Adjoint methods for quasisymmetry of vacuum fields on a surface

Adjoint methods, recently introduced to the field of stellarator optimisation [1-3], lead to significant speed-up by providing gradient information, avoiding slower gradient-free optimisation methods or expensive finite-difference evaluations of the gradient. We herein apply adjoint methods to vacuum magnetic fields, deriving the adjoint equations corresponding to two objective functions, targeting either a given rotational transform or quasisymmetry with a given helicity on a surface. To measure the deviation from quasisymmetry, a novel way of evaluating approximate flux coordinates on a single flux surface without the assumption of a neighbourhood of flux surfaces is proposed, as vacuum fields are not generally integrable. The shape gradients obtained from the adjoint formalism are evaluated numerically, checked against a finite difference evaluation and shown for several configurations. First applications to stellarator optimisation are also presented.

[1] Landreman, M., Paul, E., 2018. Nucl. Fusion 58, 076023

[2] Paul, E.J., Landreman, M., Bader, A., Dorland, W., 2018.  Nucl. Fusion 58, 076015.

[3] Paul, E.J., Abel, I.G., Landreman, M., Dorland, W., 2019.  Journal of Plasma Physics 85.