Intermediate nonlinear regimes of line-tied g mode and ballooning instability

A theoretical framework has been developed to describe the nonlinear regimes of line-tied g modes in slab geometry and ballooning instabilities in toroidal configurations. Recent experimental observation and numerical simulations demonstrate a persistence of ballooning-like filamentary structures well into the nonlinear stage of edge localized mode (ELM) activity in H-mode plasmas. Our theory is based on an expansion using two small scale lengths, the mode displacement across magnetic flux surfaces and the mode width in the most rapidly varying direction, both normalized by the equilibrium scale length. When the mode displacement across the magnetic flux surface is much less than the mode width in the most rapidly varying direction, the mode is in the linear regime. When the mode displacement grows to the order of the mode width in the rapidly varying direction, the plasma remains incompressible to lowest order, and the Cowley–Artun regime is obtained. The detonation regime, where the nonlinear growth of the mode could be finite-time singular, is accessible when the system is sufficiently close to marginal stability. At higher levels of nonlinearity, the system evolves to the intermediate nonlinear regime, when the mode displacement across the magnetic flux surface becomes comparable to the mode width in the same direction. During this phase, the nonlinear growth of the mode in the parallel and perpendicular directions are coupled, and sound wave physics contributes to nonlinear stability. The governing equations for the line-tied g mode and the ballooning instability in the intermediate nonlinear regime have been derived. A remarkable feature of the nonlinear equations is that solutions of the associated local linear mode equations continue to be valid solutions into the intermediate nonlinear regime in a Lagrangian reference frame. This property has been confirmed in the full ideal MHD simulations of both the line-tied g mode in a shearless slab and the ballooning instability in a tokamak, and may help explain the growth and persistence of the filamentary structures observed in ELM experiments well into the nonlinear phase.