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Nonlinear evolution of surface gravity waves over highly variable depth.

William Artiles1, André Nachbin

  • 1Instituto de Matemática Pura e Aplicada, Est. D Castorina 110, Jardim Botânico, Rio de Janeiro, RJ 22460-320, Brazil.

Physical Review Letters
|December 17, 2004
PubMed
Summary

New nonlinear evolution equations generalize existing Boussinesq models for surface gravity waves in variable-depth fluids. These equations accommodate complex topography and enable efficient numerical solutions using Fourier methods.

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Area of Science:

  • Fluid dynamics
  • Nonlinear wave propagation
  • Computational mathematics

Background:

  • Existing Boussinesq equations have limitations in modeling waves over complex bathymetry.
  • Generalizing these models is crucial for accurate simulation of surface gravity waves in variable environments.

Purpose of the Study:

  • To derive novel nonlinear evolution equations for finite-amplitude surface gravity waves.
  • To generalize existing Boussinesq systems to include highly variable and complex fluid topography.
  • To develop equations suitable for efficient numerical implementation.

Main Methods:

  • Expansion of a Fourier-type operator in a wave steepness parameter.
  • Formulation over a periodically extended domain.
  • Development of variable coefficient Boussinesq-type equations.

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Main Results:

  • New nonlinear evolution equations are derived, generalizing Matsuno's and Nachbin's models.
  • The equations handle finite-amplitude waves in two-dimensional, incompressible, inviscid fluids with finite, variable depth.
  • Topography can vary across broad scales and possess complex, multiply-valued profiles.

Conclusions:

  • The derived equations offer a more versatile framework for studying surface gravity waves.
  • The formulation facilitates efficient numerical solvers, particularly those employing fast Fourier transform algorithms.
  • This work advances the modeling of water waves in complex, natural environments.