Numerical study of shallow water models with variable topography

Abstract

In this thesis we develop a model for the long-time horizontal circulation in a shallow lake. The goal is to have a model that can capture the large-scale features of the circulation yet can be run quickly and cheaply. We start with shallow water models and add relevant physical terms: Coriolis force, wind shear, bottom drag, viscosity, and nonhomogeneous boundary conditions. The resulting equations are similar to the two-dimensional Navier Stokes equations and can be analyzed with similar methods. We pose the equations in a weak form and show that they are well-posed. We then discretize the equations. We use the finite element method for the spatial discretization and show that our choice of elements satisfies stability criteria. Next we test our model. We first consider problems with analytically tractable behavior and verify that our model produces correct results. Then we model Lake Erie, both with no wind and with a steady wind. We compare the results of our model to experimentally obtained measurements of the currents. Our results compare well under conditions of no wind or of steady wind, but not as well when the wind is variable.