  We investigate the geometry of a family of equations in two dimensions which
interpolate between the Euler equations of ideal hydrodynamics and the inviscid
surface quasi-geostrophic equation. This family can be realised as geodesic
equations on groups of diffeomorphisms. We show precisely when the
corresponding Riemannian exponential map is non-linear Fredholm of index 0. We
further illustrate this by examining the distribution of conjugate points in
these settings via a Morse theoretic approach.
