## Hamiltonian description of baroclinic Rossby waves.(English. Russian original)Zbl 0642.76030

Sov. Phys., Dokl. 32, No. 8, 626-627 (1987); translation from Dokl. Akad. Nauk SSSR 295, 1061-1064 (1987).
Summary: A. Weinstein [Phys. Fluids 26, 388-390 (1983; Zbl 0516.76125)] has shown that the equation for the current function $$\psi$$ (x,y,t) of barotropic Rossby waves is a Hamiltonian system. We shall show similarly that the same is true also for the current function $$\psi$$ (x,y,z,t) of baroclinic Rossby waves which satisfy the equation $(1)\quad \frac{\partial \Omega}{\partial t}+\frac{\partial (\psi,\Omega)}{\partial (x,y)}+\beta \frac{\partial \psi}{\partial x}=0,\quad \Omega =\Delta \psi +\frac{\partial}{\partial z}\frac{f^ 2_ 0}{N^ 2}\frac{\partial \psi}{\partial z},$ where x and y are the horizontal Cartesian coordinates and z the vertical Cartesian coordinates (y is the meridional coordinate); $$f_ 0$$ is the local value of the Coriolis parameters; $$\beta =\partial f/\partial y$$ is the quasiconstant derivative of the Coriolis parameter with respect to the meridional coordinate; $$\Delta$$ is the Laplace operator in x, y; $$N=N(z)$$ is the Brunt-Väisälä frequency; and the quantity $$\Omega +f_ 0+\beta y$$ is the potential vorticity. We shall show that Eq. (1) can be reduced to the Hamiltonian form $\frac{\partial \Omega}{\partial t}=\{\Omega,{\mathcal H}\};\quad {\mathcal H}=\int [| \nabla \psi |^ 2+\frac{f^ 2_ 0}{N^ 2}(\frac{\partial \psi}{\partial z})^ 2]d\quad x dy dz,$ where $${\mathcal H}$$ is a Hamiltonian, and the curly brackets indicate the so-called Poisson bracket, which in the present case is defined for any functionals F[$$\Omega$$ ] and G[$$\Omega$$ ] by the formula $\{F,G\}=\{F,G\}_ 0+\{F,G\}_ 1=\int (\Omega +\beta y)\frac{\partial (\delta F/\delta \Omega,\delta G/\delta \Omega)}\quad {\partial (x,y)}dx dy dz;$
$\{F,G]_ 1=\beta \int \frac{\delta F}{\delta \Omega}\frac{\delta}{\delta x}\frac{\partial G}{\partial \Omega}dx\quad dy dz,$ where $$\{F,G\}_ 0$$ is the value of $$\{$$ F,G$$\}$$ for $$\beta =0$$, whereas the bracket $$\{F,G\}_ 1$$ with constants is the so-called Gardner bracket from the theory of integrable systems, for which the method of introducing normal canonical variables is known.

### MSC:

 76B65 Rossby waves (MSC2010) 70H15 Canonical and symplectic transformations for problems in Hamiltonian and Lagrangian mechanics 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)

Zbl 0516.76125