Neoclassical Toroidal Viscosity Induced Rotation in Tokamaks and Quasi-symmetric Stellarators*

Non-axisymmetric magnetic perturbations generate variations in $|B|$ along a field line that induce non-ambipolar radial transport and a global toroidal force on the plasma, known as neoclassical toroidal viscosity [NTV]. A strong correlation exists between the flow evolution physics of tokamaks and quasi-helically symmetric [QHS] stellarators. In QHS-mode, there exists a helical symmetry angle $\alpha \equiv m\theta - n \zeta$, with $m,n$ fixed integers that is analogous to the poloidal direction in tokamaks. As a result, there exists a direction of near helical symmetry and thus least flow damping along $\vec{e}_h$ such that $\vec{e}_h \cdot \vec{\nabla}\alpha = 0$, analogous to the toroidal tokamak direction. In this paper, a model analytic 'toroidal' rotation equation is developed which smoothly transitions between previously asymptotic low-collisionality regimes [1], while incorporating both electron and ion NTV. In particular, the transition from ion to electron dominated NTV is presented in a single equation for the first time. This research will facilitate future comparison between NTV-induced rotation in QHS stellarators and tokamaks. [1] K.C.~Shaing, Phys.~Plasmas, \textbf{10}, 1443 (2003).