A robust unstructured grid discretization for 3-dimensional hydrostatic flows in spherical geometry: A new numerical structure for ocean general circulation modeling

Current models of the general circulation of the global oceans employ a spatial discretization of the relevant hydrodynamic fields on Cartesian rectilinear grids. For many applications, significant benefit would be expected to accrue from the versatility offered by unstructured grids. However, until very recently, available numerical methods for performing integrations on unstructured grids could not conserve discrete dynamical invariants, a numerical model characteristic that is important for robust ocean simulations over large space and time scales (as needed, for instance, in climate modeling applications). Our purpose in this paper is to describe such a conservative discretization scheme for rotating hydrostatic Boussinesq fluid dynamics on general triangular tessellations of the sphere, and to demonstrate its properties in a number of simulations that incorporate realistic ocean basin geometry. Several different implicit time discretizations are possible, each of which exhibits a form of exact numerical energy conservation in the absence of dissipation. The properties of this new numerical methodology are validated through analysis of a sequence of unforced nonlinear dynamical problems, which clearly demonstrate the capacity of the model to resolve the geostrophic adjustment process and the onset of baroclinic instability in collapsing density fronts. As a final test, a number of ocean modeling experiments with realistic climatological and wind-stress forcing are performed in order to investigate the manner in which different mesh structures and resolutions influence the simulated phenomenology. As the theoretical properties of the numerical methodology suggest, it is thereby shown to be both robust and stable. The further work that will be required to implement this structure in a state-of-the-art oceanic general circulation model, as well as other potential applications of the techniques, are discussed in the concluding section of the paper.