Traveling internal waves in a two-layer shallow medium with variable bathymetry and current: Surface and near-bottom flows

Abstract

In the linear approximation, we examine the one-dimensional problem of long internal wave propagation without reflection in a stationary two-layer flow with a free boundary in a channel of variable depth and width, in the limiting cases of surface and near-bottom currents. We use the shallow-water approximation and assume that liquids in the layers are ideal, immiscible, and having a small relative density difference, and that the flow is only in one of the layers. To filter fast processes (such as surface gravity waves), the flow velocity is assumed to have the same order of magnitude as the speed of internal waves. Both surface and near-bottom flows are divided into three classes, in accordance with conditions providing wave propagation without reflection. The properties of the flows in each of the classes are examined, and it is shown that the global reflectionless near-bottom flows do exist. The main results and conclusions are illustrated by a few particular solutions. A detailed comparison with the first part of the study concerning the flows with currents in both layers made it possible to clarify and refine some of the results obtained there, and to formulate questions for further investigation. The results obtained may be of interest for understanding of those natural processes in surface and near-bottom flows, which are significantly influenced by internal waves.