Linearisability of divergence-free fields along invariant 2-tori

Abstract

We find conditions under which the restriction of a divergence-free vector field B to an invariant toroidal surface S is linearisable. 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) = ∂nB on a toroidal surface S mutually invariant to B and ∇ × B. The right hand side ∂nB 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 vanishing with B|S/ B 2 |S being linearisable. We are calling the latter the semi-linearisability of B (with proportionality B 2 |S). The non-vanishing 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 linearisable. The linearisability of B|S is fundamental to the so-called magnetic coordinates, which are central to the theory of magnetically confined plasmas.