Lateral boundary conditions

Usually, a land mask is defined on the horizontal numerical grid. This mask is denoted by $ a^z$ for T-points, $ a^u$ for U-points and $ a^v$ for V-points with $ a^z$, $ a^u$, and $ a^v$ being integer fields. A T-point is either a land point ($ a^z=0$) or a water point ($ a^z>0$). All U- and V-points surrounding a land point are defined as closed boundary and masked out: $ a^u=0$ and $ a^v=0$. The velocities on such closed boundaries are always set to 0.

Open boundaries are defined by $ a^z>1$ for T-points. Forced boundary points are marked by $ a^z=2$ and passive boundary points by $ a^z=3$. All other T-points are characterised by $ a^z=1$. For velocity points, three different types are defined at the open boundaries. U-points are classified by $ a^u=3$ if both the T-points east and west are open boundary points and by $ a^u=2$ if one adjacent T-point is an open boundary point and the other an open water point with $ a^z=1$. The same is carried out for V-points: They are classified by $ a^v=3$ if both the T-points south and north are open boundary points and by $ a^v=2$ if one adjacent T-point is an open boundary point and the other an open water point with $ a^z=1$. U-points which are adjacent to T-points with $ a^z=2$ and which are not denoted by $ a^u=2$ or $ a^u=3$ are the external U-points and are denoted by $ a^u=4$. The same holds for V-points: Those which are adjacent to T-points with $ a^z=2$ and which are not denoted by $ a^v=2$ or $ a^v=3$ are the external V-points and are denoted by $ a^v=4$.

Figure 8: 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.
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For a simple example of grid point classification, see figure 9.

Figure 9: Classification of grid points for a simple $ 5 \times 5$ configuration ( $ i_{\max}=j_{\max}=5$). On the locations for T-, U- and V-points, the values of $ a^z$, $ a^u$, and $ a^v$, respectively, are written. The northern and eastern boundaries are closed and the western and southern boundaries are open and forced.
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When the barotropic boundary forcing is carried out by means of prescribed surface elevations only, then the surface elevation $ \zeta$ is prescribed in all T-points with $ a^z=2$. For passive boundary conditions ($ a^z=3$), where the curvature of the surface elevation is zero normal to the boundary, the surface slope is simply extrapolated to the boundary points. For a boundary point $ (i,j)$ at the western boundary this results e.g. in the following calculation for the boundary point:

$\displaystyle \zeta_{i,j}=\zeta_{i+1,j}+(\zeta_{i+1,j}-\zeta_{i+2,j})= 2\zeta_{i+1,j}-\zeta_{i+2,j}.$ (48)

kklingbe 2017-10-02