In this paper we present a general framework to construct 1D width averaged models when the flow is constrained -e.g. by topography- to be almost 1D. We start from two dimensional shallow water equations, perform an asymptotic expansion of the fluid elevation and velocity field in the spirit of wave diffusive equations and establish a set of 1D equations made of a mass, momentum and energy equations which are close to the one usually used in hydraulic engineering. We show that in some special cases, like the U-shaped river bed, that our set of equations reduces to the classical 1d shallow water equations. Out of these configurations, there is an O (1) deviation of our model from the classical one.

Le modèle original proposé par les auteurs (CHAZEL et al., 2009) pour les vagues en zone côtière est mis en œuvre et validé sur deux cas expérimentaux. Nous considérons d’abord les expériences de DINGEMANS (1994), étudiant la propagation de vagues régulières au-dessus d’une barre trapézoïdale immergée. Les résultats numériques démontrent une excellente capacité du modèle à reproduire les effets de levée, les interactions fortement non-linéaires, ainsi que la génération puis la propagation de composantes harmoniques d’ordres élevés après la barre. Ensuite, nous vérifions la capacité du modèle à propager des vagues irrégulières non-déferlantes sur une bathymétrie plus complexe (essai 26 de BECQ-GIRARD et al., 1999). L’analyse et la comparaison des spectres de variance montrent que les résultats du modèle sont également en très bon accord avec les mesures expérimentales. Abstract: The new model recently proposed by the authors (CHAZEL et al., 2009) to simulate waves in the coastal zone is applied and validated on two experimental cases. Firstly the set of experiments by DINGEMANS (1994) is considered, with the propagation of regular waves over a submerged trapezoidal bar. The numerical results show the excellent capabilities of the model to reproduce the shoaling effects, as well as the generation and the further propagation of higher harmonics after the bar. Then we check the ability of the model to propagate irregular non-breaking waves over a more complex bottom profile (experiment 26 by BECQ-GIRARD et al., 1999). The analysis and comparison of variance spectra show that the model results are also in very good agreement with experimental measurements. Keywords: Waves, Nonlinear waves, Dispersive waves, Coastal waves, Wave models, Boussinesq-type model, Two-layer modelling technique.

Recent field studies over low sloping beaches have shown that infragravity waves could dissipate a significant part of their energy in the inner surf zone. This phenomenon and the associated short- and long-wave transformations are not well-understood. In this paper, we assess the ability of the fully nonlinear Boussinesq-type model introduced in Bonneton et al. (2011) to reproduce short and long wave transformation in a case involving a strong infragravity wave dissipation close to the shoreline. This validation study, based on van Dongeren et al. (2008)'s laboratory experiments, suggests that the model is able to predict infragravity wave breaking as well as the complex interactions between short and long waves in the surf zone.

In this paper, a new method to handle wave breaking in fully non-linear Boussinesq-type models is presented. The strategy developed to treat wave breaking is based on a reformulation of the set of governing equations (namely Serre Green–Naghdi equations) that allows us to split them into a hyperbolic part in the conservative form and a dispersive part. When a wave is ready to break, we switch locally from Serre Green–Naghdi equations to Non-linear Shallow Water equations by suppressing the dispersive terms in the vicinity of the wave front. Thus, the breaking wave front is handled as a shock by the Non-linear Shallow Water equations, and its energy dissipation is implicitly evaluated from the mathematical shock-wave theory. A simple methodology to characterize the wave fronts at each time step is first described, as well as appropriate criteria for the initiation and termination of breaking. Extensive validations using laboratory data are then presented, demonstrating the efficiency of our simple treatment for wave breaking.

The fully nonlinear and weakly dispersive Green-Naghdi equations for shallow water waves of large amplitude is studied. An hybrid finite volume and finite difference splitting approach is proposed. Numerical validations are then performed in one horizontal dimension.

We investigate here the ability of a Green-Naghdi model to reproduce strongly nonlinear and dispersive wave propagation. We test in particular the behavior of the new hybrid finite-volume and finite-difference splitting approach recently developed by the authors and collaborators on the challenging benchmark of waves propagating over a submerged bar. Such a configuration requires a model with very good dispersive properties, because of the high-order harmonics generated by topography-induced nonlinear interactions. We thus depart from the aforementioned work and choose to use a new Green-Naghdi system with improved frequency dispersion characteristics. The absence of dry areas also allows us to improve the treatment of the hyperbolic part of the equations. This leads to very satisfying results for the demanding benchmarks under consideration.

