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 O ( μ 2 ) and weakly nonlinear O ( μ N ) are presented, the analytical and numerical properties of O ( μ 2 ) and O ( μ 4 ) 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 O ( μ 2 ) model has dispersion accuracy equivalent to a Padé [2,2] approximation with one extra free-parameter. The optimal O ( μ 4 ) 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 O ( μ 4 ) model shows excellent agreement to experimental data.