Abstract In contrast to two-dimensional oceanic internal waves which have been extensively investigated, there are relatively few theoretical studies on three-dimensional internal waves. The most remarkable theory describing three-dimensional internal waves is the Kadomtsev–Petviashvili (KP) equation. Nevertheless, two shortcomings – unidirectional propagation and anisotropy – limit its application in some general cases. In the current paper, via an asymptotic analysis, we derive an isotropic and bidirectional model, the modified Benney–Luke equation, for a two-layer fluid with bottom topography which can vary both in time and space, therefore tides and currents can be incorporated into the internal wave problem due to the relative motion. In the derivation, the assumption of incompressibility and long waves are invoked and the effects of the Earth’s rotation and dissipation are ignored. Based on this model, the generation of nonlinear internal waves by background currents and barotropic tides flowing over topography are investigated. The resulting equation aims to facilitate dynamical analyses, as well as the interpretation of in-situ observational data and laboratory experimental results.

中文翻译：

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内波各向同性模型的推导及其在波浪产生中的应用

摘要 相对于已被广泛研究的二维海洋内波，关于三维内波的理论研究相对较少。描述三维内波最显着的理论是 Kadomtsev-Petviashvili (KP) 方程。然而，两个缺点——单向传播和各向异性——限制了它在一些一般情况下的应用。在当前的论文中，通过渐近分析，我们推导出了一个各向同性和双向模型，即修正的 Benney-Luke 方程，适用于底部地形在时间和空间上都可以变化的两层流体，因此可以合并潮汐和洋流由于相对运动而进入内波问题。在推导中，调用了不可压缩性和长波的假设，而忽略了地球自转和耗散的影响。在此模型的基础上，研究了背景电流和流经地形的正压潮汐产生的非线性内波。由此产生的方程旨在促进动力学分析，以及对现场观测数据和实验室实验结果的解释。