A Boussinesq model for wave transformation in coastal regions, with applications to a submerged coral reef

Abstract

Based on a set of Boussinesq-type equations with improved linear dispersion characteristics in deeper water the first part of the thesis describes the development of a computational model in a single horizontal dimension (2-D). The model can be used to simulate the evolution of relatively long, weakly nonlinear waves in water of constant or variable depth provided the bed slope is of the same order of magnitude as the ratio of the mean water depth and a typical wave length. The numerical solution method is based on the finite difference method and the computations are advanced in time by using a fourth-order accurate predictor-corrector method. A special technique is employed which allows the incident wave field to be generated inside the computational domain. A Fourier method is used to prescribe a form of the incident regular wave field which satisfies the governing equations on a horizontal bottom. Scattered waves leaving the fluid domain are absorbed in the vicinity of the model boundary by employment of damping terms in the mass and momentum equations. This ensures that the wave reflection from the boundary is insignificant.The phcise and amplitude portraits of the numerical solution are considered, and examples are given illustrating that the model conserves well basic properties such as the total mass and energy within the computational domain. The model is used to study the transformation of waves in water of variable depth. The results compare well with both existing laboratory measurements and analytical theory.For practical simulations, e.g. wave evolution inside a proposed harbour, a numerical model is often required which covers two horizontal dimensions (3-D). Consequently, the model is extended to include the second horizontal dimension. Since the formulation is very general, waves can be propagated in virtually any geometry. The analytical manipulations required to generate the incident wave field internally become quite substantial in a formulation covering two horizontal dimensions, and the wave generation concept is therefore generalized and implemented in a simple and efficient way. A number of computational examples are given. These serve as a partial verification of the model. The second part of the thesis considers the effect of spilling wave breaking and the development of waves in the surf zone. The effect of spilling wave breaking is incorporated into the two-dimensional model using the concept of surface rollers. Based on the assumption of a vertical redistribution of the horizontal velocity in a breaking wave a new set of equations is derived. The temporal development of the surface roller thickness is determined heuristically using an existing method. Although the mathematical basis is rather weak and the physical description is very crude the model has the potential to describe a variety of processes such as the fluctuating breaking point caused by random waves breaking on a beach and the important conversion of potential energy to kinetic energy in the outer region of the s...