M. Antuono and M. Brocchini, Beyond Boussinesq-type equations: Semi-integrated models for coastal dynamics, Physics of Fluids, vol.12, issue.1, p.16603
DOI : 10.1016/j.coastaleng.2008.10.012

E. Audusse, F. Bouchut, M. Bristeau, R. Klein, and B. Perthame, A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows, SIAM Journal on Scientific Computing, vol.25, issue.6, pp.2050-2065, 2004.
DOI : 10.1137/S1064827503431090
URL : http://perso-math.univ-mlv.fr/users/bouchut.francois/publications/hydroproj.pdf

P. Azerad, J. L. Guermond, and B. Popov, Well-Balanced Second-Order Approximation of the Shallow Water Equation with Continuous Finite Elements, SIAM Journal on Numerical Analysis, vol.55, issue.6, pp.3203-3224
DOI : 10.1137/17M1122463
URL : https://hal.archives-ouvertes.fr/hal-01815500

S. Barré-de and A. J. Venant, 1871 Théorie du mouvement non-permanent des eaux, avec application aux crues desrivì eres etàetà l'introduction des marées dans leur lit, C. R. Acad

, Sci. Paris, vol.73, pp.147-154

S. Beji and J. Battjes, Experimental investigation of wave propagation over a bar, Coastal Engineering, vol.19, issue.1-2, pp.151-162
DOI : 10.1016/0378-3839(93)90022-Z

P. Bonneton, F. Chazel, D. Lannes, F. Marche, and M. Tissier, A splitting approach for the fully nonlinear and weakly dispersive Green???Naghdi model, Journal of Computational Physics, vol.230, issue.4, pp.1479-1498
DOI : 10.1016/j.jcp.2010.11.015
URL : https://hal.archives-ouvertes.fr/hal-00482564

J. Boussinesq, 1872 Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Mathématiques Pures et Appliquées.Deuxì eme Série, vol.17, pp.55-108

M. Brocchini, 2013 A reasoned overview on Boussinesq-type models: the interplay between physics, mathematics and numerics An improved Serre model: Efficient simulation and comparative evaluation, Proc. R. Soc. A 469, pp.404-423, 2018.

Q. Chen, R. A. Dalrymple, J. T. Kirby, A. B. Kennedy, and M. C. Haller, , 1999.

, Boussinesq modeling of a rip current system, J. Geophys. Res, vol.104, pp.20617-20637

Q. Chen, J. T. Kirby, R. A. Dalrymple, A. B. Kennedy, and A. Chawla, , 2000.

R. Cienfuegos, E. Barthelemy, and P. Bonneton, Boussinesq modeling of wave transformation, breaking and runup. II: 2D Wave-breaking model for Boussinesqtype equations including roller effects in the mass conservation equation, J. Waterw. Port Coast. J. Waterway Port Coast. Ocean Eng, vol.126, issue.136, pp.48-56, 2010.

B. Cockburn and C. W. Shu, The local discontinuous Galerkin method for timedependent convection-diffusion systems, SIAM J. Numer. Anal, vol.141, pp.2440-2463, 1998.

B. Cockburn and C. W. Shu, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, Journal of Scientific Computing, vol.16, issue.3, pp.173-261, 2001.
DOI : 10.1023/A:1012873910884

J. F. Debieve, H. Gouin, and J. Gaviglio, Evolution of the Reynolds stress tensor in a shock wave?turbulence interaction, Indian. J. Technol, vol.20, pp.90-97, 1982.

A. Duran and F. Marche, Abstract, Communications in Computational Physics, vol.185, issue.03, pp.721-760, 2015.
DOI : 10.1017/S0022112067002605

A. Duran and F. Marche, A discontinuous Galerkin method for a new class of Green???Naghdi equations on simplicial unstructured meshes, Applied Mathematical Modelling, vol.45, pp.840-864
DOI : 10.1016/j.apm.2017.01.030
URL : https://hal.archives-ouvertes.fr/hal-01303217

A. G. Filippini, M. Kazolea, and M. Ricchiuto, A flexible genuinely nonlinear approach for nonlinear wave propagation, breaking and run-up, Journal of Computational Physics, vol.310, pp.381-417
DOI : 10.1016/j.jcp.2016.01.027

