Closed orbits of MHD equilibria with orientation-reversing symmetry

TL;DR

This paper investigates the structure of MHD equilibria with orientation-reversing symmetries, demonstrating that under certain conditions, all orbits on nested tori are necessarily periodic, using topological methods.

Contribution

It generalizes the understanding of periodic orbit structures in MHD equilibria to include non-reflection orientation-reversing symmetries, introducing a topological index for these cases.

Findings

All orbits on nested tori are periodic under orientation-reversing isometries.

The techniques are primarily topological, involving an index for diffeomorphisms.

The results extend the symmetry analysis beyond reflection symmetries.

Abstract

As a generalisation of the periodic orbit structure often seen in reflection or mirror symmetric MHD equilibria, we consider equilibria with other orientation-reversing symmetries. An example of such a symmetry, which is a not a reflection, is the parity transformation $(x,y,z) \mapsto (-x,-y,-z)$ in $\mathbb{R}^3$. It is shown under any orientation-reversing isometry, that if the pressure function is assumed to have toroidally nested level sets, then all orbits on the tori are necessarily periodic. The techniques involved are almost entirely topological in nature and give rise to a handy index describing how a diffeomorphism of $\mathbb{R}^3$ alters the poloidal and toroidal curves of an invariant embedded 2-torus.