Evolution of tsunami-like isolated waves shoaling over a plane slope

Abstract

Numerical experiments are conducted to simulate Gaussian-shaped waves shoaling over a uniformly sloping bed. A higher-order pseudo-spectral method is implemented to solve the fully nonlinear Euler water-wave problem for a variable depth. Unlike a solitary wave, the amplitude and wave breadth of a Gaussian waveform can be assigned independently. Therefore, it is convenient to use a Gaussian as a generic tsunami waveform. Although the rate of shoaling is consistent with Green's law only for small-amplitude waves propagating over intermediate sloping beds, the shoaling processes, in general, are complex and the behaviors cannot be accurately predicted by a single law. Several factors are involved in the complexity. When the slope is sufficiently mild, a Gaussian wave transforms into a series of solitons, hence leading to the adiabatic shoaling process. In some instances, the initial rate of shoaling even surpasses the adiabatic rate. This occurs when a soliton emerges from the initial conditions with an amplitude greater than the original Gaussian wave amplitude. In general, there is a trend of a faster shoaling rate when the nonlinearity effect of the Gaussian is greater than that of the corresponding solitary wave, whereas the rate of shoaling for Gaussian waves is slower when the frequency dispersion is greater than that of the solitary wave. The model is further applied to simulate the 2011 Heisei tsunami event, and the shoaling behavior is analyzed by comparing the numerical results with the recorded data in the field.