The Geometry of Flux Surfaces with Quasi-Poloidal Symmetry*

We investigate the question of exact quasi-symmetry on a single flux surface, without relying on near-axis methods. We focus on the case of zero toroidal current and quasi-poloidal symmetry (QP). This case is physically significant because it results in zero bootstrap current, reduced neoclassical transport, and the potential ability to support flows that suppress turbulence. It also leads to some simplification, as the field lines are geodesics on flux surfaces. Furthermore, QP has been shown to be impossible near the axis, so a novel framework for QP surfaces is required. Thus, we develop a reduced set of equations for the local geometry of an exact QP flux surface. Under further global considerations, we show that, for an irrational, toroidal surface, the surface equations are equivalent to the vortex filament equation, an integrable PDE. In other words, the ‘evolution’ of field lines along a surface is analogous to that of a vortex filament. Finally, we perturb around magnetic-mirror-like surfaces of revolution, indicating the geometrical features that allow for a surface to ‘bend’.