Advertisement

Secondary Bifurcation

  • Huijun Yang
Part of the Applied Mathematical Sciences book series (AMS, volume 85)

Abstract

In Chapter 5 we analytically investigated the bifurcation properties of the evolution of the wave packet in the symmetric basic flow due to symmetric and asymmetric topographies, using the Rossby wave packet approximation, δ-surface approximation of the earth’s surface, and the WKB method. The results show that the topological structure of the evolution of a Rossby wave packet varies with basic flows and the topography. The subcritical and supercritical bifurcations, as well as the reverse subcritical and supercritical bifurcations, were found analytically. The effect of a zonal basic flow on the bifurcation differs from that of a meridional basic flow. The mixed scale equilibrium states were only found associated with the asymmetric topography.

Keywords

Wave Packet Bifurcation Diagram Basic Flow Scale State Bifurcation Parameter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Andronov, A.A., Vitt, E.A., and Khaiken, S.E. (1966). The Theory of Oscillators. Pergamon Press, Oxford.Google Scholar
  2. Elberry, R. L. (1968). A high-rotating general circulation model experiment with cyclic time changes. Atmospheric Science Paper No. 134, Colorado State University, Fort Collins.Google Scholar
  3. Erneux, T., and Reiss, E. (1983). Splitting of steady multiple eigenvalues may lead to periodic cascading bifurcation. SIAM J. Appl. Math. 43, 613–624.CrossRefGoogle Scholar
  4. Fultz, D., Long, R.R., Owens, G.V., Bohan, W., Kaylor, R., and Weil, J. (1959). Studies of thermal convection in a rotating cylinder with some implications for large-scale atmospheric motions. Meteorol. Monogr. No. 21.Google Scholar
  5. Gollub, J.P., and Benson, S.V. (1980). Many routes to turbulent convection. J. Fluid Mech. 100, 449–470.CrossRefGoogle Scholar
  6. Gruber, A. (1985). The wave number-frequency spectra of the 200 mb wind field in the tropics. J. Atmos. Sci. 32, 1615–1625.CrossRefGoogle Scholar
  7. Guckenheimer, J. and Holmes, P. (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, New York.Google Scholar
  8. Hart, J.E. (1984). A laboratory study of baroclinic chaos on the f-plane. Tellus 37A, 286–296.Google Scholar
  9. Hide, R. (1958). An experimental study of thermal convection in a rotating liquid. Phil. Trans. Roy. Soc. London 250A, 441–478.CrossRefGoogle Scholar
  10. Krishnamurti, R. (1970). On the transition to turbulent convection. J. Fluid Mech. 42, 295–320.CrossRefGoogle Scholar
  11. Krishnamurti, R. (1973). Some further studies on the transition to turbulent convection. J. Fluid Mech. 60, 285–303.CrossRefGoogle Scholar
  12. Landau, L.D. (1944). Turbulence. Dokl. Akad. Nauk. USSR 44, 339–340.Google Scholar
  13. Landau, L.D., and Lifshitz, F.M. (1959). Fluid Mechanics. Pergamon Press, London.Google Scholar
  14. Landau, L.D., and Lifschitz, F.M. (1987). Fluid Mechanics, 2nd ed. Pergamon, Oxford.Google Scholar
  15. Lorenz, E.N. (1963). Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141.CrossRefGoogle Scholar
  16. McGuirk, J.P., and Reiter, E.R. (1976). A vacillation in atmospheric energy parameters. J. Atmos. Sci. 33, 2079–2093.CrossRefGoogle Scholar
  17. Nicolis, G., and Prigogine, I. (1977). Self-Organization in Nonequilibrium Systems. John Wiley, New York.Google Scholar
  18. Pedlosky, J., and Frenzen, C. (1980). Chaotic and periodic behavior of finite amplitude baroclinic waves. J. Atmos. Sci. 37, 1177–1196.CrossRefGoogle Scholar
  19. Pfeffer, R.L., and Chiang, Y. (1967). Two kinds of vacillation in rotating laboratory experiments. Mon. Wea. Rev. 95, 75–82.CrossRefGoogle Scholar
  20. Reiss, E. (1983). Cascading bifurcations. SIAM J. Appl. Math. 43, 57–65.CrossRefGoogle Scholar
  21. Sparrow, C. (1982). The Lorenz Equations: Bifurcation, Chaos, and Strange Attractors. Springer-Verlag, New York.Google Scholar
  22. Webster, P.J., and Keller, J.L. (1975). Atmospheric variations: Vacillations and index cycles. J. Atmos. Sci. 32, 1283–1300.CrossRefGoogle Scholar
  23. Weng, H. Barcilon, A., and Magnan, J. (1986). Transitions between baro-clinic flow regimes. J. Atmos. Sci. 43 1760–1777.Google Scholar
  24. Yang, H. (1988a). Global behavior of the evolution of a Rossby wave packet in barotropic flows on the earth’s 6-surface. J. Atmos. Sci. 45, 113–126.CrossRefGoogle Scholar
  25. Yang, H. (1988b). Bifurcation properties of the evolution of a Rossby wave packet in barotropic flows on the earth’s 6-surface. J. Atmos. Sci. 45, 3667–3683.CrossRefGoogle Scholar
  26. Yang, H. (1988c). Secondary bifurcation of the evolution of a Rossby wave packet in barotropic flows on the earth’s 6-surface. J. Atmos. Sci. 45, 3684–3699.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Huijun Yang
    • 1
  1. 1.Geophysical Fluid Dynamics InstituteThe Florida State UniversityTallahasseeUSA

Personalised recommendations