Shallow water equations and Poincaré waves

Abstract

Shallow water equations, fundamental in fluid dynamics, offer a concise framework for studying the dynamics of a fluid layer with constant depth. They serve as a versatile tool to model various natural phenomena, including tsunami propagation, river flow, tidal patterns, and atmospheric behaviors. Derived from the more intricate Navier-Stokes equations, the shallow water equations assume a thin fluid layer compared to its horizontal extent, allowing the neglect of the vertical velocity component and approximating pressure as hydrostatic in the vertical direction. This simplification enables integration over the vertical dimension, facilitating a more tractable analysis.

The primary focus of this report lies in the exploration of shallow water equations across diverse domains, with a specific emphasis on understanding the behavior of solutions, notably the emergence of Poincaré waves. Poincaré waves represent a specific subset of solutions characterized by their frictionless and Coriolis-dependent nature. Within this broader context, attention will be directed towards more specialized solutions, including Rossby, Kelvin, and gravity-Rossby mixed waves. These distinctive waves hold significant relevance in oceanography, where they play a pivotal role in the formation of oceanic currents, exemplified by the Gulf Stream. Additionally, they contribute to the genesis of the El Niño phenomenon, a climatic event with profound implications for Earth’s climate.

Delving deeper into the analysis, a specific subset of solutions within the Poincaré space emerges as solitons. These solitons, characterized by their lack of damping effects, serve as a valuable metric for evaluating the efficiency of numerical integration schemes. Notably, any observed damping effects in solitons can be attributed solely to numerical errors, making them a reliable benchmark for assessing the accuracy of the chosen numerical integration approach.