TY - JOUR

T1 - Energy of hydrodynamic and magnetohydrodynamic waves with point and continuous spectra

AU - Hirota, M.

AU - Fukumoto, Y.

N1 - Funding Information:
The authors are grateful to Professor Zensho Yoshida for useful discussions and suggestions and to Professor Eliezer Hameiri for stimulating our interest in this problem. M. H. was supported by Research Fellowships of the Japan Society for the Promotion of Science (JSPS) for Young Scientists and is supported by the 21st Century COE program “Development of Dynamic Mathematics with High Functionality” at the Faculty of Mathematics, Kyushu University. Y. F. was supported in part by a grant-in-aid for scientific research from the Japan Society for the Promotion of Science.

PY - 2008

Y1 - 2008

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=50849137489&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=50849137489&partnerID=8YFLogxK

U2 - 10.1063/1.2969275

DO - 10.1063/1.2969275

M3 - Article

AN - SCOPUS:50849137489

VL - 49

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 8

M1 - 083101

ER -