Philippe Guyenne

Solitary water wave interactions

This article concerns the pairwise nonlinear interaction of solitary waves in the free surface of a body of water lying over a horizontal bottom. Unlike solitary waves in many completely integrable model systems, solitary waves for the full Euler equations do not collide elastically; after interactions, there is a nonzero residual wave that trails the post-collision solitary waves. In this report on new numerical and experimental studies of such solitary wave interactions, we verify that this is the case, both in head-on collisions (the counterpropagating case) and overtaking collisions (the copropagating case), quantifying the degree to which interactions are inelastic. In the situation in which two identical solitary waves undergo a head-on collision, we compare the asymptotic predictions of Su and Mirie [J. Fluid Mech. 98, 509 (1980)] and Byatt-Smith [J. Fluid Mech. 49, 625 (1971)], the wavetank experiments of Maxworthy [J. Fluid Mech. 76, 177 (1976)], and the numerical results of Cooker, Weidman, and ...

Hamiltonian long–wave expansions for water waves over a rough bottom

This paper is a study of the problem of nonlinear wave motion of the free surface of a body of fluid with a periodically varying bottom. The object is to describe the character of wave propagation in a long–wave asymptotic regime, extending the results of R. Rosales & G. Papanicolaou (1983 Stud. Appl. Math. 68, 89–102) on periodic bottoms for two–dimensional flows.We take the point of view of perturbation of a Hamiltonian system dependent on a small scaling parameter, with the starting point being Zakharovs Hamiltonian (V. E. Zakharov 1968 J. Appl. Mech. Tech. Phys. 9, 1990–1994) for the Euler equations for water waves. We consider bottom topography which is periodic in horizontal variables on a short length–scale, with the amplitude of variation being of the same order as the fluid depth. The bottom may also exhibit slow variations at the same length–scale as, or longer than, the order of the wavelength of the surface waves. We do not take up the question of random bottom variations, a topic which is considered in Rosales & Papanicolaou (1983). In the two–dimensional case of waves in a channel, we give an alternate derivation of the effective Korteweg–de Vries (KdV) equation that is obtained in Rosales & Papanicolaou (1983). In addition, we obtain effective Boussinesq equations that describe the motion of bidirectional long waves, in cases in which the bottom possesses both short and long–scale variations. In certain cases we also obtain unidirectional equations that are similar to the KdV equation. In three dimensions we obtain effective three–dimensional long–wave equations in a Boussinesq scaling regime, and again in certain cases an effective Kadomtsev–Petviashvili (KP) system in the appropriate unidirectional limit. The computations for these results are performed in the framework of an asymptotic analysis of multiple–scale operators. In the present case this involves the Dirichlet–Neumann operator for the fluid domain which takes into account the variations in bottom topography as well as the deformations of the free surface from equilibrium.

Numerical study of three-dimensional overturning waves in shallow water

Simulations in a three-dimensional numerical wave tank are performed to investigate the shoaling and breaking of solitary waves over a sloping ridge. The numerical model solves fully nonlinear potential flow equations with a high-order boundary-element method combined with an explicit time-integration method, expressed in a mixed Eulerian–Lagrangian formulation. Analyses of shoaling and breaking-wave profiles and kinematics (both on the free surface and within the flow) are carried out. It is observed that the transverse modulation of the ridge topography induces threedimensional effects on the time evolution, shape and kinematics of breaking waves. Comparisons of two- and three-dimensional results in the middle cross-section of the ridge, however, show remarkable similarities, especially for the shape and dynamics of the plunging jet.

A High-Order Spectral Method for Nonlinear Water Waves over Moving Bottom Topography

We present a numerical method for simulations of nonlinear surface water waves over variable bathymetry. It is applicable to either two- or three-dimensional flows, as well as to either static or moving bottom topography. The method is based on the reduction of the problem to a lower-dimensional Hamiltonian system involving boundary quantities alone. A key component of this formulation is the Dirichlet-Neumann operator which, in light of its joint analyticity properties with respect to surface and bottom deformations, is computed using its Taylor series representation. We present new, stabilized forms for the Taylor terms, each of which is efficiently computed by a pseudospectral method using the fast Fourier transform. Physically relevant applications are displayed to illustrate the performance of the method; comparisons with analytical solutions and laboratory experiments are provided.

