Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Discrete and continuous random water wave dynamics

Pages: 1603 - 1633, Volume 28, Issue 4, December 2010      doi:10.3934/dcds.2010.28.1603

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André Nachbin - Instituto Nacional de Matemática Pura e Aplicada/IMPA, Est. D. Castorina, 110, J. Botânico, Rio de Janeiro, RJ 22460-320, Brazil (email)

Abstract: This article reviews recent work with emphasis on deducing random dynamical systems for wave dynamics in the presence of highly disordered forcing by the topography. It is shown that the long wave reflection process generated by potential theory is the same as the one generated by a hydrostatic model. The standard (hydrostatic) shallow water equations are not the correct asymptotic approximation to the Euler equations when the topography is nonsmooth, rapidly varying and of large amplitude. Nevertheless the reflection process (statistically speaking) is shown to be the same.
   New results are presented where the potential theory (probabilistic) results for reflection process are tested against Monte Carlo simulations with a hydrostatic Navier-Stokes numerical model. This numerical model is formulated in dimensional variables and was tested in real applications. The challenge in this part of our work was to set the numerical data accordingly with the regime of interest, and compare numerical results with those of the stochastic theory. Statistics with numerically reflected signals were produced through a Monte Carlo simulation. These reflected signals were averaged and compared to results given by the stochastic theory. Very good agreement is observed. Further experiments were performed in an exploratory fashion, hoping to stimulate new research from the Discrete and Continuous Dynamical Systems' readership.

Keywords:  Water waves, dispersion and scattering, nonlinear interaction, Partial Differential Equations, Central limit and other weak theorems.
Mathematics Subject Classification:  Primary: 76B15, 35Q; Secondary: 60F05.

Received: October 2009;      Revised: February 2010;      Available Online: June 2010.