Nonlinear Tearing Modes in Current-Vortex Sheets

The linear and nonlinear development of instabilities and Alfv\'en 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 size. We find that the polarization current induced by the equilibrium flow slows the nonlinear growth of the tearing mode. The saturation of the magnetic island is hastened somewhat for $r > 1$, slowed for $r < 1$, and unmodified for $r=1$. Finally, we find that, in the presence of Alfv\'en 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.