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

Michael G Brown, Frederick D. Tappert, Sekhar E R B Sundaram

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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 + ε cos (2πy/L) cos (2πy/L)). The expectation of chaotic ray trajectories is confirmed via the construction of Poincare 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.

Original languageEnglish (US)
Pages (from-to)35-46
Number of pages12
JournalJournal of Fluid Mechanics
Volume227
StatePublished - Jun 1991

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ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Physics and Astronomy(all)
  • Condensed Matter Physics

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