Numerical Modeling of the Influence of the Beach Profile on Wave Run-Up

An analysis of the run-up over different beach profiles is performed to evaluate the influence of the seabed shape on shore flooding. The analysis was carried out on the basis of numerical solutions of the nonlinear shallow water equations. The chosen solver was shown to provide reliable (both quantitatively and qualitatively) run-up results by comparing numerical solutions (of both solitary and regular waves) with the only available analytical solution forced by a localized topographic change. The run-up patterns on both a natural beach profile and three simpler and schematic profiles derived from it were evaluated. Different wave conditions (both random and groups) were used for a total amount of 96 different cases of inundation. Results are expressed in terms of both maximum ($Z_{\max }$) and steady-state ($Z_{\mathrm{steady}}$) run-up. It is found that both types of run-up depend on the offshore variable $\sqrt{H_0{L}_0}$, as suggested by several available studies, and that, for all tested cases, random waves induce the largest $Z_{\max }$ compared with the wave groups. The largest $Z_{\max }$ is induced by the composite-planar profile for both random waves and groups. An important similarity is found for both $Z_{\max }$ and $Z_{\mathrm{steady}}$ induced by all wave types over the natural and equilibrium profiles, giving further support to the use of an equilibrium profile as representative of the natural profile. Attempts at finding an equivalent planar beach highlight unavoidable difficulties in choosing such a profile and reinforce the idea that the concept of an equivalent planar beach cannot bridge dynamics occurring from the depth of closure to the run-up. More likely, such a concept is only suitable to describe dynamics that are local to the foreshore.