9 - Numerical Methods for Solving Shallow Water Equations

Summary

Introduction

A major concern while performing weather prediction using numerical methods is to appropriately represent the geostrophic adjustment process, which reflects the broad balance between the mass fields and the wind fields. It is known that the gravity inertia waves are closely associated with the geostrophic adjustment process.

Linear One-dimensional Shallow Water Equations without Rotation

The linear one-dimensional shallow water equations without rotation represent the linear one-dimensional gravity wave and are given as follows:

where u is the zonal (east–west)component of fluid velocity, h is the perturbation height of the fluid, g is the acceleration due to gravity, and H is the mean depth of the fluid, which is considered constant. Equations (9.1) and (9.2) correspond to a system of hyperbolic partial differential equations. If one were to eliminate h from Equations (9.1) and (9.2), by differentiating Equation (9.1) with respect to t and differentiating Equation (9.2) with respect to x, and subtracting one equation from the other, one ends up with the following equation in u:

One would also obtain an equation in h identical to Equation (9.3), if one were to eliminate u from Equations (9.1) and (9.2). If one were to assume, wave type solutions for u and h, such as

and substitute them in Equations (9.1) and (9.2), one obtains the following expression for the phase speed of the gravity wave

Equation (9.4) indicates that there are two gravity waves that travel in opposite directions (i.e., in the positive and the negative x axis).

The following sections outline various methods to solve the one-dimensional linear shallow water model using the method of finite differences

Solution of Linear One-dimensional Shallow Water Equations without Rotation

Broadly, the methods of solving the linear one-dimensional shallow water equations without rotation can be divided into explicit and implicit finite difference methods.

Explicit schemes: Leapfrog scheme (non-staggered)

Utilizing central difference finite difference schemes for spatial derivatives and time derivatives, the latter known as the leapfrog scheme, for Equations (9.1) and (9.2), one gets

The stability of the leapfrog time integration scheme is determined by substituting the following Equations (9.7) and (9.8) into Equations (9.5) and (9.6).