Locking to an error field in the presence of real tearing frequencies due to finite β

We present NIMROD simulations to study error field locking behavior in plasmas having marginally stable tearing modes with real frequencies. Real frequency modes are caused by pressure gradient, curvature, and parallel dynamics in the resistive-inertial (RI) and viscoresistive (VR) tearing regimes. A periodic cylinder with a hollow pressure profile is used to model the toroidal effects. In the linear regime, raising β stabilizes the 2/1 tearing mode and leads to real frequencies. Stabilization is due to both the positive pressure gradient in the outer region and favorable curvature in the tearing layer. Linear simulations with rotation and an error field of magnitude ε show that the maximum reconnected flux occurs at the tearing mode phase velocity. In the nonlinear simulations, the plasma flow at the rational surface locks to a rotation just above the tearing mode phase velocity. Increasing ε further results in a large island that flattens the pressure, causing further nonlinear growth and terminating the propagation of the real frequency of the tearing mode. We describe the interplay of three nonlinear effects: current flattening, pressure flattening, and locking.