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

Zakharov, V. E.; Monin, A. S.; Piterbarg, L. I.

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.)

Keywords:

barotropic Rossby waves; Hamiltonian system; baroclinic Rossby waves; Laplace operator; Poisson bracket; Gardner bracket; integrable systems; normal canonical variables

Citations:

Zbl 0516.76125