Linearisability of divergence-free fields along invariant 2-tori

TL;DR

This paper establishes conditions under which divergence-free vector fields on invariant tori can be simplified to linear form, broadening the understanding of magnetic field behavior in plasma confinement without requiring strict assumptions like flux functions.

Contribution

It introduces weaker conditions for linearisability of divergence-free fields on invariant tori, utilizing cohomological equations and pseudo-analytic function theory, extending Arnold's Structure Theorems.

Findings

Field $B$ is either zero or semi-linearisable on invariant tori.

Semi-linearisability depends on solutions to a cohomological equation involving $B$.

A Diophantine condition ensures full linearisability of $B$ on the surface.

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.