## Spatial discretisation

For the spatial discretisation, a staggered C-grid is used, see Arakawa and Lamb (1977). The grid consists of prism-shaped finite volumes with the edges aligned with coordinates. The reference grid for the tracer points (from now on denoted by T-points) is shown in figures 6 and 8. The velocity points are located such that the corresponding velocity components are centralised on the surfaces of the T-point reference box, the -velocity points (from now on U-points) at the western and eastern surfaces, the -velocity points (from now on V-points) at the southern and northern surfaces and the -velocity points (from now on W-points) at the lower and upper surfaces. The indexing is carried out with -indices in eastern (-) direction, with -indices in northern (-) direction and with -indices in upward (-) direction, such that each grid point is identified by a triple . A T-point and the corresponding eastern U-point, the northern V-point and the above W-point have always the same index, see figures 6 and 8. The different grid points cover the following index ranges:

 (46)

On the T-points, all tracers such as temperature , salinity , the general tracers and the density are located. All turbulent quantities such as eddy viscosity and eddy diffusivity are located on the W-points.

 [lc][][1.4] [lc][][1.4] [lc][][1.4] [lc][][1.4]

For curvilinear grids, several arrays for spatial increments and have to be defined:

 (47)

where . The superscripts in (47) indicate whether a or is centrered at a T-, U-, V-, or X-point, respectively. For the locations of the corner points , see figure 7.

kklingbe 2017-10-02