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, ∼ |ξ|/Leq 1, and n−1 ∼ k‖/k⊥ 1, with ξ denoting the plasma displacement, Leq the characteristic equilibrium scale, k‖ and k⊥ the dominant wavenumber of perturbation parallel and perpendicular to equilibrium magnetic field lines respectively. When ∼ n−1, the CowleyArtun regime is recovered where plasma is to the lowest order incompressible [S. C. Cowley and M. Artun, Phys. Rep., 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 n−1. In the intermediate nonlinear phase when ∼ n−1/2, the lowest order plasma compression [∼ ∇ · ξ ∼ 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 ξ and plasma compression may contribute to a nonlinear stabilization. The corresponding governing model equations for this intermediate nonlinear phase are derived.