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 $B{|}_S/\Vert B{\Vert }^2{|}_S$ being linearisable. We are calling the latter the semi-linearisability of $B$ (with proportionality $\Vert B{\Vert }^2{|}_S$). 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.

Comments: 26 Pages