D. R. Fuhrman and P. A. Madsen, Simulation of nonlinear wave run-up with a high-order Boussinesq model, Coastal Engineering, vol.55, issue.2, pp.139-154, 2008.
DOI : 10.1016/j.coastaleng.2007.09.006

S. Gavrilyuk and H. Gouin, Geometrical evolution of the Reynolds stress tensor. Int, 2012.

, J. Eng. Sci, vol.59, pp.65-73

A. E. Green and P. M. Naghdi, A derivation of equations for wave propagation in water of variable depth, Journal of Fluid Mechanics, vol.338, issue.02, pp.237-246
DOI : 10.1017/S0022112076002425

M. Hattori and T. Aono, Experimental Study on Turbulence Structures Under Breaking Waves, Coastal Engineering in Japan, vol.8, issue.1, pp.97-116, 1985.
DOI : 10.1080/05785634.1985.11924408

C. Higgins, M. B. Parlange, and C. Meneveau, Energy dissipation in large-eddy simulation: dependence on flow structure and effects of eigenvector alignments, Atmospheric turbulence and mesoscale meteorology Fedorovich, R. Rotunno & B. Stevens), pp.51-69, 2004.
DOI : 10.1017/CBO9780511735035.005

. Cambridge, C. Hinterberger, J. Fröhlich, and W. Rodi, Three-dimensional and depth-averaged large-eddy simulation of some shallow water flows A. 2015 CFD Simulations of Wave Propagation and Shoaling over a Submerged Bar, Kazakova, M. & Richard, G. L. 2018 A new model of shoaling and breaking waves, pp.857-872, 2007.

M. Kazolea, A. I. Delis, I. K. Nikolos, and C. Synolakis, An unstructured finite volume numerical scheme for extended 2D Boussinesq-type equations, E. 2014 Numerical treatment of wave breaking on unstructured finite volume approximations for extended Boussinesq-type equations, pp.42-66
DOI : 10.1016/j.coastaleng.2012.05.008

, J. Comput. Phys, vol.271, pp.281-305

M. Kazolea and M. Ricchiuto, On wave breaking for Boussinesq-type models. Ocean Model, pp.16-39, 2018.
DOI : 10.1016/j.ocemod.2018.01.003
URL : https://hal.archives-ouvertes.fr/hal-01698300

J. T. Kirby, G. T. Klonaris, C. D. Memos, and T. Karambas, Boussinesq models and their application to coastal processes across a wide range of scales 2013 A Boussinesq-type model including wave-breaking terms in both continuity and momentum equations, J. Waterway Port Coast. Ocean Eng. Ocean Eng, vol.142, issue.57, pp.128-140, 2016.

A. N. Kolmogorov and A. N. Kolmogorov, The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds Numbers, The equations of turbulent motion in an incompressible fluid, pp.299-303, 1941.
DOI : 10.1098/rspa.1991.0075

L. Krivodonova, J. Xin, J. F. Remacle, N. Chevaugeon, and J. Flaherty, Izvestia Akad. Sci. USSR Phys, vol.6, pp.56-58, 2004.

, Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws, Appl. Numer. Math, vol.48, pp.323-338

D. Lannes and F. Marche, A new class of fully nonlinear and weakly dispersive Green???Naghdi models for efficient 2 D simulations, Journal of Computational Physics, vol.282, pp.238-268
DOI : 10.1016/j.jcp.2014.11.016
URL : https://hal.archives-ouvertes.fr/hal-00932858

L. Métayer, O. Gavrilyuk, S. Hank, and S. , A numerical scheme for the Green???Naghdi model, Journal of Computational Physics, vol.229, issue.6, pp.2034-2045
DOI : 10.1016/j.jcp.2009.11.021

Q. Liang and F. Marche, Numerical resolution of well-balanced shallow water equations with complex source terms, Advances in Water Resources, vol.32, issue.6, pp.873-884, 2009.
DOI : 10.1016/j.advwatres.2009.02.010
URL : https://hal.archives-ouvertes.fr/hal-00799080

D. K. Lilly, The representation of small-scale turbulence in numerical simulation experiments, Proc. IBM Scientific Computing Symp. on Environmental Sciences, 1967.

