Geometry and Curvature of Diffeomorphism Groups with H* Metric and Mean Hydrodynamics

In [HMR1], Holm, Marsden, and Ratiu derived a new model for the mean motion of an ideal fluid in Euclidean space given by the equation V̇ (t)+∇U(t)V (t)−α 2 [∇U(t)] ·△U(t) = −grad p(t) where divU = 0, and V = (1 − α2△)U . In this model, the momentum V is transported by the velocity U , with the effect that nonlinear interaction between modes corresponding to length scales smaller than α is negligible. We generalize this equation to the setting of an n dimensional compact Riemannian manifold. The resulting equation is the Euler-Poincaré equation associated with the geodesic flow of the H1 right invariant metric on D μ, the group of volume preserving Hilbert diffeomorphisms of class H. We prove that the geodesic spray is continuously differentiable from TD μ(M) into TTD s μ(M) so that a standard Picard iteration argument proves existence and uniqueness on a finite time interval. Our goal in this paper is to establish the foundations for Lagrangian stability analysis following Arnold [A]. To do so, we use submanifold geometry, and prove that the weak curvature tensor of the right invariant H1 metric on D μ is a bounded trilinear map in the H topology, from which it follows that solutions to Jacobi’s equation exist. Using such solutions, we are able to study the infinitesimal stability behavior of geodesics.