Comparison Between Boussinesq and Shallow-Water Models in Predicting Solitary Wave Runup on Plane Beaches

Abstract

The nonlinear shallow water equations and Boussinesq equations have been widely used for analyzing the solitary wave runup phenomenon. In order to quantitatively assess the merits and limitations of these two approaches, a shock-capturing scheme has been applied to numerically solve both sets of equations for predicting the runup processes over plane beaches. The analytical and experimental data available in the literature have been used as references in the assessment. In this study, the uniform sloping beach is preceded by a length of flat seafloor. When incident solitary waves are specified close to the slope, the two approaches are found to produce almost identical results for nonbreaking waves. For breaking waves, the Boussinesq equations give a better representation of the wave evolution prior to the breaking point, whereas they overestimate the short undulations accompanying the breaking process. If incoming solitary waves need to travel a long distance over a flat bed before reaching the slope, the shallow water equations cannot capture the correct waveform transformation, and the predicted runup depends heavily on the length of the flat-bed section. As this length approaches zero, however, the shallow water equations somehow give roughly the same maximum runup heights as those predicted by the Boussinesq model. The bed friction has little effect on the runup for small waves, but becomes important for large waves. The roughness coefficient needs to be calibrated to reproduce the measured runup heights of breaking waves.