Integrability, Normal Forms, and Magnetic Axis Coordinates

TL;DR

This paper proves the existence of smooth normal forms for integrable magnetic fields near elliptic and hyperbolic axes, advancing understanding of magnetic confinement in plasma physics.

Contribution

It establishes the existence of smooth normal forms near magnetic axes, including the construction of Hamada and Boozer coordinates, confirming conjectured smoothness properties.

Findings

Normal forms exist near elliptic and hyperbolic axes.

Hamada and Boozer coordinates are constructed near elliptic axes.

Results confirm smoothness conjectures for MHD equilibrium solutions.

Abstract

Integrable or near-integrable magnetic fields are prominent in the design of plasma confinement devices. Such a field is characterized by the existence of a singular foliation consisting entirely of invariant submanifolds. A regular leaf, known as a flux surface,of this foliation must be diffeomorphic to the two-torus. In a neighborhood of a flux surface, it is known that the magnetic field admits several exact, smooth normal forms in which the field lines are straight. However, these normal forms break down near singular leaves including elliptic and hyperbolic magnetic axes. In this paper, the existence of exact, smooth normal forms for integrable magnetic fields near elliptic and hyperbolic magnetic axes is established. In the elliptic case, smooth near-axis Hamada and Boozer coordinates are defined and constructed. Ultimately, these results establish previously conjectured smoothness properties for smooth solutions of the magnetohydrodynamic equilibrium equations. The key arguments are a consequence of a geometric reframing of integrability and magnetic fields; that they are presymplectic systems.