Nonlinear growth of a line-tied g mode near marginal stability

A theoretical framework is 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, ϵ∼∣ξ∣∕Leq⪡1 and n−1∼k‖∕k⊥⪡1, with ξ denoting the plasma displacement, Leq the characteristic equilibrium length scale, and k‖ and k⊥ the dominant wavenumbers parallel and perpendicular to the equilibrium magnetic field, respectively. A particular nonlinear regime is described through the imposition of the ordering ϵ∼n−1. This regime corresponds to the nonlinear phase previously described by S. C. Cowley and M. Artun [Phys. Rep. 283, 185 (1997)], where the plasma is to lowest order incompressible and the lowest-order Lagrangian compression ∇0∙ξ is zero. In this regime, nonlinearity modifies the envelope equation of the linear global mode. The detonation regime, where the nonlinear growth of the mode dominates the linear response and becomes finite-time singular, is a narrower subset of the Cowley-Artun regime. However, at sufficient amplitude the validity of this regime breaks down and subsequently transitions to an intermediate nonlinear regime where the ordering ϵ∼n−1∕2 holds. In this regime, the lowest-order Lagrangian compression is nonzero [∇0∙ξ∼O(1)]. Direct magnetohydrodynamic simulations with both a finite difference code and NIMROD code indicate that the mode remains bounded in magnitude with a slightly reduced growth in the nonlinear phase relative to the linear growth rate. During the intermediate nonlinear phase, nonlinearity directly modifies the growth of linear local modes. The corresponding governing equations for the intermediate nonlinear phase are derived.