Explosive instabilities and detonation in magnetohydrodynamics

Abstract

Many plasma systems exhibit large-scale explosive events. Solar flares, magnetic substorms and tokamak disruptions are examples of large-scale explosive events. Since the rate at which the stability boundary is crossed is usually slow, systems rarely achieve a large linear growth rate. Thus explosive events almost always require nonlinear destabilization to achieve the fast time-scales that are observed. A new mechanism for explosive behavior is demonstrated in a nonlinear MHD model of the line tied Rayleigh-Taylor instability. In this mechanism the system crosses the instability threshold in a small region of space. The nonlinearity is destabilizing and broadening causing the linear instability to develop fingers and broaden into the linearly stable region. A front forms separating the disturbed and undisturbed regions. Because the nonlinearity is destabilizing the linearly stable region is, in fact, meta stable. The energy in the fingers is large enough to destabilize the meta-stable region — a process we call detonation. In the simple system analyzed a finite time singularity occurs where the displacement becomes singular like ( t 0 − t) −2.1. the energy like ( t 0 − t) −6.4 and the destabilized region width like ( t 0 − t) −0.4. The analysis of this problem is simplified by expanding around the marginal stability point. The behavior of the displacement along the field line is determined at low order. At higher order the behavior across the field is determined to be the solution of a two-dimensional nonlinear equation. The coefficients in this equation depend on field line averages of the behavior along the field. The same nonlinear equation can be shown to govern the nonlinear behavior near marginal stability of small perpendicular wavelength instabilities in more complicated geometries where line tying or ballooning are present. Thus, the mechanism can be considered generic. Analytic and numerical results will be shown. The relevance of this mechanism to solar flares, magnetic substorms and tokamak disruptions will be discussed briefly.