Action-angle variables for the continuous spectrum of ideal magnetohydrodynamics

Action-angle variables corresponding to singular (or improper) eigenmodes are rigorously formulated for the Alfv́n 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 nonparallel to the magnetic field or supersonic at some places brings about singular eigenmodes with negative energy. The Alfv́n 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 nonsingular eigenmode with the opposite sign of energy.