Interfacial waves modulated by linear shear flow of the upper layer in a two-layer fluid with arbitrary layer depths

Abstract

Both surface and internal freak waves can be regarded as special interfacial waves. Using a two-layer model, we investigated the influence of linear shear flow (LSF) in the upper layer on interfacial waves. Specially, the model was designed to study the effects of wind shear on surface freak waves and LSF on internal freak waves. Based on the model, a nonlinear Schrödinger equation was derived to describe interfacial-wave evolution. The unstable regions where interfacial freak waves occur were identified via analysis of modulational instability. According to these unstable regions, the elevation of interfacial freak waves was studied using the Peregrine Breather solution. It is found that the steepnesses and heights of surface freak waves decrease under positive vorticity and increase under negative vorticity during supercritical up-flow. In contrast, they increase under positive vorticity and decrease under negative vorticity during supercritical down-flow. The reason is that negative vorticity which has a convergent effect on the waves is easy to excite surface freak waves under supercritical up-flow, whereas positive vorticity has a convergent effect under supercritical down-flow. In addition, the steepnesses and heights of internal freak waves decrease under positive vorticity and uniform down-flow, whereas increase under negative vorticity and uniform up-flow. The convergent effect of negative vorticity and uniform up-flow promote the generation of internal freak waves.