Interaction of a large amplitude interfacial solitary wave of depression with a bottom step

Abstract

This paper is devoted to the study of the transformation of a finite-amplitude interfacial solitary wave of depression at a bottom step. The parameter range studied goes outside the range of weakly nonlinear theory (the extended Korteweg–de Vries or Gardner equation), and we describe various scenarios of this transformation in terms of the incident wave amplitude and the step height. The dynamics and energy balance of the transformation are described. Several numerical simulations are carried out using the nonhydrostatic model based on the fully nonlinear Navier–Stokes equations in the Boussinesq approximation. Three distinct runs are discussed in detail. The first simulation is done when the ratio of the step height to the lower layer thickness after the step is about 0.4 and the incident wave amplitude is less than the limiting value estimated for a Gardner solitary wave. It shows the applicability of the weakly nonlinear model to describe the transformation of a strongly nonlinear wave in this case. In the second simulation, the ratio of the step height to the lower layer thickness is the same as that in the first run but the incident wave amplitude is increased and then its shape is described by the Miyata–Choi–Camassa solitary wave solution. In this case, the process of wave transformation is accompanied by shear instability and the billows that result in a thickening of the interface layer. In the third simulation, the ratio of the step height to the thickness of the lower layer after the step is 1.33, and then the same Miyata–Choi–Camassa solitary wave passes over the step, it undergoes stronger reflection and mixing between the layers although Kelvin–Helmholtz instability is absent. The energy budget of the wave transformation is calculated. It is shown that the energy loss in the vicinity of the step grows with an increase of the ratio of the incident wave amplitude to the thickness of the lower layer over the step.