Action-angle representation of waves and instabilities in flowing plasmas

Action-angle representation of linear fluctuation in plasma is formally performed by taking the Lagrange-Hamilton theory as the basis and invoking the recent spectral technique [Hirota and Fukumoto, J. Math. Phys. 49 083101 (2008)], which is shown to be applicable equally to the magnetohydrodynamic (MHD) system and the Vlasov-Maxwell system. This formalism leads to a natural definition of the wave energy (as well as the wave action) for each eigenmode and continuum mode. Negative energy mode is generally responsible for onsets of various instabilities in flowing plasmas. It is shown that the canonical form of the linearized system, formulated by Hamilton’s principle of least action, facilitates the consideration of the action-angle representation, and the spectral technique becomes more sophisticated than the previous work that based on the noncanonical form.