Invariant tori for the pressure-jump Hamiltonian

TL;DR

The paper addresses the existence of invariant tori for the pressure-jump Hamiltonian governing non-axisymmetric magnetohydrostatic interfaces. It recasts the problem as a positive-definite Hamiltonian on $T^*\Sigma$ and applies Converse KAM via the Maupertuis metric to identify phase-space regions through which invariant tori cannot pass, then maps these regions to $( [P], \boldsymbol\iota )$-space to determine eliminated intervals of rotational transform; it also explores Liouville (Liouville-type) metrics on tori as a broader class of integrable interfaces that could support solutions for all but two winding ratios. The main contributions are (i) a rapid computational approach to certify nonexistence regions in both phase space and parameter space, (ii) an extension of the framework to continuous-time dynamics and conjugate-point criteria, and (iii) a discussion of Liouville metrics that may realize interfaces with near-total solvability. This work provides a constructive tool for guiding stepped-pressure MHS equilibria construction and broadens the scope of potential interface geometries via integrable torus metrics, with implications for non-axisymmetric configurations beyond axisymmetric limits.

Abstract

A major achievement of Dewar and coworkers is the SPEC code to construct stepped-pressure equilibria in magnetohydrostatics without axisymmetry. Their existence had been proved by Bruno and Laurence. As part of the procedure of Bruno and Laurence, it is required to solve the Hamilton-Jacobi equation for a magnetic potential on the outside of an interface given the field on the inside and the pressure-jump across the interface. For non-axisymmetric interface, it was understood that solutions with insufficiently irrational rotational transform might not exist, and examples have been given for which there are no solutions at all for large enough pressure-jump. The present paper gives a method to compute regions in the phase space for the pressure-jump Hamiltonian through which no invariant tori pass. The paper also shows how to present the results as regions in the space of pressure-jumps and outer rotational transform for which there is no solution of the Hamilton-Jacobi equation. The method is expected to reach arbitrarily close to the full non-existence region with enough computational work, so what is left over can be relied on to be mostly invariant tori. The paper also brings to attention a class of metrics on tori that are not necessarily axisymmetric yet have integrable geodesic flow. They could give interfaces with solutions for all but finitely many rotational transforms.