Numerical representation of internal waves propagation

Similar to surface waves propagating at the interface of two fluid of different densities (like air and water), internal waves in the oceanic interior travel along surfaces separating waters of different densities (e.g. at the thermocline). Due to their key role in the global distribution of (physical) diapycnal mixing and mass transport, proper representation of internal wave dynamics in numerical models should be considered a priority since global climate models are now configured with increasingly higher horizontal/vertical resolution. However, in most state-of-the-art oceanic models, important terms involved in the propagation of internal waves (namely the horizontal pressure gradient and horizontal divergence in the continuity equation) are generally discretized using very basic numerics (i.e. second-order approximations) in space and time. In this paper, we investigate the benefits of higher-order approximations in terms of the discrete dispersion relation (in the linear theory) on staggered and nonstaggered computational grids. A fourth-order scheme discretized on a C-grid to approximate both pressure gradient and horizontal divergence terms provides clear improvements but, unlike nonstaggered grids, prevents the use of monotonic or non-oscillatory schemes. Since our study suggests that better numerics is required, second and fourth order direct space-time algorithms are designed, thus paving the way toward the use of efficient high-order discretizations of internal gravity waves in oceanic models, while maintaining good stability properties (those schemes are stable for Courant numbers smaller than 1). Finally, important results obtained at a theoretical level are illustrated at a discrete level using two-dimensional (x,z) idealized experiments