Marion Tissier, Philippe Bonneton, Fabien Marche, Florent Chazel, David Lannes


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.


Fully nonlinear Boussinesq equations; Wave breaking; Run-up; Hybrid method; Shock capturing schemes;

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Audusse, E., F. Bouchut, M.-0. Bristeau, R. Klein and B. Perthame. 2004. A fast and stable wellbalanced scheme with hydrostatic reconstruction for shallow water flows. SIAM Journal on Scientific Computing, 25(6), 2050-2065.

Berthon, C. and F. Marche, A positive preserving high order VFRoe scheme for shallow water equations: A class of relaxation schemes. 2008. SIAM Journal of Scientific Computing, 30(5), 2587-2612.

Bonneton, P. 2007. Modelling of periodic wave transformation in the inner surf zone, Ocean Engineering, 34, 1459-1471.

Bonneton, P., F. Chazel, D. Lannes, F. Marche and M. Tissier. 2010a. A splitting approach for the fully nonlinear and weakly dispersive Green-Naghdi model, in correction to Journal of Computational Physics.

Bonneton, P., E. Barthélemy, J.D. Carter, F. Chazel, R. Cienfuegos, D. Lannes, F. Marche, M. Tissier. 2010b. Fully nonlinear weakly dispersive modelling of wave propagation, breaking and run-up, in correction to European Journal of Mechanics, B/Fluids.

Bonneton P., N. Bruneau, F. Marche, B. Castelle. 2010c. Large-scale vorticity generation due to dissipating waves in the surf zone, DCDS-S, 13(4), 729-738.

Brocchini, M. and N. Dodd. 2008. Nonlinear Shallow Water Equation Modeling for Coastal Engineering, J. Wtrwy., Port, Coast., and Oc. Engrg., 134(2), 104-120.

Bruno, D., F. De Serio and M. Mossa. 2009. The FUNWAVE model application and its validation using laboratory data, Coastal Engineering, 56(7), 773-787.

Castelle, B, H. Michallet, V. Marieu, F. Leckler, B. Dubardier, A. Lambert, C. Berni, P. Bonneton, E. Barthélemy and F. Bouchette, 2010. Laboratory experiment on rip current circulations over a moveable bed: drifter measurements, in revision to Journal of Geophysical Research.

Chazel, F., D. Lannes, F. Marche. 2010. Numerical simulation of strongly nonlinear and dispersive waves using a Green-Naghdi model, Journal of Scientific Computing, DOI: 10.1007/s10915-010-9395-9.

Cienfuegos, R., E. Barthélemy, and P. Bonneton. 2006. A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-type equations. Part I: Model development and analysis. Int. J. Numer. Meth. Fluids, 56, 1217-1253.

Cienfuegos, R., E. Barthélemy and P. Bonneton. 2010. A wave-breaking model for Boussinesq-type equations including roller effects in the mass conservation equation. J. Wtrwy., Port, Coast., and Oc. Engrg., 136(1), 10-26.

Cox, D. 1995. Experimental and numerical modelling of surf zone hydrodynamics. PhD thesis, University of Delaware, Newark, Del.

Kennedy, A.B., Q. Chen, J.T. Kirby and R.A. Dalrymple. 2000. Boussinesq modelling of wave transformation, breaking and runup. I:1D. J. Wtrwy.,Port,Coast., and Oc. Engrg.,119, 618-638.

Kobayashi, N., G. De Silva and K. Watson. 1989. Wave transformation and swash oscillation on gentle and steep slopes, Journal of Geophysical Research, 94, 951-966.

Lannes, D., and P. Bonneton. 2009. Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation, Physics of Fluids, 21(1).

Madsen, P.A., 0.R. Sørensen and H.A. Schäffer. 1997. Surf zone dynamics simulated by a Boussinesq type model. Part I. Model description and cross-shore motion of regular waves, Coastal Engineering, 32, 255-287.

Marche, M., P. Bonneton, P. Fabrie and N. Seguin. 2007. Evaluation of well-balanced bore-capturing schemes for 2D wetting and drying processes, Int. J. Num. Meth. Fluids, 53(5), 867-894.

Schäffer, H.A., P. A. Madsen and R. Deigaard. 1993. A Boussinesq model for waves breaking in shallow water, Coastal Engineering, 20, 185-202.

Synolakis, C. E. 1987. The run-up of solitary waves. Journal of Fluid Mechanics, 185, 523-555.

Ting, F. and J. Kirby. 1994. Observation of undertow and turbulence in a laboratory surf zone. Coastal Engineering, 24, 51-80.