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.


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