Optimizing stellarators for large flows

Plasma flow is damped in stellarators because they are not intrinsically ambipolar, unlike tokamaks, in which the flux-surface averaged radial electric current vanishes for any value of the radial electric field. Only quasisymmetric stellarators are intrinsically ambipolar, but exact quasisymmetry is impossible to achieve in non-axisymmetric toroidal configurations. By calculating the violation of intrinsic ambipolarity due to deviations from quasisymmetry, one can derive criteria to assess when a stellarator can be considered quasisymmetric in practice, i.e. when the flow damping is weak enough. Let us denote by $\alpha$ a small parameter that controls the size of a perturbation to an exactly quasisymmetric magnetic field. Recently, it has been shown that if the gradient of the perturbation is sufficiently small, the flux-surface averaged radial electric current scales as $\alpha^2$ for any value of the collisionality. It was also argued that when the gradient of the perturbation is large, the quadratic scaling is replaced by a more unfavorable one. In this paper, perturbations with large gradients are rigorously treated. In particular, it is proven that for low collisionality a perturbation with large gradient yields, at best, an $O(|\alpha|)$ deviation from quasisymmetry. Heuristic estimations in the literature incorrectly predicted an $O(|\alpha|^{3/2})$ deviation.