Pressure Driven Currents Near Magnetic Islands in mD MHD Equilibria: Effects of Pressure Variation Within Flux Surfaces and of Symmetry

Authors:  A. Reiman

Abstract: In toroidal, magnetically  confined plasmas, the heat and particle transport is strongly anisotropic, with transport along the field lines sufficiently  strong relative to cross-field transport that the equilibrium  pressure can generally be regarded as constant on the flux surfaces in much ofthe plasma.  The regions near small magnetic islands, and those near the X-lines of larger islands, are exceptions,  having a significant  variation of the pressure within the flux surfaces. It is shown here that the variation of the equilibrium  pressure within the flux surfaces in those regions has significant consequences  for the pressure driven currents. It is further shown that the consequences are strongly affected by the symmetry of the magnetic field if the field is invariant under combined reflection in the poloidal and toroidal angles. (This symmetry property is called "stellarator symmetry".)  In non-stellarator-symmetric equilibria,  the pressure-driven currents have logarithmic singularities  at the X-lines.  In stellarator-symmetric MHD equilibria,  the singular components of the pressure-driven currents vanish.  These equilibria are to be contrasted with equilibria  having B $⋅$$▽$p  = 0, where the singular components  of the pressure-driven currents vanish regardless of the symmetry. They are also to be contrasted with 3D MHD equilibrium  solutions that are constrained  to have simply nested flux surfaces, where the pressure-driven current goes like 1 I x  near rational surfaces, wherex  is the distance from the rational surface.  (Except in the case of quasi-symmetric flux surfaces.)  For the purpose of calculating the pressure-driven currents near magnetic islands, we work with a closed subset of the MHD equilibrium  equations that involves only perpendicular force balance, and is decoupled from parallel force balance.  It is not correct to use the parallel component of the conventional MHD force balance equation,  B ⋅ $▽$p = 0, near magnetic islands. Small but nonzero values of B ⋅$▽$p are important  in this region, and small non-MHD contributions to the parallel force balance equation cannot be neglected there. Two approaches  are pursued to solve our equations for the pressure driven currents.   First, the equilibrium  equations are applied to an analytically tractable magnetic field with an island, obtaining explicit expressions  for the rotational transform and magnetic coordinates,  and for the pressure-driven current and its limiting behavior near the X-line. The second approach  utilizes an expansion about the X-line to provide a more general calculation  of the pressure-driven current near an X-line and of the rotational transform near a separatrix.  The study presented in this paper is motivated, in part, by tokamak experiments  with nonaxisymmetric magnetic  perturbations, where significant differences are observed between the behavior of stellarator-symmetric and non-stellarator-symmetric configurations with regard to stabilization  of edge localized modes (ELMs) by resonant magnetic perturbations  (RMPs). Implications  for the coupling  between neoclassical  tearing modes (NTMs), and for magnetic island stability calculations, are also discussed.

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