Normal Forms and Near-Axis Expansions for Beltrami Magnetic Fields

TL;DR

This paper develops a formal series transformation to Birkhoff-Gustavson normal form for toroidal magnetic fields near an axis, using Hamiltonian dynamics and Floquet theory to analyze and explicitly compute the magnetic field expansion.

Contribution

It introduces a unified approach for vacuum and force-free fields, employing Bishop's coordinates and Hamiltonian methods to derive a well-defined, explicit magnetic field expansion near the axis.

Findings

Normal form transformation is valid to all orders.

Explicit expansion computed to degree four.

Applicable to elliptic, hyperbolic, and resonant cases.

Abstract

A formal series transformation to Birkhoff-Gustavson normal form is obtained for toroidal magnetic field configurations in the neighborhood of a magnetic axis. Bishop's rotation-minimizing coordinates are used to obtain a local orthogonal frame near the axis in which the metric is diagonal, even if the curvature has zeros. We treat the cases of vacuum and force-free (Beltrami) fields in a unified way, noting that the vector potential is essentially the Poincar\'e-Liouville one-form of Hamiltonian dynamics, and the resulting magnetic field corresponds to the canonical two-form of a nonautonomous one-degree-of-freedom system. Canonical coordinates are obtained and Floquet theory is used to transform to a frame in which the lowest-order Hamiltonian is autonomous. The resulting magnetic axis can be elliptic or hyperbolic, and resonant elliptic cases are treated. The resulting expansion for the field is shown to be well-defined to all orders, and is explicitly computed to degree four. An example is given for an axis with constant torsion near a 1:3 resonance.