On the distribution of heat in fibered magnetic fields

TL;DR

This paper analyzes how temperature distributes in strongly magnetized plasmas with structured magnetic fields, showing it mainly varies across invariant surfaces and confirming a physical conjecture about non-integrability effects.

Contribution

It proves that temperature distribution in fibered magnetic fields is primarily across invariant tori and extends results to nearly integrable fields, confirming a conjecture on non-integrability.

Findings

Temperature is well approximated by a function varying across invariant surfaces.

Results hold for nearly integrable fields up to a critical non-integrability size.

The proof relies on ergodicity and Diophantine conditions for magnetic field lines.

Abstract

We study the equilibrium temperature distribution in a model for strongly magnetized plasmas in dimension two and higher. Provided the magnetic field is sufficiently structured (integrable in the sense that it is fibered by co-dimension one invariant tori, on most of which the field lines ergodically wander) and the effective thermal diffusivity transverse to the tori is small, it is proved that the temperature distribution is well approximated by a function that only varies across the invariant surfaces. The same result holds for "nearly integrable" magnetic fields up to a "critical" size. In this case, a volume of non-integrability is defined in terms of the temperature defect distribution and related the non-integrable structure of the magnetic field, confirming a physical conjecture of Paul-Hudson-Helander. Our proof crucially uses a certain quantitative ergodicity condition for the magnetic field lines on full measure set of invariant tori, which is automatic in two dimensions for magnetic fields without null points and, in higher dimensions, is guaranteed by a Diophantine condition on the rotational transform of the magnetic field.