The Regional Ocean Modeling System : A Split-Explicit , Free-Surface , Topography-Following-Coordinate Ocean Model

Following Blumberg & Mellor, 1987, Bleck & Smith, 1990 and Killworth et. al, 1991, split-explicit free-surface version have been developed for the three major types: sigma, isopycnic and Z−coordinate ocean models. Besides the implied time-scale separation, the splitting into barotropic and baroclinic modes typically relies on the smallness of free-surface elevation relatively to the total depth of the water column, and smallness of density variations in Boussinesque sense. Once separate time stepping algorithms are used for the barotropic mode and the three-dimensional part, the continuity equation holds only within the order of accuracy, but not exactly for the baroclinic time step with the discrepancy associated with details of time evolution of barotropic mass fluxes and free surface within the baroclinic step, which are typically averaged out to avoid numerical instability. At the same time, conservative flux-divergent discretizations for the tracer equations (i.e. heat content, salinity, etc.) require that the implied three-dimensional continuity equation holds exactly on the time-discrete level in order to simultaneously maintain volume conservation and constancy preservation properties for tracers. The inconsistency in discrete continuity equation causes biases in tracer values proportional to the tracer value itself, the ratio of free-surface elevation change to the depth of water column, and a factor depending on the difference in temporal discretization errors of barotropic and baroclinic modes. Although the effect is typically small for basin-scale configurations, in coastal regions, estuaries and fjords, where the free-surface disturbance is not very small relative to the total depth, the error may be significant enough to cause non-physical behavior of the solution and even numerical instability. In the present study we propose a split-explicit hydrodynamic kernel of the model which utilized consistent averaging of barotropic mode to allow both exact conservation and constancy preservation properties for the tracers. Due to mathematical feedback between the momentum and density equations (via temperature and salinity) for the baroclinic mode, and, similarly, between free surface and barotropic momentum equations, it is advantageous to treat both modes in such a way that after time step for the momentum equation is complete, the computed velocities participate immediately in the computation of tracers and vice versa (as opposite to advancing all equations for one time step altogether at once). This leads to a new family of time-stepping algorithms, which introduce forward-backward feedback into the best known synchronous schemes. This allows an increased time step due to the enhanced internal stability of the algorithm without sacrificing its accuracy. Additional gain in stability is achieved by improving accuracy of mutual orthogonality of the discretised barotropic and baroclinic modes, which is implemented via redesign of the barotropic sub-model in such a way that it retains information about stratification via two-way vertically averaged density.