Action-angle variables for the continuous spectrum of ideal magnetohydrodynamics

Hirota, Makoto; Fukumoto, Yasuhide*

Action-angle variables corresponding to singular (or improper) eigenmodes are rigorously formulated for the Alfvn and slow (or cusp) continuous spectra of ideal magnetohydrodynamics. The perturbation energy is then transformed into the normal form, namely, the eigenfrequency multiplied by the action variable. It is shown that the Laplace transform approach expedites this action-angle formulation more efficiently than the existing ones devoted to other kinds of continuous spectra. The presence of flow that is either non-parallel to the magnetic field or supersonic at some places brings about singular eigenmodes with negative energy. The Alfvn and slow singular eigenmodes are neutrally stable even in the presence of any external potential fields, but may cause instability when coupled with another singular or non-singular eigenmode with the opposite sign of energy.