Hawassi-ab modelling and simulation of fully dispersive nonlinear waves above bathymetry

Abstract

SummaryWater waves propagating from the deep ocean to the coast show large changes in the profile, wave speed, wave length, wave height and direction. The fascinating processes of the physical wave phenomena give challenges in the study of water waves. The motion can exhibit qualitative differences at different scales such as deep water versus shallow water, long waves versus short waves. Therefore, the existing mathematical models are restricted to the limiting cases. This dissertation concerns the development of an accurate and efficient model that can simulate wave propagation in any range of wave lengths, in any water depth and moreover can deal with various inhomogeneous problems such as bathymetry and walls, leading to wave structure interactions.The derivation of the model is based on a variational principle of water waves. The resulting dynamic equations are of Hamiltonian form for wave elevation and surface potential with non-local operators applied to the canonical surface variables. The Hamiltonian is the total energy, i.e the sum of kinetic energy and potential energy. Since the kinetic energy cannot be expressed explicitly in the basic variables an approximation is required. The corresponding approximated Hamiltonian leads to approximated Hamilton equations.The approximate Hamilton equations are expressed in pseudo-differential operators applied to the surface variables. The pseudo-differential operator has a physical interpretation related to the phase velocity. The phase velocity as function of wave length is specified by a dispersion relation. Dispersion is one of the most important physical properties in the description of water waves. Accurate modelling of dispersion is essential to obtain high-quality wave propagation results.Using spatial-spectral methods and a straightforward numerical implementation, accurate and fast performance of the model can be obtained. Moreover, the spatialspectral implementation with the global pseudo-differential operators or a generalization with global Fourier integral operators (FIO) can retain the exact dispersion property of the model. Other numerical implementations with local differential operators such as finite difference or finite element methods require that the dispersion is approximated by an algebraic function. Such an approximation leads to restrictions on the range of wave lengths that are modelled correctly.To deal with practical applications, several extensions of the model are impleviii mented. The model with localization methods in the global FIO can deal with localized effects such as breaking waves, partially or fully reflective walls, submerged bars, run-up on shores, etc. The inclusion of a fixed-structure in the spatial-spectral setting is a challenging task. The method as presented here perhaps serves as a first contribution in this topic. An extended eddy viscosity breaking model and a breaking kinematic criterion are used for the wave breaking mechanism. The extended eddy viscosity breaking model can deal with fully dispersive waves. The...

Freak wave

“…This is explainable as in the bi-chromatic wave experiment many breaking waves occur that are not accounted for in the nonlinear AB equation. The model and laboratory experiments about breaking waves are discussed in detail by Kurnia [2016], Kurnia et al [2015].…”