Spatial discretisation

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)$.
\fbox{
\includegraphics[width=12cm,bbllx=126,bblly=462,bburx=513,bbury=714]{./figures/Picsh.ps}
}

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 $ 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. 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 8. The different grid points cover the following index ranges:

\begin{displaymath}\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}\end{displaymath} (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 7: Grid layout and indexing of corner points for curvilinear grids.
[lc][][1.4] $ (x_{i-1,j},y_{i-1,j})$ [lc][][1.4] $ (x_{i,j},y_{i,j})$ [lc][][1.4] $ (x_{i,j-1},y_{i,j-1})$ [lc][][1.4] $ (x_{i-1,j-1},y_{i-1,j-1})$

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

\begin{displaymath}\begin{array}{rcl} \Delta x^c_{i,j}&=&\left\vert\left\vert \f...
...12(X_{i,j+1}-X_{i,j-1})\right\vert\right\vert \ \ \end{array}\end{displaymath} (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 7.

kklingbe 2017-10-02