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.

中文翻译：

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在任意层深的两层流体中，上层的线性剪切流调制的界面波

表面和内部的畸变波都可以被认为是特殊的界面波。使用两层模型，我们研究了上层中的线性剪切流（LSF）对界面波的影响。特别是，该模型旨在研究风切变对表面畸形波的影响以及LSF对内部畸形波的影响。基于该模型，推导了非线性Schrödinger方程来描述界面波的演化。通过分析调制不稳定性，可以确定发生界面畸形波的不稳定区域。根据这些不稳定区域，使用Peregrine Breather解决方案研究了界面畸形波的升高。研究发现，在超临界向上流动过程中，表面畸变波的陡度和高度在正涡度下减小，而在负涡度下增加。相反，在超临界向下流动期间，它们在正涡度下增加而在负涡度下减少。原因是，对波具有收敛作用的负涡旋在超临界向上流动时容易激发表面畸变波，而在超临界向下流动时正涡旋具有收敛作用。此外，内部畸变波的陡度和高度在正涡度和均匀下流时减小，而在负涡度和均匀上流时增加。负涡度和均匀上升流的收敛效应促进了内部畸形波的产生。原因是，对波具有收敛作用的负涡旋在超临界向上流动时容易激发表面畸变波，而在超临界向下流动时正涡旋具有收敛作用。此外，内部畸变波的陡度和高度在正涡度和均匀下流时减小，而在负涡度和均匀上流时增加。负涡度和均匀上升流的收敛效应促进了内部畸形波的产生。原因是，对波具有收敛作用的负涡旋在超临界向上流动时容易激发表面畸变波，而在超临界向下流动时正涡旋具有收敛作用。此外，内部畸变波的陡度和高度在正涡度和均匀下流时减小，而在负涡度和均匀上流时增加。负涡度和均匀上升流的收敛效应促进了内部畸形波的产生。