Philippe Bonneton

A splitting approach for the fully nonlinear and weakly dispersive Green-Naghdi model

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, which could be adapted to many physical models that are dispersive corrections of hyperbolic systems. 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.

Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation

A general method for the derivation of asymptotic nonlinear shallow water and deep water models is presented. Starting from a general dimensionless version of the water-wave equations, we reduce the problem to a system of two equations on the surface elevation and the velocity potential at the free surface. These equations involve a Dirichlet-Neumann operator and we show that all the asymptotic models can be recovered by a simple asymptotic expansion of this operator, in function of the shallowness parameter (shallow water limit) or the steepness parameter (deep water limit). Based on this method, a new two-dimensional fully dispersive model for small wave steepness is also derived, which extends to uneven bottom the approach developed by Matsuno \cite{matsuno3} and Choi \cite{choi}. This model is still valid in shallow water but with less precision than what can be achieved with Green-Naghdi model, when fully nonlinear waves are considered. The combination, or the coupling, of the new fully dispersive equations with the fully nonlinear shallow water Green-Naghdi equations represents a relevant model for describing ocean wave propagation from deep to shallow waters.

Wave-Breaking Model for Boussinesq-Type Equations Including Roller Effects in the Mass Conservation Equation

We investigate the ability of a 1D fully nonlinear Boussinesq model including breaking to reproduce surf zone waves in terms of wave height and nonlinear intraphase properties such as asymmetry and skewness. An alternative approach for wave-breaking parameterization including roller effects through diffusive-type terms on both, the mass conservation and momentum equations is developed and validated on regular wave and solitary wave experiments as an attempt to improve wave height and left-right asymmetry estimates. The new approach is able to reproduce wave height decay, and intraphase nonlinear properties within the entire surf zone of spilling breakers without requiring temporal evolution of model parameters.

Circulation and suspended sediment transport in a coral reef lagoon: The south-west lagoon of New Caledonia

The south-west lagoon of New Caledonia is a wide semi-open coral reef lagoon bounded by an intertidal barrier reef and bisected by numerous deep inlets. This paper synthesizes findings from the 2000-2008 French National Program EC2CO-PNEC relative to the circulation and the transport of suspended particles in this lagoon. Numerical model development (hydrodynamic, fine suspended sediment transport, wind-wave, small-scale atmospheric circulation) allowed the determination of circulation patterns in the lagoon and the charting of residence time, the later of which has been recently used in a series of ecological studies. Topical studies based on field measurements permitted the parameterisation of wave set-up induced by the swell breaking on the reef barrier and the validation of a wind-wave model in a fetch-limited environment. The analysis of spatial and temporal variability of suspended matter concentration over short and long time-scales, the measurement of grain size distribution and the density of suspended matter (1.27 kg l(-1)), and the estimation of erodibility of heterogeneous (sand/mud, terrigenous/biogenic) soft bottoms was also conducted. Aggregates were shown to be more abundant near or around reefs and a possible biological influence on this aggregation is discussed. Optical measurements enabled the quantification of suspended matter either in situ (monochromatic measurements) or remotely (surface spectral measurements and satellite observations) and provided indirect calibration and validation of a suspended sediment transport model. The processes that warrant further investigation in order to improve our knowledge of circulation and suspended sediment transport in the New Caledonia lagoon as well as in other coral reef areas are discussed, as are the relevance and reliability of the numerical models for this endeavour.

Numerical modelling of bore propagation and run-up on sloping beaches using a MacCormack TVD scheme

A McCormack TVD scheme is presented for the computation of Saint-Venant equations, in the context of coastal hydrodynamics. The dam-break problem on wet and dry bottoms is used to evaluate and discuss the performances of the scheme. A run-up simulation on a sloping beach is then presented and a comparison using an analytical solution is made. Finally, a bore propagation on a sloping beach is computed.

