Negative energy waves and MHD stability of rotating plasmas

Eigenmode analysis of ideal magnetohydrodynamic (MHD) systems with flows is performed. It is shown that the energy of stable oscillatory modes (waves) can be both positive and negative. Negative energy waves always correspond to non-symmetric modes which are nonuniform along the direction of the flow. Coupling of negative and positive energy waves is shown to be a universal mechanism of non-symmetric MHD instabilities in flowing media. To study the stability of non-symmetric modes, a new variational approach is developed based on Lyapunov theory. This approach provides a sufficient and (under some assumptions) necessary stability condition. Specific examples are given to illustrate the developed approach.