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 $[B,\nabla\times B] = 0$. Relaxing the need for a first integral of $B$ (also known as a flux function), we assume the existence of a solution $u : S \to \mathbb{R}$ to the cohomological equation $B|_S(u) = \partial_n B$ on a toroidal surface $S$ mutually invariant to $B$ and $\nabla \times B$. The right hand side $\partial_n B$ 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 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.