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Cyclic recurrence in nonlinear unidirectional ocean waves

Published online by Cambridge University Press:  21 April 2006

Peter J. Bryant
Peter J. Bryant
Affiliation:
Mathematics Department, University of Canterbury, Christchurch, New Zealand
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Abstract

A fully nonlinear model is developed for the unidirectional propagation of periodic gravity wave groups in deep water, in which the shape of the group envelopes changes cyclically. It is intended to describe the slow-time evolution of wave groups on the open ocean surface, and to generalize the cyclic recurrence that can occur during the sideband modulation of Stokes waves and Schrödinger wave groups. The weak nonlinear interactions are shown to concentrate the wave energy at the centre of each group at regular intervals, causing the waves there to be of greater height locally in space and time. This is suggested as one mechanism for the local wave breaking that is observed on the open ocean surface. The cyclically recurring wave groups may be interpreted as the limit-cycle stage in a progression from uniform wave groups to chaos on the forced, damped, ocean surface.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 192 , July 1988 , pp. 329 - 337
Copyright
© 1988 Cambridge University Press

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References

Bryant P. J. 1982 Modulation by swell of waves and wave groups on the ocean. J. Fluid Mech. 114, 443–466.Google Scholar
Bryant P. J. 1985 Doubly periodic progressive permanent waves in deep water. J. Fluid Mech. 161, 27–42.Google Scholar
Dold, J. W. & Peregrine D. H. 1986 Water-wave modulation. School of Mathematics, University of Bristol, Rep. AM-86–03.
Lake B. M., Yuen H. C., Rungaldier, H. & Ferguson W. E. 1977 Nonlinear deep-water waves: theory and experiment. Part 2. Evolution of a continuous wave train. J. Fluid Mech. 83, 49–74.Google Scholar
Miles J. 1984 Strange attractors in fluid dynamics. Adv. Appl. Mech. 24, 189–214.Google Scholar
Newell A. C. 1986 Chaos and turbulence: Is there a connection. Mathematics Applied to Fluid Mechanics and Stability: Proc. Conf. Dedicated to Richard C. DiPrima (Ed. D. A. Drew & J. E. Flaherty), pp. 157–189. SIAM.
Phillips O. M. 1985 Spectral and statistical properties of the equilibrium range in windgenerated gravity waves. J. Fluid Mech. 156, 505–531.Google Scholar
Yuen, H. C. & Ferguson W. E. 1978 Relationship between Benjamin–Feir instability and recurrence in the nonlinear Schrödinger equation. Phys. Fluids 21, 1275–1278.Google Scholar