A pressure-Poisson based Boussinesq-type phase resolving wave model

Understanding the hydraulic response during high energy events such as hurricanes and tsunamis is a problem of significant importance to coastal communities, as well as industries which have infrastructure along coastal waters. Due to the large range of temporal and spatial scales present during these events it isn't possible to fully study them in a laboratory setting and thus computational models are the standard approach to simulating hurricanes and tsunamis. The current suite of ocean circulation models used in the coastal hydraulics community solve the hydrostatic depth--averaged shallow water equations (SWE) which are adept at representing the hydraulic response to the hurricane wind and pressure forces and the ensuing storm surge runup along the coast but are unable to capture the higher frequency gravity and infragravity waves that are generated nearshore. Through the use of Boussinesq scaling a model is developed for resolving non--hydrostatic pressure profiles in nonlinear wave systems over varying bathymetry. In contrast to standard Boussinesq--type models, the model developed here focuses on solutions to the well known pressure-- Poisson problem, with the advantage that the pressure--Poisson equations are a stand alone set of equations which do not exhibit mixed space/time derivatives. The result is a Boussinesq--type pressure--Poisson model (PPBOUSS) that can be solved in a separate module and then coupled back into the SWE model. This allows for the straightforward modification of established SWE solvers to turn them into fully resolved nearshore models with very little structural change to the underlying SWE model and without the need to solve mixed space/time derivatives. Use of a Green-- Naghdi type polynomial expansion for the pressure profile in the vertical axis reduces the dimensionality of the pressure--Poisson problem to only two dimensions, significantly reducing computational cost. The resulting model shows rapid convergence properties with increasing order of polynomial expansion which can be greatly improved through the application of asymptotic rearrangement. An optimum choice of basis functions in the Green--Naghdi expansion provides significant improvements in the dispersion, shoaling and nonlinear properties of the model, for example achieving fourth order dispersive accuracy in a formally second order model, and eighth order dispersive accuracy in a formally fourth order model. Various numerical approaches to solving and coupling the PPBOUSS model are discussed including application of finite difference and finite element methods. Demonstration of the improvement in nearshore accuracy through application of the PPBOUSS model is shown through coupling with the unstructured mesh Discontinuous Galerkin Shallow Water Equation Model (DGSWEM). A straightforward numerics based wave breaking algorithm is employed in the nearshore and wave runup is captured using a globally mass conservative wetting/drying algorithm specifically designed for discontinuous Galerkin finite element models. The model is verified and validated using analytical and experimental results for both the fully nonlinear O(mu2) and weakly nonlinear O(mu4) implementations.