On 2D Harmonic Extensions of Vector Fields and Stellarator Coils

TL;DR

This paper studies the mathematical problem of extending vector fields harmonically in 2D domains with analytic boundaries, with applications to optimizing coil placement in stellarator magnetic confinement devices.

Contribution

It establishes existence conditions for harmonic extensions of vector fields and provides bounds on the extension distance relevant to stellarator coil design.

Findings

Harmonic extension exists under specific boundary conditions.

A lower bound on extension distance is derived using the Cauchy--Kovalevskaya Theorem.

The method generalizes Hadamard's result on the Laplacian Cauchy problem.

Abstract

We consider a problem relating to magnetic confinement devices known as stellarators. Plasma is confined by magnetic fields generated by current-carrying coils, and here we investigate how closely to the plasma they need to be positioned. Current-carrying coils are represented as singularities within the magnetic field and therefore this problem can be modelled mathematically as finding how far we can harmonically extend a vector field from the boundary of a domain. For this paper we consider two-dimensional domains with real analytic boundary, and prove that a harmonic extension exists if and only if the boundary data satisfies a combined compatibility and regularity condition. Our method of proof uses a generalisation of a result of Hadamard on the Cauchy problem for the Laplacian. We then provide a lower bound on how far we can harmonically extend the vector field from the boundary via the Cauchy--Kovalevskaya Theorem.