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Canonical variables for Rossby and drift waves in plasma. (English. Russian original) Zbl 0642.76130
Sov. Phys., Dokl. 32, No. 7, 560-561 (1987); translation from Dokl. Akad. Nauk SSSR 295, 86-90 (1987).
The simplest model describing Rossby waves in the atmosphere of a rotating planet is the equation \[ (1)\quad (\partial \Omega /\partial t)+\alpha (\frac{\partial \psi}{\partial x})=\{\psi,\Omega \}. \] Here \(\psi\) is the stream function, \(\Omega =\Delta \psi -v\) \(2\psi\) is the vortex density, and \(\{\psi,\Omega \}=\psi_ x\Omega_ y-\Omega_ x\psi_ y\). Equation (1) is also used to describe drift waves in a plasma in a nonuniform magnetic field. The stream function used here differs in sign from that used in geophysical hydrodynamics. To apply the results derived below to the function \(-\psi,\) one need only formally change the sign of the constant \(\alpha\). As proved earlier, Eq. (1) is a Hamiltonian system, which can be written in the form \(\partial \Omega /\partial t=[\Omega,H]\). Here \(H=1/2\int ((\nabla \psi)\) \(2+\nu\) \(2\psi\) 2)dr is the energy of the liquid which plays the role of the Hamiltonian, and the symbol [F,G] denotes the Poisson bracket defined for the functionals of \(\Omega\) (x,y) according to the formula (2) \([F,G]=\int (\Omega -\alpha y)\{\delta F/\delta \Omega\), \(\delta\) G/\(\delta\Omega\}\) dr. In this paper we discuss the problem of introducing canonical variables for the system (1), i.e., the problem of diagonalizing Poisson’s bracket (2).

76X05 Ionized gas flow in electromagnetic fields; plasmic flow
70H15 Canonical and symplectic transformations for problems in Hamiltonian and Lagrangian mechanics
82D10 Statistical mechanical studies of plasmas
76B65 Rossby waves (MSC2010)
[1] Zakharov, V. E.; Kuznetsov, E. A.: Sov. sci. Rev. C. 4, 62 (1984)
[2] Weinstein, A.: Phys. fluids. 26, 388 (1983)
[3] Hasegawa, A.; Mima, K.: Phys. fluids. 21, 87 (1978)
[4] Arnold, V. J.: Graduate texts in mathematics. Mathematical methods of classical mechanics 60 (1978)
[5] Weinstein, A.: J. diff. Geom.. 18, 523 (1983)
[6] Zakharov, V. E.: Pis’ma mat. Teor. fiz.. No. No. 2, 86 (1968)
[7] Voronovich, A. G.: Izv. akad. Nauk. SSSR fiz. Atmosf. okeana. No. No. 1, 82 (1979)
[8] Petviashvili, V. I.: Pis’ma zh. Eksp. teor. Fiz.. 32, 632 (1984)
[9] . Fizika okeana 2 (1978)
[10] Holm, D. D.: Phys. fluids. 29, 7 (1986)
[11] Monin, A. S.; Piterbarg, L. I.: Dokl. akad. Nauk SSSR gidromekh.. 295, 816 (1987)
[12] Zakhavov, V. E.; Monin, A. S.; Piterbarg, L. I.: Dokl. akad. Nauk SSSR gidromekh.. 295, 1061 (1987)
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