This talk was given at the 4th Asian-Pacific Conference on Plasma Physics on October 28 2020. I am sharing it here because it provides a rapid introduction to the business of MHD equilibria calculation.

#### Abstract

Renewed interest in stellarator design has sparked questions on the existence and accessibility of three-dimensional magneto-hydrodynamics (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.

The goal of this talk is to illustrate the essential difficulties 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. Whether resonance can be avoided in order to form a smooth sequence of MHD equilibrium states is an interesting line of inquiry, closely related to the study of stationary solutions to Euler equations.