third order terms significantly influence interior circulations.

Series expansions break down near boundaries and complex topography.

Abstract

The series expansion of the residual-mean eddy streamfunction and the quasi-Stokes streamfunction are compared up to third order in buoyancy perturbation, both formally and by using several idealised eddy-permitting zonal channel model experiments. In model configurations with flat bottom, both streamfunctions may be well approximated by the first one or two leading order terms in the ocean interior, although terms up to third order still significantly impact the implied interior circulations. Further, differences in both series expansions up to third order remain small here. Near surface and bottom boundaries, on the other hand, the leading order terms differ and are initially of alternating sign and of increasing magnitude such that the low order approximate expressions break down there. In more realistic model configurations with significant topographic features, physically inconsistent recirculation cells also appear in the ocean interior and are not effectively reduced by the next higher order terms. A measure indicating an initially increasing or decreasing series expansion is proposed.