Wave-action conservation law for eigenmodes and continuum modes

Invariance of wave-action for eigenmodes and continuum modes around quasistationary equilibrium state is investigated in a general framework that allows for the ideal magnetohydrodynamic system and the Vlasov-Maxwell system. By utilizing the averaging method for the variational principle, the wave-action of each mode is shown to be conserved if its frequency (spectrum) is sufficiently separated from other ones, whereas some conservative exchange of the wave-action may occur among the modes with close frequencies. This general conservation law is, as an example, demonstrated for a situation where the Landau damping (or growth) occurs due to a resonance between an eigenmode and a continuum mode. The damping (or growth) rate is closely related to the spectral linewidth (equal to phase mixing rate) of the continuum mode, which can be estimated by the invariance of wave-action without invoking the conventional analytic continuation of the dispersion relation.