Geometry and curvature of diffeomorphism groups with H-1 metric and mean hydrodynamics

Helm, Marsden, and Ratiu (Adv. in Math. 137 (1998), 1-81) derived a new mod el for the mean motion of an ideal fluid in Euclidean space given by the eq uation V over dot(t) + del(U(t))V(t) - alpha(Z)[del U(t)](t).Delta U(t) = - grad p(t) where div U = 0, and V = (1 - alpha(2)Delta) U. In this model, t he momentum V is transported by the velocity U, with the effect that nonlin ear interaction between modes corresponding to length scaler smaller than a is negligible. We generalize this equation to the setting of an n-dimensio nal compact Riemannian manifold. The resulting equation is the Euler-Poinca re equation associated with the geodesic flow of the H-1 right invariant me tric on D-mu(S), the group of volume preserving Hilbert diffeomorphisms of class H-S. We prove that the geodesic spray is continuously differentiable from TDmus(M) into TTDmuS(M) so that a standard Picard iteration argument p roves existence and uniqueness on a finite time interval. Our goal in this paper is to establish the foundations for Lagrangian stability analysis fol lowing Arnold (Ann. Inst. Grenoble 16 (1966), 319-361). To do so, we use su bmanifold geometry, and prove that the weak curvature tensor of the right i nvariant H-1 metric on D-mu(S), is a bounded trilinear map in the H-S topol ogy, 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. (C) 1998 Academic Press.