Nonlinear Evolution of Internal Ideal MHD Modes Near the Boundary of Marginal Stability

Abstract

A family of ideal MHD equilibria is considered introducing the concept of a driving parameter λ the increase of which beyond a certain threshold λ0 drives the plasma from a linearly stable to an unstable state. Using reductive perturbation theory, the nonlinear ideal MHD equations of motion are expanded in the neighbourhood of λ0 with respect to a small parameter ε. An appropriate scaling for the expansions is derived from the linear eigenmode problem. Integrability conditions for the reduced nonlinear equations yield nonlinear amplitude equations for the marginal mode. Nonlinearly, the instabilities are either oscillations about bifurcating equilibria, or they are explosive. In the latter case, the stability limit depends on the amplitude of the perturbation and is shifted into the linearly stable regime. Generally bifurcation of dynamically connected equilibria is observed at λ0