Nonlinear effects of variable bathymetry and free surface on mini-tsunamis generated by a moving ship

The mini-tsunami phenomenon denotes a newly observed upstream wave generated by a ship traveling across a depth change at subcritical speed. In this study, a high-order spectral model with variable bathymetry (HOS-VB) is developed to study the effects of bottom and free-surface nonlinearities on the generation and propagation of mini-tsunamis over a range of depth-based Froude number Fr0, average water depth to ship draft ratio h₀/d, bathymetric slope variationγ, and channel width to ship width ratio W/B. The analyses show that the wave amplitude of the mini-tsunamis is increased by both bottom and free-surface nonlinearities and that the influence of free-surface nonlinearities on the wave amplitude is more significant in a narrower channel. Second-order free-surface nonlinearities provide the most important contribution. Compared with linear results, second-order free-surface nonlinearities can more than double the maximum wave amplitude and induce a substantial increase in the mean free-surface level for a narrow channel at large Froude number. The wavelength at half-crest height is also increased by second-order free-surface nonlinearities, and this effect is enhanced as h₀/d decreases. Moreover, the increasing trend of wave amplitude withFr0and the decreasing trend of wave amplitude with h₀/d are more significant as the free-surface nonlinearity increases.