Nonlinear growth of the tearing mode

Abstract

The resistive tearing mode is analyzed in the nonlinear regime; nonlinearity is important principally in the singular layer around $\text{k·B\hspace{0.17em}=\hspace{0.17em}0}$ . In the case where the resistive skin time ${\tau }_s$ is much longer than the hydromagnetic time ${\mathrm{\tau \hspace{0.17em}}}_H$ , exponential growth of the field perturbation is replaced by algebraic growth like $t^2$ at an amplitude of order ${\mathrm{(\tau \hspace{0.17em}}}_H{\mathrm{\hspace{0.17em}/\hspace{0.17em}\tau \hspace{0.17em}}}_S{\mathrm{\hspace{0.17em})}}^{\text{4/5}}$ . Application of the theory to the unstable tearing modes of a tokamak with a shrinking current channel yields good agreement with the observed amplitudes of the $\text{m\hspace{0.17em}≥\hspace{0.17em}2}$ oscillations. The analysis excludes the very long wavelength mode, and $\text{m\hspace{0.17em}=\hspace{0.17em}1}$ in the tokamak, for which the “constant‐Ψ” approximation is invalid.