Nested invariant tori foliating a vector field and its curl: toward MHD equilibria and steady Euler flows in toroidal domains without continuous Euclidean isometries

TL;DR

This paper investigates the existence of special vector fields on toroidal surfaces that are relevant for magnetic confinement in fusion reactors, providing new mathematical solutions and insights into MHD equilibria and Euler flows without continuous symmetries.

Contribution

It establishes the existence of smooth solutions to a PDE-based problem on toroidal surfaces, relevant for modeling non-symmetric MHD equilibria and steady Euler flows.

Findings

Existence of smooth solutions foliated by non-invariant toroidal surfaces.

Explicit construction of solutions as equilibria of anisotropic MHD.

Relevance to magnetic field design in stellarators.

Abstract

This paper studies the problem of finding a three-dimensional solenoidal vector field such that both the vector field and its curl are tangential to a given family of toroidal surfaces. We show that this question can be translated into the problem of determining a periodic solution with periodic derivatives of a two-dimensional linear elliptic second-order partial differential equation on each toroidal surface, and prove the existence of smooth solutions. Examples of smooth solutions foliated by toroidal surfaces that are not invariant under continuous Euclidean isometries are also constructed explicitly, and they are identified as equilibria of anisotropic magnetohydrodynamics. The problem examined here represents a weaker version of a fundamental mathematical problem that arises in the context of magnetohydrodynamics and fluid mechanics concerning the existence of regular equilibrium magnetic fields and steady Euler flows in bounded domains without continuous Euclidean isometries. The existence of such configurations represents a key theoretical issue for the design of the confining magnetic field in nuclear fusion reactors known as stellarators.