Abstract
Quasi-symmetry of a steady magnetic field means integrability of first-order guiding-center motion by a spatial symmetry. Here, we derive many restrictions on the possibilities for a quasi-symmetry. We also derive an analog of the Grad–Shafranov equation for the flux function in a quasi-symmetric magnetohydrostatic field.
Highlights
The concept of quasi-symmetry was introduced by Boozer (1983) and distilled into a design principle for stellarators by Nührenberg and Zille (1988)
A fundamental step was made by Burby and Qin (2013), who stated necessary and sufficient local conditions for integrability of guidingcenter motion in terms of a continuous symmetry of three differential forms derived from the magnetic field and made clear that quasisymmetry can be separated from the issue of whether the magnetic field is MHS or not
Perturbative calculations of Garren and Boozer (1991), make it look very likely that the only possibility for exact quasisymmetry for MHS fields with bounded magnetic surfaces is axisymmetry
Summary
The concept of quasi-symmetry was introduced by Boozer (1983) and distilled into a design principle for stellarators by Nührenberg and Zille (1988). A fundamental step was made by Burby and Qin (2013), who stated necessary and sufficient local conditions for integrability of guidingcenter motion in terms of a continuous symmetry of three differential forms derived from the magnetic field and made clear that quasisymmetry can be separated from the issue of whether the magnetic field is MHS or not. We prove many consequences of quasi-symmetry and thereby restrictions on possible quasi-symmetric fields. In the case of a quasi-symmetric MHS field, we derive a generalization of the axisymmetric Grad–Shafranov (GS) equation. Burby and Qin (2013) built in an assumption that a quasi-symmetry must be a circle action We relax this requirement, though prove that under some mild conditions, it is a circle action. Throughout the paper, we will assume enough smoothness that the equations we write make sense, at least in a weak sense
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