6.5.1 Dirichlet boundary condition on the streamfunction

The no-normal flow boundary condition implies that next to lateral boundaries,

$$\hat{n} \cdot H(\overline{u},\overline{v}) \hat{n} \cdot \hat{z} \wedge \nabla_{h}\psi$$ =

$$-\hat{t} \cdot \nabla_{h} \psi$$ =

= 0, (6.34)

where $\hat{n}$ is a unit vector pointing outwards from the boundary, with the interior of the closed domain to the left, and $\hat{t} = \hat{z} \wedge \hat{n}$ is a unit vector parallel to the path traversing the boundary. This constraint says that the streamfunction is a constant along the boundaries. In the parlance of applied mathematics, such boundary conditions are known as Dirichlet conditions. In general, the streamfunction can be a different time dependent constant along the different closed boundaries

$$\psi = \mu_{r}(t),$$ (6.35)

where $\mu_{r}(t)$ is a time dependent number, and r = 1,2,...Rlabels the particular boundary (an island label), with R the total number of islands. The interpretation afforded by Stokes' Theorem, discussed in Section 6.1, indicates that $\mu_{r}(t)$ represents the time dependent volume transport circulating around the island with label r.

The presence of Dirichlet boundary conditions on the streamfunction indicates that the streamfunction at one point along the boundary is identical to the streamfunction at another point, which can generally be thousands of kilometres away. Such non-local forms of information can be quite problematical in the solution of the streamfunction on parallel machines. The discussions in Smith, Dukowicz, and Malone (1992), Dukowicz, Smith, and Malone (1993), and Dukowicz and Smith (1994) highlight this point.