Numerical simulations of Rossby–Haurwitz waves

Simulations of Rossby–Haurwitz waves have been carried out using four different highresolutionnumerical shallow water models: a spectral model, two semi-Langrangian modelspredicting wind components and potential vorticity respectively, and a finite-volume model ona hexagonal–icosahedral grid. The simulations show that (i) unlike the nondivergent case, theshallow water Rossby–Haurwitz wave locally generates small-scale features and so has a potentialenstrophy cascade, and (ii) contrary to common belief, the zonal wavenumber 4 Rossby–Haurwitz wave is dynamically unstable and will eventually break down if initially perturbed.Implications of these results for the use of the Rossby–Haurwitz wave as a numerical modeltest case are discussed. The four models tested give very similar results, giving confidence inthe accuracy and robustness of the results. The most noticeable difference between the modelsis that truncation errors in the hexagonal–icosahedral grid model excite the Rossby–Haurwitzwave instability, causing the wave to break down quickly, whereas for the other models in theconfigurations tested the instability is excited only by roundoff error at worst, and the Rossby–Haurwitz wave breaks down much more slowly or not at all.