Electric field effects in a quasisymmetric stellarator

It was recently shown [1] that the radial electric field $E_r$ can alter the neoclassical ion flow, heat flux, bootstrap current, and residual zonal flow in a tokamak, even if the ExB speed is small compared to the ion thermal speed. Here we show these calculations can be adapted to a quasisymmetric stellarator. The finite-$E_r$ effects can be understood as finite-orbit-width effects associated with variation in the electrostatic potential over an orbit. The tokamak calculation uses $B_{poloidal}/B$ as a small parameter, and in the stellarator case, we use the fact that an analogous ratio of field components can be small. The quasisymmetry calculation also exploits conservation of the ``helical momentum'' $\psi_*$, which is used in place of the toroidal or poloidal flux as the radial variable. The calculations generalize the model collision operator in [1], which keeps only velocity-space derivatives normal to the trapped-passing boundary, even as this boundary is significantly shifted by a large ExB drift. We show $E_r$ in HSX may be sufficient for these effects to be significant.\\[4pt] [1] G Kagan and P J Catto, Plasma Phys. Control. Fusion, 52, 055004 (2010).