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Analytic model for a weakly dissipative shallow-water undular bore.

G A El1, R H J Grimshaw, A M Kamchatnov

  • 1Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3T, United Kingdom. g.el@lboro.ac.uk

Chaos (Woodbury, N.Y.)
|October 29, 2005
PubMed
Summary

This study models steady undular bore formation using a viscous Kaup-Boussinesq shallow water system. The research reveals the bore

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Area of Science:

  • Fluid dynamics
  • Nonlinear wave phenomena
  • Mathematical physics

Background:

  • Shallow water systems are fundamental in fluid dynamics.
  • Undular bores are complex wave phenomena observed in shallow water.
  • Previous studies often focused on dissipationless models.

Purpose of the Study:

  • To model the formation of a steady-profile undular bore.
  • To analyze the role of viscosity in bore formation.
  • To compare viscous and dissipationless scenarios.

Main Methods:

  • Utilized the integrable Kaup-Boussinesq shallow water system with a viscous term.
  • Employed a modified finite-gap integration technique for the Ablowitz-Kaup-Newell-Segur (AKNS) scheme.
  • Reduced the Whitham system to a first-order differential equation solved numerically.

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Main Results:

  • Obtained an asymptotic profile for the steady undular bore.
  • Described the local oscillatory structure using periodic solutions.
  • Demonstrated consistency between Whitham equations and conservation laws.

Conclusions:

  • Viscosity leads to the formation of a steady-profile undular bore.
  • The modified Whitham system accurately describes viscous bore dynamics.
  • Results offer insights into wave behavior in viscous shallow water.