Optimized stellarator configurations and their analytical properties are obtained using a near-axis expansion approach. Such configurations are associated with good confinement as the guiding center particle trajectories and neoclassical transport are isomorphic to those in a tokamak. This makes them appealing as fusion reactor candidates. Using a direct coordinate approach, where the magnetic field and flux surface functions are found explicitly in terms of the position vector at successive orders in the distance to the axis, the set of ordinary differential equations for first and second order quasisymmetry is derived. Examples of quasi-axisymmetric shapes are constructed using a pseudospectral numerical method. Finally, the direct coordinate approach is used to independently verify two hypotheses commonly associated with quasisymmetric magnetic fields, namely that the number of equations exceeds the number of parameters at third order in the expansion and that the near-axis expansion does not prohibit exact quasisymmetry from being achieved on a single flux surface.
Optimizing stellarators for large flows