Including the parallel mass flow in calculating the steady-state solutions and stability of the momentum balance equations for a quasisymmetric stellarator
The Helically Symmetric Experiment (HSX) is a quasisymmetric stellarator with minimal parallel viscous damping in a helical direction. The parallel flow (Vǁ) along the magnetic field is similarly weakly damped by viscosity. In this paper, the self-consistent steady-state parallel and poloidal momentum balance equations are used to show that a large Vǁ on the order of the ion thermal velocity can increase the ion resonant radial electric field (Er) beyond the value calculated using the typical approximation that Vǁ is zero. By altering the damping of Vǁ, either by degrading the quasisymmetry or varying the neutral density, the ion resonant Er can shift in a controllable fashion. It is shown explicitly that there exist stable and unstable steady-state solutions in the two-dimensional space of Vǁ and Er. A stability analysis of each solution is performed by calculating the eigenvalues and eigenvectors of the Jacobian. The unstable solution corresponds to a saddle point in which the eigenvalues have opposite signs. The analysis leads to the conclusion that unstable solutions occur when the derivative of the total poloidal damping with respect to Er is positive. A hysteresis in Er and Vǁ is observed when the radial current density is linearly increased to a maximum and then decreased back to zero. Jumps in the radial electric field and the parallel flow are observed as the radial current density drives the evolution from one stable point to the next. This result is similar to experimental data observed on several devices.