8 - Shallow Water Equations

Summary

Introduction

The physical process of advection and the various numerical methods that treat advection were discussed in Chapter 6. In this chapter, an important process, “wave propagation,” that is observed in the atmosphere will be examined extensively. It is important to note that wave propagation is inherently contained in the governing equations for atmospheric motion. Hence, when the governing equations of atmospheric motion are integrated in time, it is realized as the excitation and propagation of waves in space. It is desirable that the propagation of these waves in the numerical solution is fairly close to the observed wave propagation. To ensure this, it is important to study the various numerical methods that are available to treat the various terms in the governing equations of atmospheric motion that represent wave propagation. The shallow water equations are a simplified set of equations that govern the horizontal propagation of gravity and or inertia–gravity waves.

The shallow water equations are widely employed in the study of atmosphere and oceans. The utility of employing the shallow water equations stems from the fact that both the atmosphere and the oceans are essentially thin fluids, with their vertical extent being very much smaller than their horizontal extent; hence, both the atmosphere and the oceans are inherently “shallow fluids.” Furthermore, the inherent nonlinearity of the dynamics of the atmosphere and the oceans are manifest in the shallow water equations.

The shallow water equations describe the evolution of a hydrostatic, constant density (homogeneous) and incompressible fluid flow on the surface of the planet Earth and, hence, these equations are equally applicable for both the atmosphere and the oceans. The applicability of the hydrostatic equation is restricted to situations in which the aspect ratio of the fluid flow, the ratio of the vertical scale to the horizontal scale, is small. There exists a hydrostatic balance when acceleration due to gravity balances the vertically directed pressure gradient force in the vertical momentum equation, resulting in negligible vertical fluid accelerations.