The Near Axis Expansion in a General Curve Frame for Quasi-Isodynamic Stellarators

Rapid computation of highly accurate quasi-symmetric stellarators was made possible through a direct solution to the magneto-hydrodynamic equations and quasi-symmetry using the near axis expansion (Landreman & Sengupta, J. Plasma Phys. 85 (2019) 815850601). This solution makes use of the Frenet frame of the magnetic axis to describe the position, which becomes discontinuous when curvature approaches 0, preventing the construction of quasi-isodynamic (QI) stellarators using this method. QI stellarators have several interesting traits, such as insensitivity to the pressure profile, and further exploring them can build on the success of W7-X and the island divertor. Previous work on this issue involved finding QI stellerators which had the curvature of the magnetic axis be exactly 0 at certain points, limiting the number of shapes which could be used (Plunk, Landreman & Helander, J. Plasma Phys. 85 (2019) 905850602). An alternative method of solving the issue associated with the Frenet frame, employed in coil construction (Singh et al., J. Plasma Phys. 86 (2020) 905860404), is to change the frame to one which behaves smoothly at these problematic points, allowing for more flexibility in the shape of the magnetic axis. In this work, we re-derive the equations of the near axis expansion up to second order with a general frame to allow for the computation of quasi-isodynamic stellarators using this method.