A boundary integral formulation for particle trajectories  in Stokes waves

André Nachbin; Roberto Ribeiro-Junior

doi:10.3934/dcds.2014.34.3135

Article Contents

2014, Volume 34, Issue 8: 3135-3153. Doi: 10.3934/dcds.2014.34.3135

This issue Previous Article A characterization of the symmetric steady water waves in terms of the underlying flow Next Article Steady-states and traveling-wave solutions of the generalized Constantin--Lax--Majda equation

A boundary integral formulation for particle trajectories in Stokes waves

André Nachbin^1, and
Roberto Ribeiro-Junior^1,

1.
Instituto Nacional de Matemática Pura e Aplicada/IMPA, Est. D. Castorina, 110, J. Botânico, Rio de Janeiro, RJ 22460-320

Received: July 2013

Revised: September 2013

Published: August 2014

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Abstract

Recently important theorems have been established presenting qualitative results for particle trajectories below a Stokes wave. A diversity of orbit patterns were described, including the case of a closed orbit when a Stokes wave propagates in the presence of an adverse current. In this work these results are revisited in a quantitative fashion through a boundary integral formulation which leads to very accurate numerical simulations of particle trajectories. The boundary integral formulation allows the accurate evaluation of the vector field of the (particle's) dynamical system, without resorting to a series expansion and a small parameter. Accurate trajectories are benchmarked against well known expansions for weakly nonlinear waves. Simulations are then performed beyond this regime. Closed orbits are found in the presence of an adverse current, as well as non-smooth trajectories that have not been reported. These occur for both adverse and favorable currents.

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Mathematics Subject Classification: Primary: 76B15, 65N; Secondary: 35Q.

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