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Nonlinear dynamics in Langmuir circulations with O(2) symmetry

Published online by Cambridge University Press:  26 April 2006

Stephen M. Cox ,
Sidney Leibovich ,
Irene M. Moroz  and
Amit Tandon
Stephen M. Cox
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Upson Hall, Cornell University, Ithaca, NY 14853-7501, USA Present address: Department of Applied Mathematics, The University of Adelaide, G.P.O. Box 498, Adelaide 5001, South Australia.
Sidney Leibovich
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Upson Hall, Cornell University, Ithaca, NY 14853-7501, USA
Irene M. Moroz
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK
Amit Tandon
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Upson Hall, Cornell University, Ithaca, NY 14853-7501, USA
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Abstract

A direct comparison is made between the dynamics obtained by weakly nonlinear theory and full numerical simulations for Langmuir circulations in a density-stratified layer having finite depth and infinite horizontal extent. In one limit, the mathematical formulation employed is analogous to that of double-diffusion phenonema with the flux of one diffusing quantity fixed at the boundaries of the layer. These problems have multiple bifurcation points, but their amplitude equations have no intrinsic (nonlinear) degeneracies, in contrast to ‘standard’ double-diffusion problems. The symmetry of the physical problem implies invariance with respect to translations and reflections in the horizontal direction normal to the applied wind stress (so-called O(2) symmetry). A multiple bifurcation at a double-zero point serves as an organizing centre for dynamics over a wide range of parameter values. This double zero, or Takens–Bogdanov, bifurcation leads to doubly periodic motions manifested as modulated travelling waves. Other multiple bifurcation points appear as double-Hopf bifurcations. It is believed that this paper gives the first quantitative comparison of dynamics of double-diffusive type predicted by rationally derived amplitude equations and by full nonlinear partial differential equations. The implications for physically observable natural phenomena are discussed. This problem has been treated previously, but the earlier numerical treatment is in error, and is corrected here. When the Stokes drift gradient due to surface waves is not constant, the analogy with the common formulations of double-diffusion problems is compromised. Our bifurcation analyses are extended here to include the case of exponentially decaying Stokes drift gradient.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 241 , August 1992 , pp. 669 - 704
Copyright
© 1992 Cambridge University Press

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