Nonlinear dynamics of the tearing mode for any current gradient

Within the traditional frame of reduced MHD, a new systematic perturbation expansion provides the equation ruling the nonlinear growth and saturation of the tearing mode for any current gradient. The small parameter is the magnetic island width $w$. The evolution equation depends on $w$ through a term $w \ln(w_0/ w)$ where $w_0$ is a nonlinear scale length which was absent in previous works. The technique is applicable to the case of an external forcing. The solution for a static forcing is computed explicitly and it exhibits three regimes in the dynamics. A simpler version of Thyagaraja's technique yields an independent confirmation of the unforced case.