Conditions for extreme wave runup on a vertical barrier by nonlinear dispersion

The runup of long strongly nonlinear waves impinging on a vertical wall can exceed six times the far-field amplitude of the incoming waves. This outcome stems from a precursory evolution process in which the wave height undergoes strong amplification due to the combined action of nonlinear steepening and dispersion, resulting in the formation of nonlinearly dispersive wave trains, i.e. undular bores. This part of the problem is first analysed separately, with emphasis on the wave amplitude growth rate during the development of undular bores within an evolving large-scale background. The growth of the largest wave in the group is seen to reflect the asymptotic time scaling provided by nonlinear modulation theory rather closely, even in the case of fully nonlinear evolution and moderately slow modulations. In order to address the effect of such a dynamics on the subsequent wall runup, numerical simulations of evolving long-wave groups are then carried out in a computational wave tank delimited by vertical walls. Conditions for optimal runup efficiency are sought with respect to the main physical parameters characterizing the incident waves, namely the wavelength, the length of the propagation path and the initial amplitude. Extreme runup is found to be strongly correlated to the ratio between the available propagation time and the shallow-water nonlinear time scale. The problem is studied in the twofold mathematical framework of the fully nonlinear free-surface Euler equations and the strongly nonlinear Serre-Green-Naghdi model. The performance of the reduced model in providing accurate long-time predictions can therefore be assessed.