Nonlinear Growth of a Line-tied g-Mode Near Marginal Stability

A theoretical framework has been developed for the study of the nonlinear gravitational ($g$-) mode of a line-tied flux tube near marginal stability. The theory is based on an expansion using two small parameters, $\epsilon\sim |{\mbox{\boldmath $\xi$}}|/L_{\rm eq}\ll 1$, and $n^{-1}\sim k_\parallel/k_\perp\ll 1$, with ${\mbox{\boldmath $\xi$}}$ denoting the plasma displacement, $L_{\rm eq}$ the characteristic equilibrium scale, $k_\parallel$ and $k_\perp$ the dominant wavenumber of perturbation parallel and perpendicular to equilibrium magnetic field lines respectively. When $\epsilon\sim n^{-1}$, the Cowley-Artun regime is recovered where plasma is to the lowest order incompressible~[S.~C. Cowley and M. Artun, Phys. Rep., {\bf 283}, 185-211 (1997)]. The detonation regime where the nonlinear growth of the mode is finite-time singular is a narrower subset of the Cowley-Artun regime. However, the validity of this regime breaks down when $\epsilon\gg n^{-1}$. In the intermediate nonlinear phase when $\epsilon\sim n^{-1/2}$, the lowest order plasma compression [$\sim\nabla\cdot{\mbox{\boldmath $\xi$}}\sim{\cal O}(1)$] is nonzero. Direct MHD simulations with both a finite difference code and NIMROD indicate that the mode remains bounded in magnitude with a slightly reduced nonlinear growth [P. Zhu, A. Bhattacharjee, and K. Germaschewski, to appear in PRL (2006); P. Zhu, C.~C. Hegna, and C.~R. Sovinec, DPP2005]. During this phase, the coupled growth of the mode amplitude ${\mbox{\boldmath $\xi$}}$ and plasma compression may contribute to a nonlinear stabilization. The corresponding governing model equations for this intermediate nonlinear phase are derived.