Nonsymmetric 3D vacuum magnetic fields with surfaces*

In this work, we present three different approaches to the question of existence of flux surfaces in a vacuum magnetic field in a low shear stellerator. Our first approach is to demonstrate that a perturbation series in the amplitude of the non-symmetric components can be carried to all orders by carefully adding certain "resonant fields" as shown in (Weitzner2014). Next, we study small perturbations of an equilibrium with closed field lines. We show that stellarator symmetry plays a crucial role in ensuring existence of surfaces to all orders in the expansion. Finally, we attempt to solve the Cauchy problem by expanding in the distance from a rational flux surface following (Weitzner2016). Since rational surface is a characteristic surface of ideal MHD, we need consistency conditions on the Cauchy data on the lowest order surface. Through a careful examination of the differences in the underlying mathematical structures of the various ideal MHD systems, we show that suppressing magnetic resonances in vacuum fields is far more complicated and involved than force free or MHD with finite beta systems.

Weitzner, H. Physics of Plasmas  (2014,2016)