Zur Stabilität eines Plasmas

Abstract

Die Stabilität von hydrodynamischen Gleichgewichtskonfigurationen wird mit Hilfe der Methode der kleinen Störungen untersucht. Es wird gezeigt, daß das Stabilitätsverhalten durch eine Differentialgleichung 2. Ordnung in der Zeit bestimmt ist, wenn man die Viskosität, den elektrischen Widerstand und die thermische Leitfähigkeit vernachlässigt. Da die Differentialgleichung selbstadjungiert ist, können einige allgemeine Theoreme abgeleitet werden, welche für alle Gleichgewichtskonfigurationen gelten. Man kann zeigen, daß der zeitliche Anstieg von Störungen unter gewissen Bedingungen beschränkt ist. Weiterhin können einige hinreichende Bedingungen für die Stabilität angegeben werden. Für den Spezialfall, daß innerhalb eines Plasmazylinders das Magnetfeld verschwindet, werden die Differentialgleichungen explizit gelöst und Bedingungen für die Stabilität abgeleitet. Schließlich wird auch gezeigt, daß die Differentialgleichung auch selbstadjungiert ist, wenn der Druck nicht isotrop ist. It is shown that the stability of hydromagnetic equilibrium as studied by the method of small perturbations is controlled by one differential equation of second order in time, if one neglects viscosity, electrical resistivity and thermal conductivity. Since the differential equation is self-adjoint some general theorems can be derived which hold for all configurations of hydromagnetic equilibrium. It is possible to show that the rates of growing are limited under certain conditions. Also some sufficient conditions of stability can be given. For a plasma cylinder, inside of which the magnetic field vanishes, the differential equations are solved explicitly and conditions for stability are given. Finally it is shown that the differential equation is also self-adjoint if the pressure is not isotropic.

Hamiltonian magnetohydrodynamics: Lagrangian, Eulerian, and dynamically accessible stability—Theory

“…[4][5][6] will be treated. This principle is an infinite-dimensional version of Lagrange's necessary and sufficient stability theorem of mechanics (see, e.g., [26]), which is applicable to natural Hamiltonians of the separable form, kinetic plus potential.…”

“…The former is the root of the necessary and sufficient hydromagnetic energy principle of Refs. [4][5][6] for static equilibria, while the latter is the root of various Eulerian sufficient conditions for stability (see e.g, Ref. [7]).…”

Global ideal magnetohydrodynamic stability analysis for the configurational space of Wendelstein 7–X

“…The CAS3D stability code 2 was designed to computationally treat the global ideal MHD stability problem in its variational form 7,8 for general 3-D toroidal plasmas, especially for truly 3-D equilibria such as of the Helias 9,6 type, which are not suitable for either the stellarator expansion approach 10 or the averaging methods. 11 On the one hand this code treats the full 3-D problem; on the other it exploits general structural features of the perturbation functions; for reference the most important of those which were employed for code development are described here in short.…”

MHD stability of a filament immersed in a solar bipolar magnetic region

“…One of the most convenient procedures for this is the energy principle of ideal MHD (Bernstein et a/., 1958;Hain et al, 1957). According to this method, we will have stability if the change in potential energy 6U( due to a small perturbation {, is positive for all possible perturbations which satisfy the required boundary conditions.…”

“…We apply our two-dimensional formalism, based on the energy principle of Bernstein et al (1958) and Hain et al (1957), to different variants of this model, and we investigate under which circumstances such a line-tying magnetic configuration is really capable of stably supporting the mass distribution of the quiescent filament.…”