Electric current dynamics in the stellarator coil winding surface model

TL;DR

This paper develops a theoretical framework for understanding current distribution patterns on stellarator coil winding surfaces, revealing conditions for the emergence of key features like centre and saddle points, which are crucial for coil optimization.

Contribution

It introduces a novel theoretical analysis of current distributions on winding surfaces, establishing conditions for the appearance of centre and saddle points in coil design.

Findings

Dichotomy principle for toroidal winding surfaces: current either has both centre and saddle points or is zero everywhere.

Opposite current orientations on boundary circles lead to centre and saddle points, with most field lines being periodic.

Harmonic potential currents produce closed poloidal orbits, simplifying coil design.

Abstract

In stellarator design, the coil winding surfaces $\Sigma\subset\mathbb R^3$ support current distributions $j$ that shape the magnetic field. This work provides a theoretical framework explaining the emergence of centre and saddle point regions, a key feature in coil optimisation. For coil winding surfaces with a toroidal shape, we prove a dichotomy principle: the current distribution has both centre and saddle point regions or is no-where vanishing. For coil winding surfaces that consist of piecewise cylinders, we show that if $j$ is oppositely oriented on the two boundary circles, centre and saddle points appear, and all but finitely many field lines of $j$ are periodic. When $j$ admits a harmonic potential, all field lines are closed poloidal orbits. These results offer insights into current patterns on winding surfaces, with implications for coil design strategies and their simplification.