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Chaos in small-amplitude surface gravity waves over slowly varying bathymetry

Published online by Cambridge University Press:  26 April 2006

Michael G. Brown ,
Frederick D. Tappert  and
Sekhar E. R. B. Sundaram
Michael G. Brown
Affiliation:
Rosenstiel School of Marine and Atmospheric Science, University of Miami, 4600 Rickenbacker Causeway, Miami, FL 33149, USA
Frederick D. Tappert
Affiliation:
Rosenstiel School of Marine and Atmospheric Science, University of Miami, 4600 Rickenbacker Causeway, Miami, FL 33149, USA
Sekhar E. R. B. Sundaram
Affiliation:
Rosenstiel School of Marine and Atmospheric Science, University of Miami, 4600 Rickenbacker Causeway, Miami, FL 33149, USA
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Abstract

We consider the motion of small-amplitude surface gravity waves over variable bathymetry. Although the governing equations of motion are linear, for general bathymetric variations they are non-separable and cannot be solved exactly. For slowly varying bathymetry, however, approximate solutions based on geometric (ray) techniques may be used. The ray equations are a set of coupled nonlinear ordinary differential equations with Hamiltonian form. It is argued that for general bathymetric variations, solutions to these equations - ray trajectories - should exhibit chaotic motion, i.e. extreme sensitivity to initial and environmental conditions. These ideas are illustrated using a simple model of bottom bathymetry, h(x,y) = h0(1 + εcos (2πx/L) cos (2πy/L)). The expectation of chaotic ray trajectories is confirmed via the construction of Poincaré sections and the calculation of Lyapunov exponents. The complexity of chaotic geometric wavefields is illustrated by considering the temporal evolution of (mostly) chaotic wavecrests. Some practical implications of chaotic ray trajectories are discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 227 , June 1991 , pp. 35 - 46
Copyright
© 1991 Cambridge University Press

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References

Benjamin, T. B. 1967 Instability of periodic wavetrains in nonlinear dispersive systems.. Proc. R. Soc. Lond. A 299, 59–75.Google Scholar
Brown, M. G. & Tappert, F. D. 1987 Catastrophe theory, caustics, and travel time diagrams in seismology. Geophys. J. R. Astr. Soc. 88, 217–229.Google Scholar
CERC (Coastal Engineering Research Center) 1977 Wave refraction In Shore Protection Manual, 3rd Edn., Vol. I, Sect. 2.3. US Government Printing Office, Washington, D.C.
Hardy, J. R. 1968 Some grid and projection problems in the numerical calculation of wave refraction. J. Geophys. Res. 73, 7083–7087.Google Scholar
Hénon, M. 1983 Numerical exploration of Hamiltonian systems. In Chaotic Behavior of Deterministic Systems: Les Houches Lectures 36 (ed. G. Iooss, R. G. H. Helleman & R. Stora), pp. 171–271. North-Holland.
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics, pp. 256–259. Pergamon.
Lichtenberg, A. J. & Lieberman, M. A. 1983 Regular and Stochastic Motion. Springer.
Lighthill, J. 1986 The recently recognized failure of predictability in Newtonian dynamics.. Proc. R. Soc. Lond. A 407, 35–50.Google Scholar
Mei, C. C. 1983 The Applied Dynamics of Ocean Surface Waves. Wiley Interscience.
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley Interscience.