On stability and instability criteria for magnetohydrodynamics.

Area of Science:

• Physics
• Plasma Physics
• Magnetohydrodynamics

Background:

• Most three-dimensional magnetohydrodynamic (MHD) equilibria exhibit indefinite second variation of energy.
• The applicability of the Arnold criterion for stability analysis is thus limited to a restricted class of equilibria.
• Investigating conditions that lead to MHD instability is crucial for understanding plasma behavior.

Purpose of the Study:

• To determine conditions under which magnetohydrodynamic (MHD) equilibria become unstable.
• To present a sufficient condition for linear instability in the Eulerian representation.
• To apply this criterion to axisymmetric toroidal equilibria and rotating MHD.

Main Methods:

• Linearization of MHD equations about an equilibrium state.
• Analysis of the spectrum of the linearized MHD equations.
• Definition of an operator using local partial differential equations (PDEs) to establish an instability criterion.

Main Results:

• A sufficient condition for linear instability is derived, relating the maximal real part of the MHD spectrum to the growth rate of a PDE-defined operator.
• Application to axisymmetric toroidal equilibria yields stronger instability conditions than previously known.
• The study confirms instability for a significant class of rotating magnetohydrodynamic (MHD) systems.

Conclusions:

• The presented instability criterion provides a robust method for identifying unstable MHD equilibria.
• The findings are particularly relevant for understanding the stability of rotating toroidal plasmas.
• Further research can explore the implications of these stronger instability conditions in various plasma confinement scenarios.