An analytic formula for surface currents generating prescribed plasma equilibrium fields

TL;DR

This paper derives an analytic formula for surface currents on a coil winding surface that generate prescribed plasma equilibrium magnetic fields, explaining observed current distribution phenomena.

Contribution

It introduces a new analytic formula for surface currents that produce specific plasma magnetic fields, enhancing understanding and potential control of plasma equilibria.

Findings

The formula precisely reproduces the desired plasma equilibrium field.

It explains why currents tend to concentrate at points of highest normal magnetic curvature.

Includes discussion on numerical approximation aspects.

Abstract

Given a plasma domain $P\subset\mathbb{R}^3$, a plasma equilibrium field $B$ on $P$ and a coil winding surface $\Sigma$ surrounding $P$, we provide an analytic formula whose output is a surface current distribution $j$ on $\Sigma$ such that $\operatorname{BS}(j)+\operatorname{BS}(\operatorname{curl}(B))=B$ in $P$, i.e. the combination of the plasma current magnetic field and the surface current magnetic field exactly produce the full plasma equilibrium field. We use this new formula to provide a theoretical explanation of the empirical phenomenon that currents tend to attain their maximal strength at points which correspond to points of highest normal magnetic curvature on the plasma boundary. Some discussions regarding aspects of numerical approximations are also included.