Rotation roots and neoclassical viscosity in quasi-symmetry

In a quasi-symmetric device, there exists a symmetry angle $\alpha_h = \theta -N\zeta/M$, such that $|B| = B_0 \left(1 - \epsilon_h \cos{M \alpha_h} \right)$ along a field-line, with several much smaller helical `sidebands.' Provided the departure from symmetry is small, i.e. $\delta B_{\rm eff}/B_0 \ll \epsilon_h$ where $\delta B_{\rm eff}/B_0$ is the effective helical sideband strength, flow damping and thus flow evolution along and `cross' the direction of symmetry in a flux surface decouple [1,2], and can be determined successively. In the context of a fluid-moment approach [3], the momentum equation in the symmetry direction is equivalent to the ambipolarity condition. Steady state rotation solutions of this equation are equivalent to ambipolar radial electric field `roots' in conventional stellarator theory and will be presented for various banana-drift neoclassical flow damping regimes [2].\\[4pt] [1] J.~D.~Callen, A.~J.~Cole, and C.~C.~Hegna, Tech.~Rep.~UW-CPTC 08-7, Univ.~of Wisconsin, http://www.cptc.wisc.edu (2009).\\[0pt] [2] A.~J.~Cole, C.~C.~Hegna, and J.~D.~Callen, Tech.~Rep.~UW-CPTC 08-8, Univ.~of Wisconsin, http://www.cptc.wisc.edu (2009).\\[0pt] [3] K.~C.~Shaing and J.~D.~Callen, Phys.~Fluids 26, 3315 (1983).