Free-surface effects induced by internal solitary waves (ISWs) propagating through a background shear current are theoretically investigated in a shallow water framework. Following the Hamiltonian approach, we implement a mathematical formulation valid for a free surface, two-layer stratified system in the presence of mobile layers. Associated to the undisturbed condition characterized by the background fluid at rest, solitons with both positive and negative polarities are considered. To reproduce the typical oceanic conditions, we focus on theoretical predictions for a density ratio set to 0.99. Although under Boussinesq conditions a rigid-lid at the top of the theoretical domain is considered a good approximation, our analysis shows that main ISWs features may change when the fluid system, forced by shear currents, is modeled with a free surface. Signs assumed by three dimensionless quantities, i.e., the deformation parameter, the background velocity, and the undisturbed amplitude, allowed us to uniquely predict how the undisturbed solitons modify their interfacial profiles and change their celerity, in response to the background forcing. Our results show that waves celerity predicted by the rigid-lid model can be lower than the one resulting from the Hamiltonian formulation, although this never occurs in the absence of mobile layers. Theoretical predictions reveal that shearing current can induce deviations from the standard ISWs free-surface profile. We discuss the role of nonhydrostatic pressures in inducing free-surface short disturbances, in the form of undulated or multihumped profiles. The typical phase-opposition between interfacial and surface displacements can be lost as the free-surface displacements assume multihumped configurations.

中文翻译：

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由剪切电流强迫的内部孤子引起的自由表面效应

在浅水框架中理论上研究了由通过背景剪切流传播的内部孤立波 (ISW) 引起的自由表面效应。遵循哈密顿方法，我们实现了一个数学公式，该公式适用于存在移动层的自由表面、两层分层系统。与以静止背景流体为特征的未受干扰条件相关，考虑了具有正极性和负极性的孤子。为了重现典型的海洋条件，我们专注于密度比设置为 0.99 的理论预测。尽管在 Boussinesq 条件下，理论域顶部的刚性盖被认为是一个很好的近似值，但我们的分析表明，当受剪切流强制的流体系统用自由表面建模时，主要的 ISW 特征可能会发生变化。由三个无量纲量假设的符号，即变形参数、背景速度和未受干扰的振幅，使我们能够唯一地预测未受干扰的孤子如何修改它们的界面轮廓并改变它们的速度，以响应背景强迫。我们的结果表明，刚性盖模型预测的波速可能低于哈密顿公式所产生的波速，尽管这在没有移动层的情况下永远不会发生。理论预测表明，剪切电流会导致与标准 ISW 自由表面轮廓的偏差。我们讨论了非静水压力在以起伏或多峰剖面的形式诱导自由表面短扰动中的作用。