The SQG Equation as a Geodesic Equation

We demonstrate that the surface quasi-geostrophic (SQG) equation given by $$\theta_t + \left<u, \nabla \theta\right>= 0,\;\;\; \theta = \nabla \times (-\Delta)^{-1/2} u,$$ is the geodesic equation on the group of volume-preserving diffeomorphisms of a Riemannian manifold $M$ in the right-invariant $\dot{H}^{-1/2}$ metric. We show by example, that the Riemannian exponential map is smooth and non-Fredholm, and that the sectional curvature at the identity is unbounded of both signs.