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.
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August 2008
Research Article|
August 20 2008
Energy of hydrodynamic and magnetohydrodynamic waves with point and continuous spectra
M. Hirota;
M. Hirota
a)
Faculty of Mathematics,
Kyushu University
, Fukuoka 812-8581, Japan
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Y. Fukumoto
Y. Fukumoto
Faculty of Mathematics,
Kyushu University
, Fukuoka 812-8581, Japan
Search for other works by this author on:
a)
Present address: Japan Atomic Energy Agency, Naka, Ibaraki 311-0193, Japan.
J. Math. Phys. 49, 083101 (2008)
Article history
Received:
December 09 2007
Accepted:
July 18 2008
Citation
M. Hirota, Y. Fukumoto; Energy of hydrodynamic and magnetohydrodynamic waves with point and continuous spectra. J. Math. Phys. 1 August 2008; 49 (8): 083101. https://doi.org/10.1063/1.2969275
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