Modulation effect of linear shear flow on interfacial waves in a two-layer fluid with finite layer depths

Abstract

The modulation effect of linear shear flow (LSF) comprising uniform and shear flows with constant vorticity on interfacial waves in a two-layer fluid with finite layer depths is studied. Herein, lower-layer LSF is focused on. A nonlinear Schrödinger equation (NLSE) modified by lower-layer LSF is derived. By comparing and analyzing the dispersion relation of upper- and lower-layer LSFs, sufficient conditions of Kelvin–Helmholtz stability are afforded to ensure the applicability of NLSE. Based on the relationship between the modulational instability (MI) of NLSE and interfacial freak waves (IFWs) represented as Peregrine breather, existence conditions of IFWs affected by LSF are presented. The convergence effect of the flow field against the wave propagation direction, e.g., uniform up-flow and positive (negative) vorticity of the lower (upper) layer, increases the MI growth rate. This subsequently increases the wave height and promotes IFW generation. However, the divergent effect of the flow field along the wave propagation direction, including uniform down-flow and negative (positive) vorticity of lower (upper) layer, inhibits IFW generation. Moreover, eight kinds of LSFs are presented under uniform flow and vorticity, among which two promote generation, two inhibit generation, and the remaining four depend on the counteracting effect between uniform flow and vorticity.