16 research outputs found
Hamiltonian models for the propagation of irrotational surface gravity waves over a variable bottom
A single incompressible, inviscid, irrotational fluid medium bounded by a
free surface and varying bottom is considered. The Hamiltonian of the system is
expressed in terms of the so-called Dirichlet-Neumann operators. The equations
for the surface waves are presented in Hamiltonian form. Specific scaling of
the variables is selected which leads to approximations of Boussinesq and KdV
types taking into account the effect of the slowly varying bottom. The arising
KdV equation with variable coefficients is studied numerically when the initial
condition is in the form of the one soliton solution for the initial depth.Comment: 18 pages, 6 figures, 1 tabl
Soliton dynamics in a strong periodic field: the Korteweg-de Vries framework
Nonlinear long wave propagation in a medium with periodic parameters is considered in the framework of a variable-coefficient Korteweg-de Vries equation. The characteristic period of the variable medium is varied from slow to rapid, and its amplitude is also varied. For the case of a piecewise constant coefficient with a large scale for each constant piece, explicit results for the damping of a soliton damping are obtained. These theoretical results are confirmed by numerical simulations of the variable-coefficient Korteweg-de Vries equation for the same piecewise constant coefficient, as well as for a sinusoidally-varying coefficient. The resonance curve for soliton damping is predicted, and the maximum damping is for a soliton whose characteristic timescale is of the same order as the coefficient inhomogeneity scale. If the variation of the nonlinear coefficient is very large, and includes the a critical point where the nonlinear coefficient equals to zero, the soliton breaks and is quickly damped
Long wave expansions for water waves over random topography
In this paper, we study the motion of the free surface of a body of fluid
over a variable bottom, in a long wave asymptotic regime. We assume that the
bottom of the fluid region can be described by a stationary random process
whose variations take place on short length scales and which
are decorrelated on the length scale of the long waves. This is a question of
homogenization theory in the scaling regime for the Boussinesq and KdV
equations. The analysis is performed from the point of view of perturbation
theory for Hamiltonian PDEs with a small parameter, in the context of which we
perform a careful analysis of the distributional convergence of stationary
mixing random processes. We show in particular that the problem does not fully
homogenize, and that the random effects are as important as dispersive and
nonlinear phenomena in the scaling regime that is studied. Our principal result
is the derivation of effective equations for surface water waves in the long
wave small amplitude regime, and a consistency analysis of these equations,
which are not necessarily Hamiltonian PDEs. In this analysis we compute the
effects of random modulation of solutions, and give an explicit expression for
the scattered component of the solution due to waves interacting with the
random bottom. We show that the resulting influence of the random topography is
expressed in terms of a canonical process, which is equivalent to a white noise
through Donsker's invariance principle, with one free parameter being the
variance of the random process . This work is a reappraisal of the paper
by Rosales & Papanicolaou \cite{RP83} and its extension to general stationary
mixing processes
Global well-posedness for a family of regularized Benjamin-type equations
In this work we prove local and global well-posedness results for the Cauchy
problem of a family of regularized nonlinear Benjamin-type equations in both
periodic and nonperiodic Sobolev spaces.Comment: 22 page
Modified Shallow Water Equations for significantly varying seabeds
In the present study, we propose a modified version of the Nonlinear Shallow
Water Equations (Saint-Venant or NSWE) for irrotational surface waves in the
case when the bottom undergoes some significant variations in space and time.
The model is derived from a variational principle by choosing an appropriate
shallow water ansatz and imposing some constraints. Our derivation procedure
does not explicitly involve any small parameter and is straightforward. The
novel system is a non-dispersive non-hydrostatic extension of the classical
Saint-Venant equations. A key feature of the new model is that, like the
classical NSWE, it is hyperbolic and thus similar numerical methods can be
used. We also propose a finite volume discretisation of the obtained hyperbolic
system. Several test-cases are presented to highlight the added value of the
new model. Some implications to tsunami wave modelling are also discussed.Comment: 34 pages, 18 figures, 65 references. Other author's papers can be
downloaded at http://www.denys-dutykh.com
Rotational waves generated by current-topography interaction
We study nonlinear free-surface rotational waves generated through the interaction of a vertically sheared current with a topography. Equivalently, the waves may be generated by a pressure distribution along the free surface. A forced Korteweg–de Vries equation (fKdV) is deduced incorporating these features. The weakly nonlinear, weakly dispersive reduced model is valid for small amplitude topographies. To study the effect of gradually increasing the topography amplitude, the free surface Euler equations are formulated in the presence of a variable depth and a sheared current of constant vorticity. Under constant vorticity, the harmonic velocity component is formulated in a simplified canonical domain, through the use of a conformal mapping which flattens both the free surface as well as the bottom topography. Critical, supercritical, and subcritical Froude number regimes are considered, while the bottom amplitude is gradually increased in both the irrotational and rotational wave regimes. Solutions to the fKdV model are compared to those from the Euler equations. We show that for rotational waves the critical Froude number is shifted away from 1. New stationary solutions are found and their stability tested numerically.</p