Pfirsch-Schluter Current in and near a Magnetic Island: Singular Behavior and Symmetry Effects.

The current along magnetic field lines that enforces quasi- neutrality is called the ``Pfirsch-Schluter current''. We show that the Pfirsch-Schluter current has, in general, a logarithmic singularity at the X-line of a magnetic island separatrix if $\nabla \cdot {\rm {\bf j}}_{\bot } $ is nonzero there. The singular component of the Pfirsch-Schluter current vanishes if the configuration is stellarator symmetric about a point on the X-line. (Symmetric with respect to simultaneous reflection in the poloidal and toroidal angles.) We consider, in particular, the case where ${\rm {\bf j}}_{\bot } $ is determined by the MHD equilibrium force-balance equation and the pressure gradient is determined by a diffusion equation. There is a critical scale length, xc, determined by the ratio of the perpendicular and parallel diffusion coefficients, such that the pressure is not flattened on flux surfaces within a distance of order xc about the X-line. The variation of pressure on flux surfaces in that region leads to a nonzero $\nabla \cdot {\rm {\bf j}}_{\bot } $ at the X-line, and a large Pfirsch-Schluter current near the X-line. This is a significant piece of physics that is absent in analytical calculations for perturbed cylindrical models having a single resonant Fourier component, and in 3D codes that have no variation in pressure within their flux surfaces.