Tsunami waves transform into a series of solitary waves or undular bores over a mild slope. Nonlinear dispersion of water waves plays an important role in modeling tsunami wave propagation and deformation over varying bottom topography. This paper presents a numerical investigation on the shoaling and runup of multi-solitary waves using the fully nonlinear Boussinesq equations solved with a high-order adaptive time-stepping TVD solver (FUNWAVE-TVD). We first simulated 1-D solitary waves traveling up a slope and then validated with solitary wave runup datasets from well-controlled laboratory experiments. Numerical results show that the runup of multi-solitary waves with uniform initial amplitude over a 1:20 slope varies with each individual wave. Then, we extended the simulations of multi-solitary wave evolution and overtaking collisions over a slope for the cases of unequal initial wave amplitude. Throughout our study, we meticulously analyzed and discussed details of wave profiles and runups considering the effects of wave breaking.
Solitary waves, by definition, are waves that maintain their shape and speed while propagating through a medium, resisting the usual tendency of waves to disperse. These waves are characterized by their stability and localized nature, often traveling over long distances without changing their fundamental structure. Solitary waves have an inherent connection with tsunamis, as tsunamis result from the runup of solitary waves, making it crucial to comprehend the dynamics of these waves and their potential collisions. Notable instances, such as the 2004 Indian Ocean tsunami and the 2011 Japan Tohoku tsunami, underscore the severe threats posed by these natural disasters, causing substantial damage and loss of life in coastal communities. Tsunamis present intrinsic dangers; therefore, analyzing the dynamics of solitary wave runup and collisions is necessary to reveal the nearshore evolution mechanism of tsunami waves. Boussinesq equations have been applied to model solitary wave runups (Kennedy et al., 2000; Zhao, 2011), but how to deal with wave breaking, sponge layer, and shock-capturing (Choi et al., 2018) in an efficient way has been a challenge for all numerical schemes.
