Preferred Magnetic Axes For Optimal Quasi-Axisymmetry

Recently, enormous progress has been made in obtaining quasisymmetry (QS) of outstanding precision through numerical optimization. Significant analytical progress has also been made possible thanks to the asymptotic expansions near the magnetic axis (NAE). A critical factor in realizing good QS through the second-order of the NAE is the choice of the magnetic axis. However, because of the complexity of the second-order NAE equations, analytical characterization of these preferred magnetic axes for optimal QS has not been possible so far. In this work, we shall attempt to answer this question for quasi-axisymmetric (QA) systems. We show that the magnetic axis is well described for small rotational transforms by the same equations that govern Euler-Kirchhoff elastic rod centerlines (Langer and Singer, SIAM review 1996, Pfefferlé et al. PoP 2018). Surprisingly, the connection to these equations can only be made partially within the NAE framework and requires several concepts from the soliton theory. We shall present analytical and numerical evidence supporting our insights for a broad range of QA stellarators.