Rectifiability of divergence-free fields along invariant 2-tori

We find conditions under which the restriction of a divergence-free vector field B to an invariant toroidal surface S is rectifiable; namely constant in a suitable global coordinate system. The main results are similar in conclusion to Arnold’s Structure Theorems but require weaker assumptions than the commutation [B,∇× B] = 0 . Relaxing the need for a first integral of B (also known as a flux function), we assume the existence of a solution u: S →ℝ to the cohomological equation B| _S(u) = ∂ _n B on a toroidal surface S mutually invariant to B and ∇× B . The right hand side ∂ _n B is a normal surface derivative available to vector fields tangent to S . In this situation, we show that the field B on S is either identically zero or nowhere zero with B| _S/‖ B‖ ^2 | _S being rectifiable. We are calling the latter the semi-rectifiability of B (with proportionality ‖ B‖ ^2 | _S ). The nowhere zero property relies on Bers’ results in pseudo-analytic function theory about a generalised Laplace-Beltrami equation arising from Witten cohomology deformation. With the use of de Rham cohomology, we also point out a Diophantine integral condition where one can conclude that B| _S itself is rectifiable. The rectifiability and semi-rectifiability of B| _S is fundamental to the so-called magnetic coordinates, which are central to the theory of magnetically confined plasmas.