An index for closed orbits in Beltrami fields

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). The most important feature of the index is its computability: it can be rigorously determined from a C1-approximation to the vector field on any meridional disc of the solid torus. As a result, we obtain a test for detecting the non-existence of a global cross-section to the 3D flow via purely 2D information.