Near-singular equilibrium currents near magnetic islands with broken symmetry

The conventional cylindrical analytical model of a magnetic island is symmetric with respect to combined reflection in the poloidal and toroidal angles, a symmetry property sometimes called 'stellarator symmetry'. There are cancellations in the pressure driven currents when an island is stellarator symmetric that are not there when the symmetry is broken. When the symmetry is broken, new aspects of the island physics emerge. The diamagnetic current driven by the scalar pressure is no longer ambipolar. The pressure-driven current generally has a logarithmic (integrable) singularity at the island separatrix. These effects manifest themselves when taking into account incomplete flattening of the pressure along magnetic field lines. A small magnetic island has only a small effect on the ambient pressure gradient, so that the pressure is not constant on the flux surfaces near the island. For larger islands, this behavior persists near the X-lines. The equilibrium solutions with a small island are to be compared with three-dimensional magnetohydrodynamic equilibrium solutions that are constrained to have simply nested flux surfaces, where the pressure-driven current goes like 1/x near rational surfaces, where x is the distance from the rational surface. This paper presents an investigation of the pressure driven current near a small magnetic island in a cylindrical magnetic field with perturbed circular flux surfaces. The conventional analytical cylindrical model of a field with a magnetic island is augmented with a resonant component that breaks the stellarator symmetry without breaking the flux surfaces. The effects of stellarator symmetry on magnetic islands in tokamaks are of particular interest in the context of the stabilization of edge localized modes by resonant magnetic perturbations. In stellarators, the design studies for the major contemporary devices have discarded half the design parameters available for optimization at the outset by assuming stellarator symmetry.