Analysis of Numerically Induced Oscillations in Two-Dimensional Finite-Element Shallow-Water Models Part II: Free Planetary Waves

The behavior of planetary (Rossby) waves in finite-element numerical models is investigated by using a quasi-geostrophic approximation to the midlatitude, $\beta$-plane, two-dimensional linear shallow-water equations. A dispersion relation analysis is employed here to determine the possible occurrence of spurious modes and to ascertain the dispersive/dissipative nature of the finite-element Galerkin mixed formulation. Three finite-element pairs are considered by using a variety of mixed interpolation schemes. For each pair the frequency or dispersion relation is obtained and analyzed, and the dispersion properties are compared analytically and graphically with the continuous case. It is shown that certain choices of mixed interpolation schemes may lead to significant phase and group velocity errors and spurious solutions in the calculation of slow Rossby waves, due to the coupling between the momentum and continuity equations. Numerical solutions of two test problems to simulate slow Rossby waves are in good agreement with the analytical results.