An index for closed orbits in Beltrami fields

Abstract

We consider the class of Beltrami fields (eigenfields of the curl operator) on three-dimensional Riemannian solid tori: such vector fields arise as steady incompressible inviscid fluids and plasmas. Using techniques from contact geometry, we construct an integer-valued index for detecting closed orbits in the flow which are topologically inessential (they have winding number zero with respect to the solid torus). This index is independent of the Riemannian structure, and is computable entirely from a C^1 approximation to the vector field on any meridional disc of the solid torus.