Resonant interaction of waves of continuous and discrete spectra in the simplest model of a stratified shear flow
- 1 Downloads
The equations of dynamics of eddy—wave disturbances of two-dimensional stratified flows in an ideal incompressible fluid that are written in a Hamiltonian form are used to study the resonant interaction of waves of discrete and continuous spectra. A gravity—shear wave generated at a jump of the density and vorticity of the undisturbed flow and a wave generated at a weak vorticity jump, which is similar to a wave of a continuous spectrum, participate in the interaction. The equations are written in terms of normal variables to obtain the system of evolution equations for the amplitudes of the interacting waves. The stability condition for eddy—wave disturbances is derived within the framework of the linear theory. It is shown that a cubic nonlinearity may lead to the stabilization of unstable disturbances if the coefficient of the nonlinear term is positive.
KeywordsOceanic Physic Discrete Spectrum Wave Disturbance Resonant Interaction Shear Wave
Unable to display preview. Download preview PDF.
- 1.P. G. Drazin and W. H. Reid, Hydrodynamic Stability (Cambridge Univ. Press, Cambridge, 1981).Google Scholar
- 3.A. D. D. Craik, Wave Interactions and Fluid Flows (Cambridge Univ. Press, Cambridge, 1985).Google Scholar
- 5.Yu. A. Stepanyants and A. L. Fabrikant, Propagation of Waves in Shear Flows (Nauka, Moscow, 1996) [in Russian].Google Scholar
- 6.I. A. Sazonov and I. G. Yakushkin, “Evolution of Disturbances in a Three-Layer Model of the Atmosphere with Shear Instability,” Izv. Akad. Nauk, Fiz. Atmos. Okeana [Izv., Atmos. Ocean. Phys. 35, 472–480 (1999) [Izv., Atmos. Ocean. Phys. 35, 427–434 (1999)].Google Scholar
- 8.L. A. Dikii, Hydrodynamic Stability and Atmospheric Dynamics (Gidrometeoizdat, Leningrad, 1976) [in Russian].Google Scholar
- 10.N. N. Romanova and I. G. Yakushkin, “Hamiltonian Description of Motions in an Ideal Stratified Fluid,” Dokl. Akad. Nauk 380, 630–634 (2001).Google Scholar
- 11.N. N. Romanova and I. G. Yakushkin, “Hamiltonian Description of Shear and Gravity Shear Waves in an Ideal Incompressible Fluid,” Izv. Akad. Nauk, Fiz. Atmos. Okeana 43, 579–590 (2007) [Izv., Atmos. Ocean. Phys. 43, 533–543 (2007)].Google Scholar
- 12.V. P. Goncharov and V. I. Pavlov, Problems of Hydrodynamics in Hamiltonian Description (Mosk. Gos. Univ., Moscow, 1993) [in Russian].Google Scholar
- 13.V. P. Goncharov, “Nonlinear Waves in Flows of Layers of Equal Vorticity,” Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 22, 468–477 (1986).Google Scholar
- 14.G. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1964; Mir, Moscow, 1977).Google Scholar
- 15.V. E. Zakharov, “Hamiltonian Formalism for Waves in Nonlinear Media with Dispersion,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 17, 431–453 (1974).Google Scholar