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Hamiltonian formalism for Rossby waves. (English) Zbl 0913.76019
Zakharov, V. E. (ed.), Nonlinear waves and weak turbulence. Transl. from the Russian. Providence, RI: AMS, American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 182 (36), 131-166 (1998).
This paper is concerned with the Hamiltonian structure of the Obukhov-Charney-Hasegawa-Mima equation $${\partial(\Delta\psi- \psi)\over \partial t}+\beta {\partial\psi\over\partial x}= J(\Delta\psi, \psi)$$. It is well-known that the Hamiltonian form of this equation is based on the noncanonical Poisson bracket $\{F, G\}= \iint \Omega(x,y)J\left({\partial F\over\partial\Omega(x, y)},{\delta G\over\delta \Omega(x,y)}\right)dx dy,$ where $$\Omega(x,y)= \Delta\psi- \psi+\beta y$$. The crucial point of the present paper is a new method used to transform the above bracket to the standard canonical form. This method is valid for vortices $$\Omega(x,y)$$ containing closed contour lines as well as an open one. Thus, the range of applicability of the canonical variables is extended to solitons and sets of solitons.
For the entire collection see [Zbl 0879.00029].

##### MSC:
 76B65 Rossby waves (MSC2010) 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 35Q35 PDEs in connection with fluid mechanics