The SQG Equation as a Geodesic Equation

TL;DR

This paper shows that the surface quasi-geostrophic (SQG) equation can be interpreted as a geodesic equation on a certain group of volume-preserving diffeomorphisms, revealing geometric properties like smoothness of the exponential map and unbounded curvature.

Contribution

It establishes a geometric interpretation of the SQG equation as a geodesic flow on a Riemannian manifold with a specific metric, and analyzes its geometric properties.

Findings

The Riemannian exponential map for the SQG equation is smooth.

The exponential map is non-Fredholm.

Sectional curvature at the identity is unbounded of both signs.

Abstract

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.