Reduction of three-dimensional, volume-preserving flows with symmetry

Abstract

We develop a general, coordinate-free theory for the reduction of volume-preserving flows with a volume-preserving symmetry on three-manifolds. The reduced flow is generated by a one-degree-of-freedom Hamiltonian which is the generalization of the Bernoulli invariant from hydrodynamics. The reduction procedure also provides global coordinates for the study of symmetry-breaking perturbations. Our theory gives a unified geometric treatment of the integrability of three-dimensional, steady Euler flows and two-dimensional, unsteady Euler flows, as well as quasigeostrophic and magnetohydrodynamic flows.