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Stellarators are generically small current and low plasma beta ($\beta=p/B^2\ll1$) devices. Often the construction of vacuum magnetic fields with goodmagnetic surfaces is the starting point for an equilibrium calculation.Although in cases with some continuous spatial symmetry flux functions canalways be found for vacuum magnetic fields, an analogous function does not, ingeneral, exist in three dimensions. This work examines several simpleequilibria and vacuum magnetic field problems with the intent of demonstratingthe possibilities and limitations in the construction of such states. Startingwith a simple vacuum magnetic field with closed field lines in a topologicaltorus (toroidal shell with a flat metric), we obtain a self-consistent formalperturbation series using the amplitude of the nonsymmetric vacuum fields as asmall parameter. We show that systems possessing stellarator symmetry allow theconstruction order by order. We further indicate the significance ofstellarator symmetry in the amplitude expansion of the full idealmagnetohydrodynamics (MHD) problem as well. We then investigate the conditionsthat guarantee neighboring flux surfaces given the data on one surface, byexpanding in the distance from that surface. We show that it is much moredifficult to find low shear vacuum fields with surfaces than force-free fieldsor ideal MHD equilibrium. Finally, we demonstrate the existence of a class ofvacuum magnetic fields, analogous to `snakes' observed in tokamaks, which canbe expanded to all orders.

Low-shear three-dimensional equilibria and vacuum magnetic fields with flux surfaces