Rigidity of MHD equilibria to smooth incompressible ideal motion near resonant surfaces

TL;DR

This paper investigates the limitations of smooth incompressible ideal plasma motions in reaching certain MHD equilibria, revealing that some boundary displacements are inherently unsupported due to resonant behaviors near rational layers.

Contribution

It demonstrates the rigidity of MHD equilibria under ideal motions and analyzes the resonant behavior near rational layers using the Hahm--Kulsrud--Taylor problem.

Findings

Certain boundary displacements are formally unsupported.

The vector field generating volume-preserving diffeomorphisms vanishes at all orders near the resonant layer.

Resonant behavior constrains the accessible MHD equilibria.

Abstract

In ideal MHD, the magnetic flux is advected by the plasma motion, freezing flux-surfaces into the flow. An MHD equilibrium is reached when the flow relaxes and force balance is achieved. We ask what classes of MHD equilibria can be accessed from a given initial state via smooth incompressible ideal motion. It is found that certain boundary displacements are formally not supported. This follows from yet another investigation of the Hahm--Kulsrud--Taylor (HKT) problem, which highlights the resonant behaviour near a rational layer formed by a set of degenerate critical points in the flux-function. When trying to retain the mirror symmetry of the flux-function with respect to the resonant layer, the vector field that generates the volume-preserving diffeomorphism vanishes at the identity to all order in the time-like path parameter.