A double-layer Boussinesq-type model for highly nonlinear and dispersive waves

Abstract

28 pages, 5 figures. Soumis à Proceedings of the Royal Society of London A.We derive and analyze in the framework of the mild-slope approximation a new double-layer Boussinesq-type model which 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 Agnon et al. (Agnon et al. 1999, J. Fluid Mech., 399 pp. 319-333) and enhanced by Madsen et al. (Madsen et al. 2003, Proc. R. Soc. Lond. A, 459 pp. 1075-1104) which consists in 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 2004, Proc. R. Soc. Lond. A, 460 pp. 2637-2669) allowing to lower 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 deep water. Finally, numerical simulations are performed to quantify the nonlinear behaviour of the model, and the results exhibit a nonlinear range of validity reaching deep water areas