  We study analytical and numerical aspects of the bifurcation diagram of
simply-connected rotating vortex patch equilibria for the quasi-geostrophic
shallow-water (QGSW) equations. The QGSW equations are a generalisation of the
Euler equations and contain an additional parameter, the Rossby deformation
length $\EE^{-1}$, which enters in the relation between streamfunction and
(potential) vorticity. The Euler equations are recovered in the limit $\EE\to
0$. We prove, close to circular (Rankine) vortices, the persistence of the
bifurcation diagram for arbitrary Rossby deformation length. However we show
that the two-fold branch, corresponding to Kirchhoff ellipses for the Euler
equations, is never connected even for small values $\EE$, and indeed is split
into a countable set of disjoint connected branches. Accurate numerical
calculations of the global structure of the bifurcation diagram and of the
limiting equilibrium states are also presented to complement the mathematical
analysis.