On the density structure of far-wake vortices in a stratified fluid

An experimental investigation of the three-dimensional density structure of far-wake vortices generated by moving a sphere in a linear saline stratification has been carried out via conductivity measurements. These measurements were performed in the vortex cores along vertical and horizontal profiles to capture the three-dimensional nature of the vortices. We first present a simple model of an isolated vortex, the calculated internal density field of which has been confirmed by the conductivity measurements performed in both turbulent and laminar wakes. The time-dependent density structure is also described, as well as the decay of the density peak within the vortices. An identical density structure corresponding to an intensification of the background stratification has been identified for far-wake vortices originating from either laminar or turbulent near-wakes. This suggests that these vortices exhibit universal features of quasi two-component structures in stratified fluids, such as large-scale vortices commonly found in the ocean.

Tidal bore dynamics in funnel-shaped estuaries

The formation and dynamics of tidal bores in funnel-shaped estuaries is investigated from both a global tidal wave scaling analysis and new quantitative field observations. We show that tidal bore occurrence in convergent estuaries can be estimated from a dimensionless scaling parameter characterizing the relative intensity of nonlinear friction versus local inertia in the momentum equation. A detailed analysis of tidal bore formation and secondary wave structure is presented from a unique long-term database (observations of more than 200 tides) acquired during four campaigns in the two main French tidal-bore estuaries: the Seine and Gironde/Garonne estuaries. We describe the effect of freshwater discharge on the global tidal wave transformation at the estuarine scale and on local tidal bore occurrence in the upper estuary. Our field data suggest that the tidal bore intensity is mainly governed by the dimensionless tidal range, which characterizes the local tidal wave nonlinearity. We also show that the secondary wavefield associated with tidal bore propagating in natural estuaries differs significantly from those associated to undular bores in rectangular channels. In particular, we observe an abrupt decrease of the whelp steepness when the Froude number goes below 1.1. This secondary field transition can explain why tidal bore occurrence in worldwide estuaries is certainly underestimated.

Lee-wave breaking over obstacles in stratified flow

Experimental results are presented on the lee-wave breaking process which occurs at low Froude numbers when uniform and strongly stratified flow approaches two-dimensional and quasi two-dimensional Gaussian-shaped obstacles. It was found that the lee-wave breaking process is essentially independent of the two-dimensional and the quasi two-dimensional shape of the obstacles. The attainment of the critical condition where the steepening wave becomes statically unstable does not mark a threshold to breakdown. Instead, the wave remains dynamically stable for several buoyancy periods, overturning into an “S”-shape with maximum overturning reaching about 55° past the vertical. It is observed that the primary instability forms a quasi two-dimensional spanwise vortex over the central portion of the obstacles and is mainly shear driven. The quasi two-dimensional spanwise vortex persists for a few buoyancy periods before undergoing a three-dimensional convective instability, similar to a Rayleigh–Taylor instability...

Infragravity‐wave modulation of short‐wave celerity in the surf zone

The cross-shore evolution of individual wave celerity is investigated using two high-resolution laboratory experiments on bichromatic waves. Individual waves are tracked during their onshore propagation and their characteristics, including celerity, are estimated. The intrawave variability in celerity is low in the shoaling zone but increases strongly after breaking. It is maximum when the infragravity-wave height to water depth ratio is the largest, that is to say close to the shoreline. There the observed range of individual wave celerity can be as large as the mean celerity value. This variability can be largely explained by the variations in water depth and velocity induced by the infragravity waves. The differences in celerity are such that they lead to the merging of the waves in the inner surf zone for most of the wave conditions considered. Again, the location at which the first waves start merging strongly correlates with the infragravity-wave height to water depth ratio. The consequences of these findings for celerity-based depth-inversion techniques are finally discussed. Surprisingly, accounting for the infragravity-wave modulation of the velocity field in the celerity estimate does not significantly improve depth estimation in the surf zone. However, it is shown that the occurrence of bore merging decreases significantly the coherence of the wavefield in the surf zone. This loss of coherence could hamper celerity estimation from pixel intensity time series and explain, at least partly, the relatively poor performance of depth-inversion techniques in the inner surf zone.

Far-Wake of a Sphere in a Stably Stratified Fluid: Characterization of the Vortex Structures

This article discusses the structure of the far-wake of a towed sphere in a saline stratification. We compare very low Froude number experiments to existing results at higher Froude numbers and investigate the vertical structure of the far-wake in terms of their vorticity and density fields. We show that the vertical propagation of vorticity is viscously dominated and propose a simple three-dimensional model for the quasi-equilibrium of the structure in terms of the density field.