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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).

##### MSC:
 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)
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