Obstructions to topological relaxation for generic magnetic fields

TL;DR

This paper demonstrates that in certain toroidal domains, a generic set of divergence-free vector fields cannot be topologically equivalent to magnetohydrostatic equilibria due to complex orbit structures and a new rigidity theorem.

Contribution

It introduces a geometric obstruction and a rigidity theorem showing that generic magnetic fields with complex dynamics cannot relax to MHS equilibria.

Findings

Residual set of vector fields with complex orbit structures

Dense nondegenerate periodic orbits prevent topological equivalence to MHS

New rigidity theorem for magnetic field relaxation

Abstract

For any axisymmetric toroidal domain $\Omega \subset \mathbf{R}^3$ we prove that there is a locally generic set of divergence-free vector fields that are not topologically equivalent to any magnetohydrostatic (MHS) equilibrium in $\Omega$. Each vector field in this set is Morse-Smale on the boundary, does not admit a nonconstant first integral, and exhibits fast growth of periodic orbits; in particular this set is residual in the Newhouse domain. The key dynamical idea behind this result is that a vector field with a dense set of nondegenerate periodic orbits cannot be topologically equivalent to a generic MHS equilibrium. On the analytic side, this geometric obstruction is implemented by means of a novel rigidity theorem for the relaxation of generic magnetic fields with a suitably complex orbit structure.