Error field penetration and locking to the backward propagating wave

Resonant field amplification or error field penetration involves driving a weakly stable tearing perturbation in a rotating toroidal plasma. In this paper it is shown that the locking characteristics for modes with finite real frequencies $\omega_{r}$ are quite different from the conventional results. A calculation of the tearing mode amplitude assuming modes with frequencies $\pm\omega_{r}$ in the plasma frame shows that it is maximized when the frequency of the stable backward propagating mode ($-\omega_{r}$) in the lab frame is zero, i.e. when $v=\omega_{r}/k$. Even more importantly, the locking torque is exactly zero at the mode phase velocity, with a pronounced peak at just higher rotation, leading to a locked state with plasma velocity $v$ just above the mode phase velocity in the lab frame. Real frequencies $\pm\omega_{r}$, leading to a $v\rightarrow-v$ symmetry, are known to occur due to the Glasser effect [A.H. Glasser, J.M. Greene, and J.M. Johnson, Phys. Fluids {\bf 19}, 567 (1976).] for modes in the resistive-inertial (RI) regime. This therefore leads to locking of the plasma velocity to just above the phase velocity. It is also shown that similar real frequencies occur over a range of parameters in the visco-resistive (VR) regime with pressure, and the locking torque is similar to that in the RI regime. Real frequencies occur due to diamagnetic effects in other tearing mode regimes and also show this effect, but without the $v\rightarrow-v$ symmetry. Nonlinear effects on the mode amplitude and torque for weakly stable modes or large error fields are discussed. Also, the possibility of applying external fields of different helicities to drive sheared flows in toroidal plasmas is discussed.