:. I. Goldstine, P. Liu, Y. S. Cho, M. J. Briggs, U. Kanoglu et al., Yorktown Heights Runup of solitary waves on a circular island, J. Fluid Mech, vol.302, pp.195-210, 1995.

X. Lu, B. Dong, B. Mao, and X. Zhang, 2015 A two-dimensional depth-integrated non-hydrostatic numerical model for nearshore wave propagation. Ocean Model, pp.187-202
DOI : 10.1016/j.ocemod.2015.11.001

P. J. Lynett, T. S. Wu, and P. L. Liu, Modeling wave runup with depth-integrated equations, Coastal Engineering, vol.46, issue.2, pp.89-107, 2002.
DOI : 10.1016/S0378-3839(02)00043-1
URL : http://coastal.usc.edu/plynett/publications/Lynett - Modeling Wave Runup 2002 CE.pdf

P. A. Madsen, R. Murray, and O. R. Sørensen, A new form of the Boussinesq equations with improved linear dispersion characteristics, Coastal Engineering, vol.15, issue.4, pp.371-388, 1991.
DOI : 10.1016/0378-3839(91)90017-B

P. A. Madsen, O. R. Sørensen, and H. A. Schäffer, Surf zone dynamics simulated by a Boussinesq type model. Part I. Model description and cross-shore motion of regular waves, Coastal Engineering, vol.32, issue.4, pp.255-287, 1997.
DOI : 10.1016/S0378-3839(97)00028-8

K. Nadaoka and H. Yagi, Shallow-Water Turbulence Modeling and Horizontal Large-Eddy Computation of River Flow, Journal of Hydraulic Engineering, vol.124, issue.5, pp.493-500, 1998.
DOI : 10.1061/(ASCE)0733-9429(1998)124:5(493)

O. Nwogu, Alternative Form of Boussinesq Equations for Nearshore Wave Propagation, Journal of Waterway, Port, Coastal, and Ocean Engineering, vol.119, issue.6, 1993.
DOI : 10.1061/(ASCE)0733-950X(1993)119:6(618)

, J. Waterw. Port Coast, vol.119, pp.618-638

O. K. Nwogu, . Asce, T. Okamoto, and D. R. Basco, Numerical prediction of breaking waves and currents with a Boussinesq model The Relative Trough Froude Number for initiation of wave breaking: Theory, experiments and numerical model confirmation, Coastal Engineering Proc. 25th Int. Conf, pp.4807-4820, 1996.

D. H. Peregrin, Long waves on a beach, Journal of Fluid Mechanics, vol.13, issue.04, pp.815-827, 1967.
DOI : 10.1029/JZ071i002p00393

S. B. Pope, Turbulent Flows Stabilized residual distribution for shallow water simulations, J. Comput. Phys, vol.228, pp.1071-1115, 2000.

G. Richard,

, Application au ressaut hydraulique et aux trains de rouleaux

G. L. Richard and S. L. Gavrilyuk, The classical hydraulic jump in a model of shear shallow-water flows, Journal of Fluid Mechanics, vol.338, pp.492-521
DOI : 10.1017/jfm.2012.96
URL : https://hal.archives-ouvertes.fr/hal-01459449

G. L. Richard and S. L. Gavrilyuk, Modelling turbulence generation in solitary waves on shear shallow water flows, Journal of Fluid Mechanics, vol.8, pp.49-74
DOI : 10.1098/rspa.1974.0072
URL : https://hal.archives-ouvertes.fr/hal-01459155

L. F. Richardson, . Cambridge, V. Roeber, and K. F. Cheung, Weather Prediction by Numerical Process 2012 Boussinesq-type model for energetic breaking waves in fringing reef environments, Coast. Eng, vol.70, pp.1-20, 1922.

