Boussinesq modelling with higher-order dispersion; derivation and numerical modelling

In this thesis the modelling of water wave propagation over uneven bottoms using Boussinesq-like models with higher-order frequency dispersion is studied. Boussinesq-like equations describe the propagation of weakly non-linear shallow water waves. As for long waves the depth-dependence of the velocity field is almost absent, the vertical coordinate z has been removed in deriving Boussinesq-like models. A problem with Boussinesq-like models which were in use some time ago, is that they do not have particularly good frequency dispersion characteristics, especially not at depths where practical problems had to be solved. Therefore, many efforts have been spent to improve the dispersion characteristics. Dingemans (1994b) has started with a higher-order dispersion relation written in the form of a rational polynomial as obtained by Schaffer and Madsen (1994). Using operator correspondence four different Boussinesq-like models with only third and lower derivatives can be derived for horizontal bottom. Uneven bottom terms can be obtained with some mathematical operations. In this manner a total number of 48 different models is obtained each with four degrees of freedom. By comparing the linear shoaling coefficient with the exact linear shoaling coefficient the degrees of freedom are optimized. It turned out that 24 models have exactly the same linear shoaling coefficient, which is very accurate up to kh=5. However, the models (may) differ in higher-order shoaling behaviour. The higher-order shoaling is invesitigated by solving the models numerically using the Keller's Box compact difference scheme. This scheme is implicit and therefore unconditionally stable and it has a second-order accuracy. The procedure is to rewrite the set of two third-order partial differential equations to a set of six first-order partial differential equations by introducing four more variables. The system of linear finite difference equations has block-diagonal structure, which fits in a 17 diagonal matrix and is solved by a Thomas algorithm. Subsequently, a weakly reflecting boundary condition is formulated by the Sommerfeld radiation condition for the classical shallow water equations together with a sponge layer resulting in a reflection coefficient of 0(10-3). The numerical models are validated against the measurements used in the intercomparison study of Dingemans (1994a). It turned out that the best correspondence with measurements is obtained by four models with essentially the same accuracy. It is concluded that the new Boussinesq-like equations with higher-order dispersion has increased the applicability of Boussinesq modelling for wave propagation over uneven bottoms and that Keller's Box method is a very suitable method for integrating Boussinesqlike equations.