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 $a^v=4$. 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.

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:

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