Semi-Analytical Solutions of Shallow Water Waves with Idealised Bottom Topographies

TL;DR

This paper introduces semi-analytical solutions for 2D shallow water equations with idealised bottom topographies using the Adomian decomposition method, enabling accurate modeling of complex geophysical flows while maintaining parameter correlations.

Contribution

It develops a novel semi-analytical approach using ADM that captures nonlinear phenomena and connects existing solution ansatzes, also deriving new solutions for geostrophic oscillations and vortices.

Findings

Demonstrates robustness of the approach across various oceanic conditions

Provides new insights into geophysical flow behaviors

Validates solutions through numerical experiments

Abstract

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 characterize various types of geophysical flows.