Analysing two-dimensional shallow water equations with idealised bottom topographies have many applications in the atmospheric and oceanic sciences; however, restrictive flow pattern assumptions have been made to achieve explicit solutions. This work employs the Adomian decomposition method (ADM) to develop semi-analytical formulations of these problems that preserve the direct correlation of the physical parameters while capturing the nonlinear phenomenon. Furthermore, we exploit these techniques as reverse engineering mechanisms to develop key connections between some prevalent ansatz formulations in the open literature as well as derive new families of exact solutions describing geostrophic inertial oscillations and anticyclonic vortices with finite escape times. Our semi-analytical evaluations show the promise of this approach in terms of providing robust approximations against several oceanic variations and bottom topographies while also preserving the direct correlation between the physical parameters such as the Froude number, the bottom topography, the Coriolis parameter, as well as the flow and free surface behaviours. Our numerical validations provide additional confirmations of this approach while also illustrating that ADM can also be used to provide insight and deduce novel solutions that have not been explored, which can be used to characterise various types of geophysical flows.

分析具有理想化底部地形的二维浅水方程在大气和海洋科学中有许多应用；然而，已经做出限制性流动模式假设以获得明确的解决方案。这项工作采用 Adomian 分解方法 (ADM) 来开发这些问题的半解析公式，在捕获非线性现象的同时保留物理参数的直接相关性。此外，我们利用这些技术作为逆向工程机制来开发公开文献中一些流行的模拟公式之间的关键联系，并推导出描述地转惯性振荡和具有有限逃逸时间的反气旋涡流的新精确解族。我们的半分析评估显示了这种方法的前景，它可以针对几种海洋变化和底部地形提供稳健的近似值，同时还可以保留弗劳德数、底部地形、科里奥利参数等物理参数之间的直接相关性，以及作为流动和自由表面行为。我们的数值验证为这种方法提供了额外的确认，同时还说明 ADM 也可用于提供洞察力并推导出尚未探索的新解决方案，这些解决方案可用于表征各种类型的地球物理流动。科里奥利参数，以及流动和自由表面行为。我们的数值验证为这种方法提供了额外的确认，同时还说明 ADM 也可用于提供洞察力并推导出尚未探索的新解决方案，这些解决方案可用于表征各种类型的地球物理流动。科里奥利参数，以及流动和自由表面行为。我们的数值验证为这种方法提供了额外的确认，同时还说明 ADM 也可用于提供洞察力并推导出尚未探索的新解决方案，这些解决方案可用于表征各种类型的地球物理流动。