A comparison of two formulations of barotropic–baroclinic splitting for layered models of ocean circulation

In numerical models of ocean circulation, it is widespread practice to split the fast and slow motions into barotropic and baroclinic subsystems, respectively. In the case of the baroclinic equations, the dependent variables can either be (1) slowly-varying baroclinic quantities, obtained from splitting the original flow variables into barotropic and baroclinic components, or (2) the original unsplit variables, which can vary on both the fast and slow time scales. In the second case, the variables in each layer are adjusted after each (long) baroclinic time step to ensure compatibility with the results produced from the barotropic equations. The second approach can be applied to the layer thickness equation to ensure exact conservation of mass within each layer. In the case of the momentum equations, the second approach amounts to replacing unresolved fast portions of Coriolis and pressure forcing with time averages of well-resolved forcing from the barotropic system. In this study, both approaches for the momentum equations are evaluated, in several test problems, by comparing to analytical solutions or to solutions computed with an unsplit code that uses short time steps. The two methods give very similar results in some simple problems for which analytical solutions are known. However, in some eddying double-gyre simulations, the formulation with unsplit variables requires a significant reduction in the baroclinic time step in order to avoid numerical difficulties that include grid noise and inaccurate representation of the flow field. In contrast, the formulation with split variables does not display such difficulties, and in those same examples it can be used with zero explicit horizontal viscosity. All of these computations employ a two-level time-stepping method that was previously developed by the author.