Izvestiya, Atmospheric and Oceanic Physics

, Volume 44, Issue 1, pp 53–63 | Cite as

Resonant interaction of waves of continuous and discrete spectra in the simplest model of a stratified shear flow

Article
  • 1 Downloads

Abstract

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.

Keywords

Oceanic Physic Discrete Spectrum Wave Disturbance Resonant Interaction Shear Wave 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    P. G. Drazin and W. H. Reid, Hydrodynamic Stability (Cambridge Univ. Press, Cambridge, 1981).Google Scholar
  2. 2.
    R. A. Cairns, “The Role of Negative Energy Waves in Some Instabilities of Parallel Flow,” J. Fluid Mech. 92, 1–14 (1979).CrossRefGoogle Scholar
  3. 3.
    A. D. D. Craik, Wave Interactions and Fluid Flows (Cambridge Univ. Press, Cambridge, 1985).Google Scholar
  4. 4.
    A. D. D. Craik and J. A. Adam, “Explosive Resonant Wave Interaction in a Three-Layer Fluid Flow,” J. Fluid Mech. 92, 15–33 (1979).CrossRefGoogle Scholar
  5. 5.
    Yu. A. Stepanyants and A. L. Fabrikant, Propagation of Waves in Shear Flows (Nauka, Moscow, 1996) [in Russian].Google Scholar
  6. 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
  7. 7.
    K. M. Case, “Stability of Inviscid Plane Couette Flow,” Phys. Fluids 3(2), 143–148 (1960).CrossRefGoogle Scholar
  8. 8.
    L. A. Dikii, Hydrodynamic Stability and Atmospheric Dynamics (Gidrometeoizdat, Leningrad, 1976) [in Russian].Google Scholar
  9. 9.
    M. Kelbert and I. Sazonov, Pulses and Other Wave Processes in Fluids (Cluwer, Dordrecht, 1996).CrossRefGoogle Scholar
  10. 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. 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. 12.
    V. P. Goncharov and V. I. Pavlov, Problems of Hydrodynamics in Hamiltonian Description (Mosk. Gos. Univ., Moscow, 1993) [in Russian].Google Scholar
  13. 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. 14.
    G. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1964; Mir, Moscow, 1977).Google Scholar
  15. 15.
    V. E. Zakharov, “Hamiltonian Formalism for Waves in Nonlinear Media with Dispersion,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 17, 431–453 (1974).Google Scholar
  16. 16.
    V. E. Zakharov and E. A. Kuznetsov, “Hamiltonian Formalism for Nonlinear Waves,” Usp. Fiz. Nauk 167, 1137–1167 (1997).CrossRefGoogle Scholar
  17. 17.
    N. N. Romanova, “Hamiltonian Description of Wave Dynamics in Nonequilibrium Media,” Nonlin. Proc. Geophys. 1, 234–240 (1994).CrossRefGoogle Scholar
  18. 18.
    N. N. Romanova, “Hamiltonian Approach to the Derivation of Evolution Equations for Wave Trains in Weakly Unstable Media,” Nonlin. Proc. Geophys. 5, 241–253 (1998).CrossRefGoogle Scholar
  19. 19.
    N. N. Romanova and S. Yu. Annenkov, “Three-Wave Resonant Interactions in Unstable Media,” J. Fluid Mech. 539, 57–91 (2005).CrossRefGoogle Scholar

Copyright information

© MAIK Nauka 2008

Authors and Affiliations

  1. 1.Oboukhov Institute of Atmospheric PhysicsRussian Academy of SciencesMoscowRussia

Personalised recommendations