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 7. The velocity points are located such that the corresponding velocity components are centralised on the surfaces of the T-point reference box, the $u$-velocity points (from now on U-points) at the western and eastern surfaces, the $v$-velocity points (from now on V-points) at the southern and northern surfaces and the $w$-velocity points (from now on W-points) at the lower and upper surfaces.

Figure 6: Layout of the model horizontal model grid in Cartesian coordinates. Shown are the reference boxes for the T-points. The following symbols are used: $+$: T-points; $\times$: U-points; $\star$: V-points; $\bullet$: X-points. The inserted box denotes grid points with the same index $(i,j)$.

Figure 7: Layout of the model vertical model grid through the U-points. Shown are the reference boxes for the T-points. The following symbols are used: $+$: T-points; $\times$: U-points; $\triangle$: W-points; $\circ$: X$^u$-points. The inserted box denotes grid points with the same index $(i,k)$. The grid in the $(j,k)$-plane through the V-points is equivalent.

The indexing is carried out with $i$-indices in eastern (${\cal X}$-) direction, with $j$-indices in northern (${\cal Y}$-) direction and with $k$-indices in upward ($z$-) direction, such that each grid point is identified by a triple $(i,j,k)$. 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 7. The different grid points cover the following index ranges:

$$\begin{array}{llll} \mbox{T-points:} & 1 \leq i \leq i_{\max}... ... & 1 \leq j \leq j_{\max}, & 0 \leq k \leq k_{\max} \end{array}$$ (46)

On the T-points, all tracers such as temperature $T$, salinity $S$, the general tracers $c^i$ and the density are located. All turbulent quantities such as eddy viscosity $\nu_t$ and eddy diffusivity $\nu_t'$ are located on the W-points.

Figure 8: Grid layout and indexing of corner points for curvilinear grids.

For curvilinear grids, several arrays for spatial increments $\Delta x$ and $\Delta y$ have to be defined:

$$\begin{array}{rcl} \Delta x^c_{i,j}&=&\left\vert\left\vert \f... ...2(X_{i,j+1}-X_{i,j-1})\right\vert\right\vert \\ \\ \end{array}$$ (47)

where $\left\vert\left\vert X_{i,j}-X_{i-1,j}\right\vert\right\vert =\left((x_{i,j}-x_{i-1,j})^2+(y_{i,j}-y_{i-1,j})^2\right) ^{1/2}$. The superscripts $c,u,v,+$ in (47) indicate whether a $\Delta x$ or $\Delta y$ is centrered at a T-, U-, V-, or X-point, respectively. For the locations of the corner points $X_{i,j}=(x_{i,j},y_{i,j})$, see figure 8.