Modelling wave transformation over coastal profiles with a nonlinear and dispersive potential model, and application to freak waves in variable water depth

Michel Benoit
Institut de Recherche sur les Phénomènes Hors Equilibre

With the objective of modeling coastal wave dynamics taking into account nonlinear and dispersive effects in variable bottom conditions, an accurate nonlinear potential flow model is developed. The model is based on the time evolution of two surface quantities: the free surface position and the free surface velocity potential (Zakharov, 1968). Following Tian and Sato (2008) a spectral approach is used to resolve vertically the velocity potential in the whole domain, by decomposing the potential using the orthogonal basis of Chebyshev polynomials.

The mathematical theory and numerical development are described, and the model is then validated with the application of several 1DH test cases including (i) the propagation of nonlinear regular wave over a submerged bar (experiments by Dingemans (1994)), (ii) the transformation of nonlinear irregular waves over a barred beach (experiments by Becq-Girard et al. (1999)), (iii) the generation of nonlinear harmonics on a submerged, with comparisons to small-scale experiments currently performed at ESPCI (Paris). All test cases results agree well with experimental data, confirming the model's ability to simulate accurately nonlinear and dispersive waves.

The model is then applied to study the formation of freak waves in coastal waters, with attention paid to the effect of the bottom slope on the statistics of these extreme waves. The simulation results are compared with laboratory experiments performed in a wave flume by Kashima et al. (2103). In particular, the reproduction of the most extreme waves in the experimental records is studied. The distribution of extreme wave heights is compared to existing formulas, such as Mori & Janssen (2006). The evolution of statistical parameters from deep to shallow water conditions is also analyzed, in particular the high-order statistical moments: skewness and kurtosis.


Benoit M., Raoult C., Yates M.L. (2014) Fully nonlinear and dispersive modelling of surf zone waves: non-breaking tests. In Coastal Engineering Proceedings, (ICCE'2014), 15-20 July 2014, Seoul (Korea), 1(34), waves.15, DOI: 10.9753/icce.v34.waves.15.

Raoult C., Benoit M., Yates M.L. (2016) Validation of a fully nonlinear and dispersive wave model with laboratory non-breaking experiments. Coastal Engineering. DOI : 10.1016/j.coastaleng.2016.04.003

Yates M.L., Benoit M. (2015) Accuracy and efficiency of two numerical methods of solving the potential flow problem for highly nonlinear and dispersive water waves. International Journal for Numerical Methods in Fluids. Vol. 77(10), pp 616-640. DOI: 10.1002/fld.3992