Why is it so hard to generate 3D MHD equilibria with smoothly nested flux surfaces?

Renewed interest in stellarator design has sparked questions on the existence and accessibility of three-dimensional magneto-hydrodynamic (MHD) equilibria with “good” nested flux-surfaces. Several numerical tools exist to obtain three-dimensional MHD equilibria. These methods aspire to produce and optimise the magnetic fields so that the field-lines lie on toroidally nested flux-surfaces, which is the basis of plasma confinement in magnetic fusion devices such as tokamaks and stellarators. Under the assumption of translational and/or rotational symmetry (isometries), the MHD equilibrium problem reduces to a two-dimensional elliptic PDE for the scalar flux-function, called the Grad–Shafranov equation. Flux-surfaces naturally correspond to the level sets of the flux-function extruded to surfaces along the direction of symmetry.

Without explicit symmetry, it can be proven, whenever the plasma current is not everywhere parallel to the magnetic field, that field-lines lie on nested tori [1,Theorem 7]. However, the embedding of these topological tori may be extremely complicated (knotted) in real space. The theorem also does not inform on the existence nor on the accessibility of such equilibrium configurations. On the other hand, force-free fields correspond to minimum energy solutions of the boundary value problem. The behaviour of their field-lines is known to be chaotic even in the simplest three-dimensional instances. Three-dimensional MHD equilibria (within the class of smooth magnetic fields) whose field-lines lie on simple nested flux-surfaces are thus rather exceptional, fine-tuned solutions.

The goal of this talk is to illustrate an essential difficulty in smoothly deforming an initial configuration with nested flux-surfaces through a family of MHD equilibria to reach a target three-dimensional configuration with equivalent flux-surfaces. The issue from a physical point of view is that flux-surfaces with periodic field-lines (rational rotational transform) are sensitive to resonant perturbations. Even the smallest amplitude leads to the formation of magnetic islands in the vicinity of resonant surfaces. Whether resonance can be avoided in order to form a smooth sequence of MHD equilibrium states is a different line of inquiry, closely related to the study of stationary solutions to Euler equations and their properties.

Taylor configuration [2] using suitable generating functions to perturb the Grad-Shafranov equation. The colour contours represent the level sets of the flux-function and the thick green curves are level sets of the current density. The fact that the two do not overlap indicates lack of force balance, occurring at all orders with respect to the boundary deformation [3].

[1] V. I. Arnold, Annales de l’Institut Fourier 10, 137 (1967).
[2] T. S. Hahm and R. M. Kulsrud, The Physics of Fluids 28, 2412 (1985).
[3] D. Pfefferlé et al., PPCF 62, 074004 (2020).