Early nonlinear regime of MHD internal modes: The resistive case

It is shown that the critical layer analysis, involved in the linear theory of internal modes, can be extended continuously into the early nonlinear regime. For the $m=1$ resistive mode, the dynamical analysis involves two small parameters: the magnetic Reynolds number S and the $m=1$ mode amplitude A, that measures the amount of nonlinearities in the system. The location of the instantaneous critical layer and the dominant dynamical equations inside it are evaluated self-consistently, as A increases and crosses some S-dependent thresholds. A special emphasis is put on the influence of the initial q-profile on the early nonlinear behavior. Predictions are given for a family of q-profiles, including the important low shear case, and shown to be consistent with recent experimental observations.