### A baroclinic instability test-case

This benchmark is a baroclinic instability, simulated with three-dimensional models by Tartinville et al. (1998). It consists of an initial cylinder of water of lower salinity than the ambient sea water. This cylinder extends on the half upper layer of the domain. An eddy forms which, after an adjustment period, develops a second order instability and splits into two smaller eddies. The domain of interest is a square box of depth 20 m and length 30 km with open boundaries. The complete description of the benchmark can be found in the original paper from Tartinville et al. (1998).

Tartinville et al. (1998) used sponge regions to approximate open boundaries. As an alternative, the mesh flexibility of the finite element method can be used to move the boundaries far enough from the region of interest to limit their influence. The following multiscale mesh, used for the SLIM simulations of the benchmark, is made up of 3742 triangles, with a resolution varying from 1 km to 100 km. It is extruded in the vertical direction to obtain 18 layers. The size of the box on the right panel is 60 × 60 km.

The dynamics of the benchmark can be observed on the following movie, showing isocontours of the salinity. At the bottom of the domain, we show the sea-surface elevation, the arrows representing the two dimensional mean velocities.

The evolution of kinetic energy and enstrophy are easy to compute and can be used as quantitative diagnostics. We observe inertial oscillations of a period of approximately 15 hours, linked to the eddy oscillations visible in video. Both kinetic energy and enstrophy starts to increase after about 150 hours, when the vortex break up in two smaller vortices. The values of the diagnostics obtained with the SLIM model (UP) can be compared to those obtained byTartinville et al. (1998) using different finite-difference models (DOWN).