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We present an extensive numerical comparison of a family of balance modelsappropriate to the semi-geostrophic limit of the rotating shallow waterequations, and derived by variational asymptotics in Oliver (2006) for smallRossby numbers ${\mathrm{Ro}}$. This family of generalized large-scalesemi-geostrophic (GLSG) models contains the $L_1$-model introduced by Salmon(1983) as a special case. We use these models to produce balanced initialstates for the full shallow water equations. We then numerically investigatehow well these models capture the dynamics of an initially balanced shallowwater flow. It is shown that, whereas the $L_1$-member of the GLSG family isable to reproduce the balanced dynamics of the full shallow water equations ontime scales of ${\mathcal{O}}(1/{\mathrm{Ro}})$ very well, all other membersdevelop significant unphysical high wavenumber contributions in theageostrophic vorticity which spoil the dynamics.

Comparison of variational balance models for the rotating shallow water equations