V. Roeber, K. F. Cheung, and M. H. Kobayashi, Shock-capturing Boussinesq-type model for nearshore wave processes, Coastal Engineering, vol.57, issue.4, pp.407-423, 2010.
DOI : 10.1016/j.coastaleng.2009.11.007

J. C. Rotta, Statistische Theorie nichthomogener Turbulenz. Z. Phys, vol.129, pp.547-572, 1951.

M. K. Sharifian, G. Kesserwani, and Y. Hassanzadeh, 2018 A discontinuous Galerkin approach for conservative modelling of fully nonlinear and weakly dispersive wave transformations . Ocean Model, pp.61-79

F. Shi, J. T. Kirby, J. C. Harris, J. D. Geiman, and S. T. Grilli, 2012 A highorder adaptive time-stepping TVD solver for Boussinesq modeling of breaking waves and coastal inundation. Ocean Model, General circulation experiments with the primitive equations: I, pp.36-51, 1963.

, The basic equations, Mon. Weather Rev, vol.91, pp.99-164

G. Stelling and M. Zijlema, An accurate and efficient finite-difference algorithm for non-hydrostatic free-surface flow with application to wave propagation, International Journal for Numerical Methods in Fluids, vol.36, issue.1, pp.1-23, 2003.
DOI : 10.1016/S0378-3839(99)00009-5

D. T. Swigler, Laboratory study investigating the three-dimensional turbulence and kinematic properties associated with a breaking solitary wave Texas A & M Uni- versity. Synolakis, C. E. 1987 The runup of solitary waves, J. Fluid Mech, vol.185, pp.523-545, 2009.

V. M. Teshukov, Gas-dynamic analogy for vortex free-boundary flows, Journal of Applied Mechanics and Technical Physics, vol.38, issue.3, pp.303-309, 2007.
DOI : 10.1007/s10808-007-0039-2

F. C. Ting and J. Kirby, Observation of undertow and turbulence in a laboratory surf zone, Coastal Engineering, vol.24, issue.1-2, pp.51-80
DOI : 10.1016/0378-3839(94)90026-4

M. Tissier, P. Bonneton, F. Marche, F. Chazel, and D. Lannes, A new approach to handle wave breaking in fully non-linear Boussinesq models, Coastal Engineering, vol.67, pp.54-66
DOI : 10.1016/j.coastaleng.2012.04.004
URL : https://hal.archives-ouvertes.fr/hal-00798996

V. V. Titov and C. E. Synolakis, Modeling of Breaking and Nonbreaking Long-Wave Evolution and Runup Using VTCS-2, Journal of Waterway, Port, Coastal, and Ocean Engineering, vol.121, issue.6, pp.308-316, 1995.
DOI : 10.1061/(ASCE)0733-950X(1995)121:6(308)

M. Tonelli and M. Petti, Finite volume scheme for the solution of 2D extended Boussinesq equations in the surf zone, Ocean Engineering, vol.37, issue.7, pp.567-582, 2010.
DOI : 10.1016/j.oceaneng.2010.02.004

M. Tonelli and M. Petti, Simulation of wave breaking over complex bathymetries by a Boussinesq model, Journal of Hydraulic Research, vol.19, issue.3, pp.473-486, 2011.
DOI : 10.1006/jcph.2000.6670

J. Veeramony and I. A. Svendsen, The flow in surf-zone waves, Coastal Engineering, vol.39, issue.2-4, pp.93-122, 2000.
DOI : 10.1016/S0378-3839(99)00058-7

G. Wei, J. T. Kirby, S. T. Grilli, and R. Subramanya, A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves, Journal of Fluid Mechanics, vol.107, issue.-1, pp.71-92, 1995.
DOI : 10.1063/1.865459

J. A. Zelt, The run-up of nonbreaking and breaking solitary waves, Coastal Engineering, vol.15, issue.3, pp.205-246, 1991.
DOI : 10.1016/0378-3839(91)90003-Y

Y. Zhang, A. B. Kennedy, A. S. Donahue, and J. J. Westerink,

C. Dawson, 2014 Rotational surf zone modeling for O(µ 4 ) Boussinesq-Green-Naghdi systems. Ocean Model, pp.43-53

Y. Zhang, A. B. Kennedy, N. Panda, C. Dawson, and J. J. Westerink, , 2013.

?. Boussinesq and . Green, Naghdi rotational water wave theory, Coast. Eng, vol.73, pp.13-27