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Discrete and continuous random water wave dynamics
This article reviews recent work with emphasis on deducing random
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
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.