### Three-dimensional component of SLIM

The three-dimensional part of the model solves the Boussinesq equations under the hydrostatic approximation.

The three-dimensional mesh is made up of prismatic elements built from the extrusion of a two-dimensional triangular mesh. The location of the degrees of freedom is chosen to ensure a stable and efficient scheme:

• The horizontal velocity rely on a completely discontinuous representation (P1DG-L1D). As the degrees of freedom are not shared between columns of prisms, each  of these columns correspond to an independent linear system. It is then not necessary to build a three-dimensional global matrix, and the memory usage is highly reduced. The parallel scaling is strong as each block system is solved separately.
• The representation of the vertical velocity is non-conforming on the horizontal, while it is discontinuous along the vertical (P1NC-L1D). The vertical velocity is obtained by a vertical integration of the horizontal velocity divergence from the sea bottom to the surface. As the latter is constant over each element, its vertical integration can result in jumps affecting the vertical velocity.  The non-conforming representation is used to limit these jumps.
• The tracers use the same element as the vertical velocity to ensure conservation and consistency (P1NC-L1D).
• For the same reason, the free-surface elevation is represented using the the horizontal component of the element used for the vertical velocity (P1NC).

To avoid a constraint on the time-step based on the celerity of gravity waves, the model uses a mode splitting approach. Vertically averaged equations are used to deduce the sea surface elevation evolution using an implicit time-stepping. The three-dimensional baroclinic mode is evolved using an implicit-explicit (IMEX) procedure in which vertical terms and coriolis are treated implicitly while horizontal advection and horizontal diffusion are explicit to allow each column of prisms to be solved independently. Tracers are fully implicit to avoid a restriction on the time step due to a large tracer diffusivity.

Three different vertical discretizations are implemented in the model.  Sigma coordinates offer an accurate representation of the bathymetry, which prevents the apparition of excessive vorticity and mixing, compared to z coordinates. However, in the presence of highly anisotropic operators whose orientation is mainly horizontal, small errors in the computation of horizontal gradients, due to the inclination of the cells and interfaces (grey cells and dashed lines on the figures), can generate spurious diffusion or biased pressure forcing term. Shaved cells offer the combined advantages of sigma and z coordinates. They do not generate spurious vorticity and, except in the bottommost cells, the horizontal gradients are exactly computed for a fluid stratified in the vertical direction.