### A baroclinic rotational test-case

This test-case is a baroclinic and rotational benchmark developed by Avlesen et al. (2001). It was designed to be “a test problem complex enough to measure fundamental qualities of numerical schemes in ocean models, but otherwise as simple as possible”. The complete description of the benchmark can be found in the original paper from Avlesen et al. (2001).

The domain is a square box of 200 km x 200 km. An initial density is set, as shown in the following figure:

The bechmark is run in two modes: the *diagnostic* mode and the *prognostic* mode. In the *diagnostic* mode, the density is constant in time, and acts as a forcing term on the momentum equation. An equilibrium between pressure forcing and coriolis acceleration develops to reach a steady-state regime. The norm and direction of the velocity in this steady-state regime is shown below:

The evolution of the kinetic energy is a useful diagnostic for a quantitative comparison with other models. This evolution shows oscillations of a period of approximately 15 hours are generated by an adjustement around an equilibrium state dictated by the constant density field. The results obtained with the SLIM model (mesh of 531 triangles extruded over 8 layers to obtain 4248 prisms) are compared with a reference finite-difference model, ** i.e.** the Bergen Ocean Model (BOM). The latter uses a higher resolution (345.600 cells). As it was designed without mode-splitting, the time step, constraint by the surface gravity waves celerity, is set to 15 seconds, while the SLIM model uses a time step of 180 seconds.

The second run is fully prognostic: the density is advected by the flow and subject to diffusion. Both density and velocity interact mutually. The kinetic energy is exponentially damped to be finally anihilated when no density gradient subsists: