Geometry and Curvature of Diffeomorphism Groups withH1Metric and Mean Hydrodynamics

Holm, Marsden, and Ratiu (Adv. in Math.137(1998), 1–81) derived a new model for the mean motion of an ideal fluid in Euclidean space given by the equationV(t)+∇U(t) V(t)−α2[∇U(t)]t·ΔU(t)=−grad p(t) where div U=0, andV=(1−α2Δ) U. In this model, the momentumVis transported by the velocityU, with the effect that nonlinear interaction between modes corresponding to length scales smaller thanαis negligible. We generalize this equation to the setting of ann-dimensional compact Riemannian manifold. The resulting equation is the Euler–Poincaré equation associated with the geodesic flow of theH1right invariant metric on Dsμ, the group of volume preserving Hilbert diffeomorphisms of classHs. We prove that the geodesic spray is continuously differentiable fromTDsμ(M) intoTTDsμ(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 (Ann. Inst. Grenoble16(1966), 319–361). To do so, we use submanifold geometry, and prove that the weak curvature tensor of the right invariantH1metric on Dsμis a bounded trilinear map in theHstopology, from which it follows that solutions to Jacobiʹs equation exist. Using such solutions, we are able to study the infinitesimal stability behavior of geodesics.