Low Collisionality Neoclassical Toroidal Viscosity in Tokamaks and Quasi-symmetric Stellarators

Non-resonant magnetic perturbations can affect plasma rotation in toroidally confined plasmas through their modification to $|B|$. Variations along a field line induce nonambipolar radial transport and produce a global neoclassical toroidal viscous force [NTV]. In this work, previously calculated radial particle fluxes for the low-collisionality ``$\nu$'' and ``$1/\nu$'' regimes [1] are unified into a single particle flux (or toroidal viscous force). Provided pitch-angle scattering dominates over collisional energy exchange, the energy component of phase space can be decoupled into independent regions $\left(E >E_c \right.$ for $\nu$ regime, $E < E_c$ for $1/\nu$ regime, with $E_c$ determined by $\left.\nu_i(E_c) =\epsilon \, \omega_E\right)$ within which the perturbed distribution function can be calculated similar to [1]. Using a technique first employed in axisymmetric neoclassical theory [2], the smoothed particle flux is constructed by summing the partial contributions from $\nu$ and $1/\nu$ banana drift effects respectively. The complete NTV force is expressed in terms of the equilibrium flows and a temperature-gradient-determined ``intrinsic'' flow. [1] K.C.~Shaing, Phys.~Plasmas, \textbf{10}, 1443 (2003). [2] K.T.~Tsang, and J.D.~Callen, Phys.~Fluids \textbf{19}, 667 (1976).