Simple numerical tests for ocean tidal models

There is a growing interest in tidal effects on the global wind-driven oceanic circulation. Tidal models used in such investigations have been verified by comparison with satellite and tide gauge data, but synthetic tests have not been published. In this paper we present three numerical tests in spherical geometry, which are suitable for testing the tidal component of global ocean models. The first test is a tsunami-like propagation of an initial Gaussian depression with no external forcing. The other two tests examine the tidal response of an ocean with an undulating bottom with four Gaussian ridges and an ocean with a flat bottom with a realistic land mask. We provide the results from six model configurations, which differ in the time-stepping scheme and computational grid used. Most of them are implemented in present-day global ocean models. Although the proposed numerical tests are simple compared to realistic simulations, their analytic solutions are not available. We thus check the conservation of time invariants to ensure that the solutions are physically meaningful. We also compare the time evolution of certain physical quantities and the differences in sea surface heights at particular time instants with respect to a reference solution. All tested time stepping schemes are suitable for tidal studies except for the Euler implicit time stepping scheme. Model configurations based on the Arakawa grids B/E use smoothing to suppress the grid-scale noise which results in an energy leakage of around 5%. The B/E-grid energy leakage is probably acceptable if we consider that tuned diffusive terms are used in real-world configurations. The C-grid and B/E-grid solutions differ in the vicinity of solid boundaries as a consequence of different boundary conditions. The B-grid and E-grid solutions are similar, unless the shape of the solid boundaries is complex due to the different shapes of the respective grid cells.