Hysteresis behavior observed in simulations of the evolution of the radial electric field and parallel flow in the HSX stellarator*

The Helically Symmetric Experiment (HSX) is a quasisymmetric stellarator with minimal parallel viscous damping in a helical direction. The parallel flow (V||) along the magnetic field is similarly weakly damped by viscosity. We use the self-consistent steady-state parallel and poloidal momentum balance equations to show that a large V|| can increase the ion resonant electric field (Er) beyond the value calculated using the typical approximation that V|| is zero. Shaing’s model of non-linear viscosity [1] and damping due to neutrals are included in the calculations. By altering the damping of V||, the ion resonant Er can shift in a controllable fashion. It is shown explicitly there are both stable and unstable steady-state solutions in the two-dimensional space of V|| and Er. A stability analysis of each solution has been performed by calculating the eigenvalues of the Jacobian. It is shown that the unstable solution corresponds to a saddle point in which the eigenvalues have opposite signs and that such solutions occur when the derivative of the total poloidal damping with respect to |Er| is negative.  These results explain experimentally observed phenomena such as the hysteresis in L-H transitions in the Large Helical Device [2].

[1] K. C. Shaing, Physics of Plasmas, 5 3841 (1993). [2] S. Kitajima et al, Nucl. Fusion, 51 083029 (2011)