# Comparison of variational balance models for the rotating shallow water equations

## TL;DR

This paper compares various variational balance models for the rotating shallow water equations, highlighting the effectiveness of the L1-model in capturing balanced dynamics over relevant time scales.

## Contribution

It provides a comprehensive numerical comparison of GLSG models, including the L1-model, for their ability to replicate shallow water flow dynamics.

## Key findings

- L1-model accurately reproduces balanced dynamics over time scales of 1/Ro
- Other models develop unphysical high wavenumber features
- L1-model is a robust choice for initial state generation

## Abstract

We present an extensive numerical comparison of a family of balance models appropriate to the semi-geostrophic limit of the rotating shallow water equations, and derived by variational asymptotics in Oliver (2006) for small Rossby numbers ${\mathrm{Ro}}$. This family of generalized large-scale semi-geostrophic (GLSG) models contains the $L_1$-model introduced by Salmon (1983) as a special case. We use these models to produce balanced initial states for the full shallow water equations. We then numerically investigate how well these models capture the dynamics of an initially balanced shallow water flow. It is shown that, whereas the $L_1$-member of the GLSG family is able to reproduce the balanced dynamics of the full shallow water equations on time scales of ${\mathcal{O}}(1/{\mathrm{Ro}})$ very well, all other members develop significant unphysical high wavenumber contributions in the ageostrophic vorticity which spoil the dynamics.