ABSTRACT
The Shallow Water Equations are frequently used as a model for both atmospheric and oceanographic circulation. They are a simple form of the equations of motion that describe the evolution of an incompressible fluid in response to gravitational and rotational accelerations. In this work we write these equations in Lagrangian coordinates, and the positions of fluid particles are identified for all time. Lagrangian coordinates are useful for following contaminant transport and for assimilation of Lagrangian data. They also give a well posed statement of the full Navier-Stokes equations in a region. These coordinates do no vary with time because the independent variables are the particles' fixed initial position (compare with Eulerian coordinates where the independent variables are the fixed spatial position), thus the numerical solution can be found with traditional methods. Both a low order finite difference method, and a high order Chebyshev pseudospectral method in space with a fourth order Runge-Kutta method in time, are used to solve the Lagrangian shallow water equations. The test cases include linear solutions (center, source, spiral sink) and a nonlinear solution. It will be shown that the particles' trajectories can be accurately computed by solving the Lagrangian shallow water equations for realistic temporal and spatial scales.