  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.
