Nonlinearly unstable interchange modes in transverse magnetic field

The nonlinear stability of the ideal magnetohydrodynamic interchange mode for plasma immersed in a constant transverse magnetic field near marginal conditions is studied. We use reduced equations for a strong axial field to find an analytic solution for the nonlinear behaviour as a function of the deviation from marginality. The study is motivated in order to assess B-field tolerances in stellarator coil design. A systematic perturbation analysis in the smallness parameter, $|b_2/B_c|^{1/2}$, is carried out, where $B_c$ is the critical transverse magnetic field for the marginally stable ideal mode, and $b_2$ is the deviation from $B_c$. The lowest order expansion yields an eigenvalue equation for the magnitude of the critical field required for marginal stability, $B_c$. The calculation is carried out to third order, including nonlinear terms, and a time-evolution equation for the amplitude is found. In the short wavelength limit we find that the system is nonlinearly unstable for large enough perturbations even if $b_2/B_c>0$ (linearly stable). This result is similar to that of Cowley and Artun\footnote{S. C. Cowley and M. Artun, Physics Reports {\bf 283}, 185 (1997).} for the marginally stable line-tied $g$-mode. If the system is driven nonlinearly unstable, the resulting growt