Low-shear three-dimensional equilibria in a periodic cylinder

Abstract

We carry out expansions of non-symmetric toroidal ideal magnetohydrodynamic (MHD) equilibria with nested flux surfaces about a periodic cylinder, in which physical quantities are periodic of period$2\unicode[STIX]{x03C0}$in the cylindrical angle$\unicode[STIX]{x1D703}$and$z$. The cross-section of a flux surface at a constant toroidal angle is assumed to be approximately circular, and data are given on the cylindrical flux surface$r=1$. Furthermore, we assume that the magnetic field lines are closed on the lowest-order flux surface, and the magnetic shear is relatively small. We extend earlier work in a flat torus by Weitzner (Phys. Plasmas, vol. 23, 2016, 062512) and demonstrate that a power series expansion can be carried out to all orders using magnetic flux as an expansion parameter. The cylindrical metric introduces certain new features to the expansions compared to the flat torus. However, the basic methodology of dealing with resonance singularities remains the same. The results, even though lacking convergence proofs, once again support the possibility of smooth, low-shear non-symmetric toroidal MHD equilibria.