Integrability, normal forms, and magnetic axis coordinates

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 entirely consisting of invariant submanifolds. A compact regular leaf (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: they are presymplectic systems.

Existence of global symmetries of divergence-free fields with first integrals

“…Proof. The proof of this lemma is similar to that of [3,Lemma 3.4]. One direction is clear; if ι u ι B µ = ν then 0 = ι B ι u ι B µ = ι B ν regardless of whether η(u) = 0.…”

“…It shows that, given a flux system (B, ν, µ), one can use the 1-form η, if it exists, to uniquely generate a vector field u which is a symmetry for the rescaled field B. It will also be shown how adapted 1-forms relate to the theory in [3], which in turn can be used to prove Thm. I.3 from a more abstract perspective.…”

“…Without substantial gradients, the core of a fusion device would not be hot enough to sustain fusion reactions and the outer region cold enough for the device's walls to resist damage. For any B admitting both a symmetry and an integral, the field is integrable in the broad sense [3] as defined in [4]. On the other hand, divergence-free fields generally admit no symmetries nor integrals and are chaotic [5,6].…”

“…It has been established [3,7] that a divergence-free vector field B in 3D admits a Noether theorem: loosely, this states that for each symmetry field of B, there corresponds a first integral of B. However, the converse is false: there exists B which has a first integral but no corresponding symmetry (see for instance Example II.12).…”

“…I. 3. In fact, motivated by the work of [12,13] we exhibit an example (Example IV.3) flux system (B, dψ, µ) on a solid torus for which an adapted 1-form exists on all level sets of ψ except the unique level set contained in the complement of the toroidal region.…”