Implementation of a fully nonlinear Hamiltonian Coupled-Mode Theory, and application to solitary wave problems over bathymetry

Abstract

This paper deals with the implementation of a new, efficient, non-perturbative, Hamiltonian coupled-mode theory (HCMT) for the fully nonlinear, potential flow (NLPF) model of water waves over smooth, single-valued, but otherwise arbitrary bathymetry Papoutsellis and Athanassoulis (2017). Applications considered herein concern the interaction of solitary waves with bottom topographies and vertical walls both in two- and three-dimensional environments. The essential novelty of HCMT is a new representation of the Dirichlet-to-Neumann operator, which is needed to close the Hamiltonian evolution equations. This new representation emerges from the treatment of the substrate kinematical problem by means of exact semi-separation of variables in the instantaneous, irregular, fluid domain, established recently by Athanassoulis and Papoutsellis (2017). The HCMT ensures an efficient dimensional reduction of the exact NLFP, being able to treat a variable bathymetry as simply as the flat-bottom case, without domain transformation. A key point for the efficient implementation of the method is the fast and accurate evaluation of the space–time varying coefficients appearing in some of its equations. In this paper, all varying coefficients are calculated analytically, resulting in a refined version of the theory, characterized by improved accuracy at significantly reduced computational time. This improved version of HCMT is first validated against existing experimental results and other computations, and subsequently applied to new solitary wave-bottom interaction problems. The latter include: (i) the investigation of a new type of “Bragg scattering” effect, appearing when a solitary wave propagates over a seabed with a sinusoidal patch, and (ii) the disintegration, focusing and reflection of a solitary wave moving over a three-dimensional bathymetry consisting of parallel banks and troughs, and impinging on a vertical wall.