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 2π in the cylindrical angle θ and z. The cross-section of a flux surface at a constant toroidal angle is assumed to be approximately circular, and data is 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 [Physics of Plasmas 23, 062512 (2016)]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, lowshear non-symmetric toroidal MHD equilibria.

Charged particle dynamics near an X-point of a non-symmetric magnetic field with closed field lines

“…On the other hand, various asymptotic expansions (Weitzner 2016;Jaquiery & Sengupta 2019; provide hints that vacuum and ideal MHD equilibria with flux surfaces and continuous rotational transform and pressure profiles might exist provided that the rotational transform is close to a low-order rational number, the magnetic field shear is low, and the boundary is chosen self-consistently. The jump from a closed-field line MHD equilibrium with exactly zero shear to a neighboring equilibrium with minimal magnetic shear might not be discontinuous Weitzner (2016).…”

Vacuum magnetic fields with exact quasisymmetry near a flux surface. Part 1. Solutions near an axisymmetric surface

“…The near-surface theory presented in this work builds on the formalism developed by Sengupta and Weitzner (Weitzner 2016;Jaquiery & Sengupta 2019;Sengupta & Weitzner 2019), which was used to study the existence of low-shear magnetic fields with surfaces and construct a global vacuum solution with closed field lines in slab geometry ). Although we discuss expansions about axisymmetric surfaces in this work, the methodology works for non-symmetric surfaces as well.…”