Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-05-09T02:41:58.642Z Has data issue: false hasContentIssue false
  • English
  • Français

Universality and strange attractors in internal-wave dynamics

Published online by Cambridge University Press:  20 April 2006

Henry D. I. Abarbanel
Henry D. I. Abarbanel
Affiliation:
Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720 Present address: University of California, San Diego, MPL, A-005, La Jolla, CA 92093.
Get access Rights & Permissions [Opens in a new window]

Abstract

We argue that the universality and statistical nature of the deep-ocean internal gravity-wave spectrum results from a strange attractor in the driven, dissipative internal-wave field. To explore this we construct a model which injects energy into the oceanic surface at a constant rate. A two-dimensional version of the model is explored analytically and numerically. For the numerical work we restrict our considerations to a few of the longest-wavelength modes. This few-mode system exhibits bifurcation into limit cycles, period doubling of the limit cycles, and chaotic, non-periodic behaviour associated with a strange attractor. In an appendix we present some discussion of the three-dimensional version of the model.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 135 , October 1983 , pp. 407 - 434
Copyright
© 1983 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arnol'D, V. I. 1978 Mathematical Methods of Classical Mechanics, $ 16. Springer.
Bretherton, F. P. 1969 Waves and turbulence in stably stratified fluids. Radio Sci. 4, fR 1279.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability, $ 103. Oxford University Press.
Eckmann, J.-P. 1981 Roads to turbulence in dissipative dynamical systems Rev. Mod. Phys. 53, 643.Google Scholar
Eriksen, C. C. 1978 Measurements and models of fine structure, internal gravity waves, and wave breaking in the deep ocean J. Geophys. Res. 83, 2989.Google Scholar
Fenstermacher, P. R., Swinney, H. L. & Gollub, J. P. 1979 Dynamic instabilities and the transition to chaotic Taylor vortex flow J. Fluid Mech. 94, 103.Google Scholar
Garrett, C. C. & Munk, W. 1975 Space time scales of internal waves: a progress report. J. Geophys. Res. 80, 291.Google Scholar
Garrett, C. C. & Munk, W. 1979 Internal waves in the ocean Ann. Rev. Fluid Mech. 11, 339.Google Scholar
Holloway, G. 1980 Oceanic internal waves are not weak waves J. Phys. Oceanogr. 10, 906.Google Scholar
Iooss, G. & Joseph, D. 1981 Elementary Stability and Bifurcation Theory. Springer.
Kuramoto, Y. & Koga, S. 1982 Anomalous period doubling bifurcations leading to chemical turbulence Phys. Lett. 92A, 1.Google Scholar
Lanford, O. E. 1980 Turbulence and strange attractors. In Hydrodynamic Instabilities and the Transition to Turbulence (ed. H. L. Swinney & J. P. Gollub). Springer.
Libchaber, A. & Maurer, J. 1981 A Raleigh-Bénard experiment: helium in a small box. Ecole Normale Preprint.
Lorenz, E. N. 1963 Deterministic nonperiodic flow J. Atmos. Sci. 20, 130.Google Scholar
Lyubimov, D. V. & Zaks, M. A. 1983 On two mechanisms of the transition to chaos in finite dimensional models of convection. Preprint of the Perm State University. USSR.
Ott, E. 1981 Strange attractors and chaotic motions of dynamical systems Rev. Mod. Phys. 53, 655.Google Scholar
Pedlosky, J. 1979 Geophysical Fluid Dynamics. Springer.
Phillips, O. M. 1977 The Dynamics of the Upper Ocean, 2nd edn. Cambridge University Press.
Robbins, K. A. 1979 Periodic solutions and bifurcation structure at high R in the Lorenz model. SIAM J. Appl. Maths 36, 457.Google Scholar
Ruelle, D. & Takens, F. 1971 On the nature of turbulence Commun. Math. Phys. 20, 167.Google Scholar
Thorpe, S. A. 1975 The excitation, dissipation, and interaction of internal waves in the deep ocean J. Geophys. Res. 80, 328.Google Scholar
Watson, K., West, B. J. & Cohen, B. J. 1976 Coupling of surface and internal gravity waves: a mode coupling model. J. Fluid Mech. 77, 185.Google Scholar
Wunsch, C. 1975 Deep ocean internal waves: what do we really know? J. Geophys. Res. 80, 339.Google Scholar