In this paper, a modification of the standard geophysical equatorial β-plane model equations, incorporating a gravitational-correction term in the tangent plane approximation, is derived. We present an exact solution to meet the modified governing equations, whose form is explicit in the Lagrangian framework and which represents internal oceanic waves in the presence of a constant underlying current. It is rigorously established, that the solution is dynamically possible, by way of analytical and degree-theoretical considerations. In the sense that the mapping it prescribes from Lagrangian to Eulerian coordinates is a global diffeomorphism. In addition, the paper an analysis of the mean flow velocities and related mass transport are presented in this paper, they are induced by certain geophysical internal waves. In particular, we examine an exact solution to the geophysical governing equations in the modified β-plane approximation at the equator which incorporates a constant underlying current.

中文翻译：

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修正赤道 β 平面近似下有潜在流的地球物理内波的精确解

在本文中，导出了标准地球物理赤道 β 平面模型方程的修正，在切平面近似中加入了重力校正项。我们提出了一个精确解来满足修改后的控制方程，其形式在拉格朗日框架中是明确的，并且在存在恒定潜在流的情况下表示内部海浪。通过分析和程度理论的考虑，严格确定该解决方案是动态可能的。从这个意义上说，它规定从拉格朗日坐标到欧拉坐标的映射是全局微分同胚。此外，本文还分析了平均流速和相关的质量传输，它们是由某些地球物理内波引起的。特别是，