Numerical simulation of three-dimensional nonlinear water waves

We present an accurate and efficient numerical model for the simulation of fully nonlinear (non-breaking), three-dimensional surface water waves on infinite or finite depth. As an extension of the work of Craig and Sulem [19], the numerical method is based on the reduction of the problem to a lower-dimensional Hamiltonian system involving surface quantities alone. This is accomplished by introducing the Dirichlet-Neumann operator which is described in terms of its Taylor series expansion in homogeneous powers of the surface elevation. Each term in this Taylor series can be computed efficiently using the fast Fourier transform. An important contribution of this paper is the development and implementation of a symplectic implicit scheme for the time integration of the Hamiltonian equations of motion, as well as detailed numerical tests on the convergence of the Dirichlet-Neumann operator. The performance of the model is illustrated by simulating the long-time evolution of two-dimensional steadily progressing waves, as well as the development of three-dimensional (short-crested) nonlinear waves, both in deep and shallow water.

High order well-balanced CDG-FE methods for shallow water waves by a Green-Naghdi model

In this paper, we consider a one-dimensional fully nonlinear weakly dispersive Green-Naghdi model for shallow water waves over variable bottom topographies. Such model describes a large spectrum of shallow water waves, and it is thus of great importance to design accurate and robust numerical methods for solving it. The governing equations contain mixed spatial and temporal derivatives of the unknowns. They also have still-water stationary solutions which should be preserved in stable numerical simulations. In our numerical approach, we first reformulate the Green-Naghdi equations into balance laws coupled with an elliptic equation. We then propose a family of high order numerical methods which discretize the balance laws with well-balanced central discontinuous Galerkin methods and the elliptic part with continuous finite element methods. Linear dispersion analysis for both the (reformulated) Green-Naghdi system and versions of the proposed numerical scheme is performed when the bottom topography is flat. Numerical tests are presented to illustrate the accuracy and stability of the proposed schemes as well as the capability of the Green-Naghdi model to describe a wide range of shallow water wave phenomena.

Numerical simulation of solitary waves on plane slopes

In this paper, we present a numerical method for the computation of surface water waves over bottom topography. It is based on a series expansion representation of the Dirichlet-Neumann operator in terms of the surface and bottom variations. This method is computationally very efficient using the fast Fourier transform. As an application, we perform computations of solitary waves propagating over plane slopes and compare the results with those obtained from a boundary element method. A good agreement is found between the two methods.

PROGRESS IN FULLY NONLINEAR POTENTIAL FLOW MODELING OF 3D EXTREME OCEAN WAVES

SG and FE acknowledge the US National Science Foundation (NSF), under grant CMS0100223 of “the Engineering/Earthquake, Hazards and Mitigation Program”, and SG also acknowledges the US Office of Naval Research, under grant N000140510068, for supporting part of this work. Jeff Harris’ help is gratefully acknowledged for providing original data for the last two figures. Frederic.Dias@cmla.ens-cachan.fr. guyenne@math.udel.edu. PG acknowledges support from the University of Delaware Research Foundation and the US NSF under grant DMS-0625931. fochesat@cmla.ens-cachan.fr. enet@alkyon.nl.

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 focus on the two-dimensional case, assuming that the bottom of the fluid region can be described by a stationary random process β(x, ω) 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 Korteweg–de Vries equations.The analysis is performed from the point of view of perturbation theory for Hamiltonian partial differential equations (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 Donskers invariance principle, with one free parameter being the variance of the random process.

SOLITARY-WAVE COLLISIONS

Experimental and theoretical results are presented for binary collisions between copropagating and counter-propagating solitary waves. The experiments provide highresolution measurements of water surface profiles at fixed times, thereby enabling direct comparisons with predictions by a variety of mathematical models. These models include the 2-soliton solution of the Korteweg-deVries equation, numerical solutions of the Euler equations, and linear superposition of KdV solitons.