Tissier, M., Bonneton, P., Marche, F., Chazel, F. And Lannes, D., 2011. Nearshore dynamics of tsunami-li ke undular bore using a fully nonlinear Boussinesq mode l. Journal of Coastal Research, SI 64 (Proceedings of the 11th International Coastal Symposium), pg ‐ pg. Szczecin, Poland, ISSN 0749-0208 When tsunami wave fronts reach the shore, they can evolve into a large range of bore types, from undul ar nonbreaking bore to purely breaking bore. It is the co mplex competition between non-linearities, dispersi ve effects and energy dissipation which will govern their tran sformations, making the prediction of their evoluti on a challenging task for numerical models. In this pape r we investigate the ability of a fully nonlinear Bo ussinesq model, SURF-WB, to predict bore dynamics in a large range of Froude numbers. The model is first applied to the formation of undular bores, and compared with l aboratory data. Its ability to predict the differen t bore shapes is then investigated. Finally, the effects of the b ore transformation on wave run-up over a sloping be ach are considered.

Tissier, M., Bonneton, P., Marche, F., Chazel, F. And Lannes, D., 2011. Nearshore dynamics of tsunami-like undular bore using a fully nonlinear Boussinesq model. Journal of Coastal Research, SI 64 (Proceedings of the 11th International Coastal Symposium), 603 - 607. Szczecin, Poland, ISSN 0749-0208 When tsunami wave fronts reach the shore, they can evolve into a large range of bore types, from undular non-breaking bore to purely breaking bore. It is the complex competition between nonlinear effects, dispersive effects and energy dissipation which governs their transformations, making the prediction of their evolution a challenging task for numerical models. In this paper we investigate the ability of a fully nonlinear Boussinesq model, SURF-WB, to predict bore dynamics for a large range of Froude numbers. The model is first applied to the formation of undular bores and further compared with laboratory data. Its ability to predict the different bore shapes is then investigated. Finally, the effects of the bore transformation on wave run-up over a sloping beach are considered.

In this paper, a fully nonlinear Boussinesq model is presented and applied to the description of breaking waves and shoreline motions. It is based on Serre Green-Naghdi equations, solved using a time-splitting approach separating hyperbolic and dispersive parts of the equations. The hyperbolic part of the equations is solved using Finite-Volume schemes, whereas dispersive terms are solved using a Finite-Difference method. The idea is to switch locally in space and time to NSWE by skipping the dispersive step when the wave is ready to break, so as the energy dissipation due to wave breaking is predicted by the shock theory. This approach allows wave breaking to be handled naturally, without any ad-hoc parameterization for the energy dissipation. Extensive validations of the method are presented using laboratory data.

The fully nonlinear and weakly dispersive Green-Naghdi model for shallow water waves of large amplitude is studied. The original model is first recast under a new formulation more suitable for numerical resolution. An hybrid finite volume and finite difference splitting approach is then proposed. The hyperbolic part of the equations is handled with a high-order finite volume scheme allowing for breaking waves and dry areas. The dispersive part is treated with a classical finite difference approach. Extensive numerical validations are then performed in one horizontal dimension, relying both on analytical solutions and experimental data. The results show that our approach gives a good account of all the processes of wave transformation in coastal areas: shoaling, wave breaking and run-up.

To describe the strongly nonlinear dynamics of waves propagating in the final stages of shoaling and in the surf and swash zones, fully nonlinear models are required. The ability of the Serre or Green Naghdi (S–GN) equations to reproduce this nonlinear processes is reviewed. Two high-order methods for solving S–GN equations, based on Finite Volume approaches, are presented. The first one is based on a quasi-conservative form of the S–GN equations, and the second on a hybrid Finite Volume/Finite Difference method. We show the ability of these two approaches to accurately simulate nonlinear shoaling, breaking and runup processes.

To describe the strongly nonlinear dynamics of waves propagating in the final stages of shoaling and in the surf and swash zones, fully nonlinear models are required. The ability of the Serre or Green Naghdi (S-GN) equations to reproduce this nonlinear processes is reviewed. Two high-order methods for solving S-GN equations, based on Finite Volume approaches, are presented. The first one is based on a quasi-conservative form of the S-GN equations, and the second on a hybrid Finite Volume/Finite Difference method. We show the ability of these two approaches to accurately simulate nonlinear shoaling, breaking and runup processes.

