The Geometry of Axisymmetric Ideal Fluid Flows with Swirl

The sectional curvature of the volume preserving diffeomorphism group of a Riemannian manifold M can give information about the stability of inviscid, incompressible fluid flows on M . We demonstrate that the submanifold of the volumorphism group of the solid flat torus generated by axisymmetric fluid flows with swirl, denoted by D μ , E ( M ) , has positive sectional curvature in every section containing the field X = u ( r ) ∂ θ iff ∂ r ( r u 2 ) > 0 . This is in sharp contrast to the situation on D μ ( M ) , where only Killing fields X have nonnegative sectional curvature in all sections containing it. We also show that this criterion guarantees the existence of conjugate points on D μ , E ( M ) along the geodesic defined by X .