A Boussinesq-scaled, pressure-Poisson water wave model

Through the use of Boussinesq scaling we develop and test a model for resolving non-hydrostatic pressure profiles in nonlinear wave systems over varying bathymetry. A Green–Nagdhi type polynomial expansion is used to resolve the pressure profile along the vertical axis, this is then inserted into the pressure-Poisson equation, retaining terms up to a prescribed order and solved using a weighted residual approach. The model shows rapid convergence properties with increasing order of polynomial expansion which can be greatly improved through the application of asymptotic rearrangement. Models of Boussinesq scaling of the fully nonlinear Oðl2Þ and weakly nonlinear OðlNÞ are presented, the analytical and numerical properties of Oðl2Þ and Oðl4Þ models are discussed. Optimal basis functions in the Green–Nagdhi expansion are determined through manipulation of the free-parameters which arise due to the Boussinesq scaling. The optimal Oðl2Þ model has dispersion accuracy equivalent to a Padé [2,2] approximation with one extra free-parameter. The optimal Oðl4Þ model obtains dispersion accuracy equivalent to a Padé [4,4] approximation with two free-parameters which can be used to optimize shoaling or nonlinear properties. In comparison to experimental results the Oðl4Þ model shows excellent agreement to experimental data. 2014 The Authors. Published by Elsevier Ltd. This is anopenaccess article under the CCBY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).