Nonlinear Tearing Modes in Current-Vortex Sheets
The linear and nonlinear development of instabilities and Alfvén resonances in a plane current-vortex sheet is presented here for sheared equilibrium profiles $\boldsymbol{B_{y0}} = \tanh(z)\boldsymbol{\hat{y}}$ and $\boldsymbol{V_{y0}} = M_0\tanh(z/r)\boldsymbol{\hat{y}}$. We extend Rutherford's nonlinear model for constant-psi magnetic islands to account for a sheared equilibrium flow and determine the flow's impact on the magnetic island's nonlinear evolution for $M_0<1$. In this regime, the flow introduces two nonlinear contributions: a polarization current term and a modification of the saturation term derived by Militello and Porcelli (2004). We find that the polarization current generated by the equilibrium flow consistently reduces the nonlinear growth rate of the tearing mode. The flow-induced modification of the saturation term, however, depends on the magnetic-to-velocity shear width ratio, $r$. For $r=1$, the saturation term is unchanged from the case without flow. For $r>1$, it increases in magnitude with $M_0$, strengthening its stabilizing effect and further reducing the island growth rate. In contrast, for $r<1$, the saturation term decreases in magnitude with increasing $M_0$ and can become positive, leading to a transition from a stabilizing to a destabilizing contribution. Nevertheless, throughout the parameter range considered, the combined effect of the polarization current and modified saturation term is to slow the growth of the magnetic island. Finally, we find that, in the presence of Alfvén resonances, the magnetic island's growth in the nonlinear regime is no longer adequately characterized by constant-psi, and the dynamics of such islands are not captured by the model.