Abstract
The semi-Lagrangian adveation scheme is applied to the meridional-plane model of the thermohaline circulation developed by Marotzke et al., whose governing equations are solved under a variety of boundary conditions. To determine the extent to which the accuracy and efficiency of the calculations depends on the numerical integration scheme, the test problem is solved independently using an explicit finite-difference (leapfrog in time, centered difference in space) method and three implicit methods: a finite difference, a finite element, and an upwind scheme. Integrations of the model to several equilibria are performed to determine the accuracy, efficiency, and stability of each integration scheme as a function of time step. For the same level of accuracy the time step used in the semi-Lagrangian scheme is found to be at least five times greater than that employed in the case of the implicit methods. The time step used in the implicit methods in turn are at least six times greater than those needed in the explicit integration of the governing equations. It is further shown that Dirichlet, Neumann, and mixed boundary conditions can be handled efficiently with the semi-Lagrangian method.
The semi-Lagrangian method is applied in the usual three-time-level and two-time-level interpolating versions as well as in a noninterpolating, three-time-level version. The two-time-level scheme further doubles the speed of the time integration step for the same level of accuracy, beyond that which is achieved using the three-time-level scheme. The noninterpolating scheme does not eliminate the damping introduced by the interpolation, as pointed out by Ritchic. Hence, the two-time-level, semi-Lagrangian advection method stands out as a viable time integration scheme for climate models that are normally run for hundreds of years and is best suited for ocean climate studies.
