Nonlinear Stability of a Class of Magnetostatic Equilibria With an Application to Solar Prominences

Abstract

We consider a particular class of three-dimensional magnetostatic equilibria in which the plasma is submitted to a vertical gravitational field and the gradient of the total (thermal+magnetic) pressure vanishes. We show analytically that an equilibrium in that class makes the energy an absolute minimum in the set of all the configurations accessible from it by an arbitrary finite deformation constrained by ideal MHD and imposed to vanish on a rigid conducting wall (line-tying condition). Along with energy conservation, this implies the nonlinear ideal stability of that equilibrium in the following sense. Suppose that a perturbation of energy w(0) is applied at time t = 0 and thus evolves by obeying the nonlinear MHD equations. Then some measure of the sizes of the plasma velocity and the deformation of the structure can be made to stay at any t 0 below an arbitrarily prescribed value by choosing w(0) small enough. Nonlinear stability also holds true for a configuration obtained by superposing an equilibrium of the previous type and a nonmagnetic equilibrium which is also an energy minimizer-for instance an equilibrium with uniform specific entropy, which is shown to have that property. Our result applies to a subset of a family of equilibria, computed by B. C. Low, which includes in particular the standard Kippenhahn-Schlüter model describing the magnetic support of solar corona prominences.

On the Support of Solar Prominence Material by the Dips of a Coronal Flux Tube

“…This model describes the local structure of the prominence and, as a result, has no transition to a hot corona and is infinite in extent in the vertical direction. This model has been shown to be both linearly and nonlinearly stable to ideal Lagrangian magnetohyrodynamic (MHD) perturbations (Kippenhahn & Schlüter 1957;Anzer 1969;Zweibel 1982;Galindo-Trejo & Schindler 1984;Aly 2011) .…”