Energy of hydrodynamic and magnetohydrodynamic waves with point and continuous spectra

Energy of waves (or eigenmodes) in an ideal fluid and plasma is formulated in the noncanonical Hamiltonian context. By imposing the kinematical constraint on perturbations, the linearized Hamiltonian equation provides a formal definition of wave energy not only for eigenmodes corresponding to point spectra but also for singular ones corresponding to a continuous spectrum. The latter becomes dominant when mean fields have inhomogeneity originating from shear or gradient of the fields. The energy of each wave is represented by the eigenfrequency multiplied by the wave action, which is nothing but the action variable and, moreover, is associated with a derivative of a suitably defined dispersion relation. The sign of the action variable is crucial to the occurrence of Hopf bifurcation in Hamiltonian systems of finite degrees of freedom [M. G. Krein, Dokl. Akad. Nauk SSSR, Ser. A 73, 445 (1950)]. Krein's idea is extended to the case of coalescence between point and continuous spectra.