Un modèle original, de type Boussinesq, basé sur une approche double-couche a récemment été proposé par les auteurs (CHAZEL et al., 2009), permettant de modéliser des vagues fortement non-linéaires et dispersives en zone proche côtière et côtière.Ce modèle présente notamment d'excellentes propriétés linéaires jusqu'à kh=10 (où k est le nombre d'onde et h la hauteur d'eau) tant pour les profils verticaux de vitesses orbitales que pour le shoaling.Après un bref résumé de la construction mathématique du modèle, nous présentons sa mise en oeuvre numérique en une dimension d'espace horizontale.Le modèle est ensuite appliqué à deux cas expérimentaux, afin d'estimer son domaine de validité non-linéaire pour des profondeurs d'eau intermédiaires à grandes.En premier lieu, nous considérons les expériences classiques de DINGEMANS (1994) étudiant la propagation de vagues régulières au-dessus d'une barre trapézoïdale immergée.Les résultats numériques obtenus démontrent une excellente capacité du modèle à reproduire les effets de shoaling, les interactions fortement non-linéaires, ainsi que la génération puis la propagation d'harmoniques d'ordre élevé après la barre.Ensuite, nous mettons en exergue la capacité du modèle à propager des vagues irrégulières non-déferlantes, en simulant une des expériences en canal de BECQ-GIRARD et al. (1999) sur une bathymétrie plus complexe, comportant également un haut fond.Dans ce deuxième cas, les résultats du modèle sont également en très bon accord avec les mesures expérimentales.

We derive and analyse, in the framework of the mild-slope approximation, a new double-layer Boussinesq-type model that is linearly and nonlinearly accurate up to deep water. Assuming the flow to be irrotational, we formulate the problem in terms of the velocity potential, thereby lowering the number of unknowns. The model derivation combines two approaches, namely the method proposed by Agnonet al.(Agnonet al.1999J. Fluid Mech.399, 319–333) and enhanced by Madsenet al.(Madsenet al.2003Proc. R. Soc. Lond. A459, 1075–1104), which consists of constructing infinite-series Taylor solutions to the Laplace equation, to truncate them at a finite order and to use Padé approximants, and the double-layer approach of Lynett & Liu (Lynett & Liu 2004aProc. R. Soc. Lond. A460, 2637–2669), which allows lowering the order of derivatives. We formulate the model in terms of a static Dirichlet–Neumann operator translated from the free surface to the still-water level, and we derive an approximate inverse of this operator that can be built once and for all. The final model consists of only four equations both in one and two horizontal dimensions, and includes only second-order derivatives, which is a major improvement in comparison with so-called high-order Boussinesq models. A linear analysis of the model is performed, and its properties are optimized using a free parameter determining the position of the interface between the two layers. Excellent dispersion and shoaling properties are obtained, allowing the model to be applied up to the deep-water valuekh=10. Finally, numerical simulations are performed to quantify the nonlinear behaviour of the model, and the results exhibit a nonlinear range of validity reaching at leastkh=3π.

In this paper we focus on the water waves problem for uneven bottoms on a two-dimensional domain. Starting from the symmetric Boussinesq systems derived in [F. Chazel, Influence of bottom topography on long water waves, M2AN 41 (4) (2007) 771-799], we recover the uncoupled Korteweg-de Vries (KdV) approximation justified in [G. Schneider, C,E. Wayne, The long-wave limit for the water-wave problem. I. The case of zero surface tension. Comm. Pure Appl. Math. 162 (3) (2002) 247-285] for flat bottoms, and in [T. Iguchi, A long wave approximation for capillary-gravity waves and an effect of the bottom, Comm. Partial Differential Equations 32 (2007) 37-85] in the context of bottoms tending to zero at infinity at a substantial rate. The goal of this paper is to investigate the validity of this approximation for more general bathymetries. We exhibit two kinds of topography for which this approximation diverges from the Boussinesq solutions. A topographically modified KdV approximation is then proposed to deal with such bathymetries, where topography-dependent terms are added to the solutions of the KdV equations. Finally. all the models involved are numerically computed and compared. (C) 2008 Elsevier Masson SAS. All rights reserved.

We focus here on the water waves problem for uneven bottoms in the long-wave regime, on an unbounded two or three-dimensional domain. In order to derive asymptotic models for this problem, we consider two different regimes of bottom topography, one for small variations in amplitude, and one for strong variations. Starting from the Zakharov formulation of this problem, we rigorously compute the asymptotics expansion of the involved Dirichlet-Neumann operator. then, following the global strategy introduced by Bona, Colin and Lannes, new symetric asymptotic models are derived for each regime of bottom topography. Solutions of these systems are proved to give good approximations of solutions of the water waves problem. These results hold for solutions that evanesce at infinity as well as for spatially periodic ones.