Metastable and explosive properties of ballooning modes in laboratory and space plasmas.

摘要

The nonlinear properties of ballooning instabilities are examined within the framework of ideal magnetohydrodynamics with diamagnetic corrections. Metastability and explosiveness, generic features of this widespread class of instability, are described by a combination of analytical and computational means. A five order multiple scale analysis is performed to derive a simplified partial differential equation describing the early nonlinear behavior of the ballooning mode envelope near marginal stability. The “detonation equation” is generically applicable to all pressure or gravity driven modes near marginal stability. Solution of the simplified detonation equation via numerical and approximate analytical methods reveals several novel nonlinear behaviors. Without diamagnetic corrections an unstable ballooning mode evolves directly to explosive instability, growing preferentially in one direction, broadening in the flux function coordinate to destabilize previously linearly stable regions of plasma, and narrowing in the other cross-field direction. With diamagnetic corrections included, a given equilibrium at marginal instability may be characterized by two equilibrium lumped parameters, one describing ideal magnetohydrodynamic stability effects, the other describing finite Larmor radius effects. A nonlinear accessibility condition is found in the two-dimensional equilibrium parameter space that determines whether linearly unstable ballooning modes evolve to explosive instability. Together, the accessibility condition and linear stability boundary divide the equilibrium parameter space into three regions. Region I is linearly stable; region II is linearly unstable and nonlinearly oscillatory; region III is linearly unstable and nonlinearly explosive, evolving similarly to the purely ideal nonlinear ballooning instability. Though oscillations occur in regions I and II, sufficiently large excitations result in explosive evolution. For equilibria evolving through marginal stability region III and explosive instability are inevitably encountered. The predictions of the simplified detonation analysis are compared to fully nonlinear ideal magnetohydrodynamic simulations with diamagnetic corrections. Simulation of the line-tied Rayleigh-Taylor-Parker instability confirms the qualitative nonlinear behaviors predicted by the simplified detonation equation analysis. In the application of the theory to laboratory plasmas, the qualitative nonlinear features of the detonation analysis are evident in Tokamak Fusion Test Reactor high β disruptions. Implications of the theory for magnetospheric substorms and solar prominences are given as well.