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 equation
where div U=0, and V=(1&: 2 2) 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 : is negligible. We generalize this equation to the setting of an n-dimensional compact Riemannian manifold. The resulting equation is the Euler Poincare equation associated with the geodesic flow of the H 1 right invariant metric on D s + , the group of volume preserving Hilbert diffeomorphisms of class H s . We prove that the geodesic spray is continuously differentiable from TD s + (M ) into TTD s + (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. Grenoble 16 (1966), 319 361). To do so, we use submanifold geometry, and prove that the weak curvature tensor of the right invariant H 1 metric on D s + is a bounded trilinear map in the H s 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.