An analytic formula for surface currents generating prescribed plasma equilibrium fields

Given a plasma domain $P\subset {\mathbb{R}}^3$, a plasma equilibrium field $B$ on $P$ and a coil winding surface $\mathrm{\Sigma }$ surrounding $P$, we provide an analytic formula whose output is a surface current distribution $j$ on $\mathrm{\Sigma }$ such that $\mathrm{BS}(j)+\mathrm{BS}(\mathrm{curl}(B))=B$ 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.