Hamiltonian representation of equations of hydrodynamics and its use in describing wave movements in currents with shear

A procedure for the Hamiltonian formulation of the equations of motion is proposed for hydrodynamic-type media. It is shown that if the relations, besides the Euler equations, have a linearly functional dependence on the hydrodynamic velocity v, then the Klebsch representation establishing the relationship between v and the canonical variables may be found in a general form. The fundamental aspects of the formulation are illustrated for a uniform incompressible fluid with a shear flow. The normal variables of the problem are found for the special case of a semiinfinite medium and a bilinear flow profile. For the one-dimensional case, the nonlinear long waves are described using the Benjamin-Ono and Riemann equations. Estimates are presented for characteristic values of the parameters of the atmospheric boundary layer.