6.5.2 Separating the streamfunction's boundary value problem

The purpose of this section is to develop an algorithm for solving the streamfunction's boundary value problem (BVP). To do so, reconsider the horizontal momentum equations written in the form

 

$${\bf u}_{t} = - \nabla_{h} (p/\rho_{o}) + {\bf G},$$ (6.36)

where the vector ${\bf G}$ represents all the remaining terms given in equations (4.1) and (4.2). Taking the vertical average of this equation, substituting the definition (6.6) for the barotropic streamfunction, and using the expression (6.14) for the hydrostatic pressure gradient, yields

$$\frac{1}{H} \hat{z} \wedge \nabla_{h} \psi_{t} -\frac{1}{\rho_{o}} \biggl( \nabla_{h} (p_{a}+p_{l}) + \overline{\nabla_{h}p_{b}} \biggr) + \overline{{\bf G}}.$$ = (6.37)

Taking the curl of this equation eliminates the atmospheric and lid pressures

$$\nabla_{h} \wedge \left( \frac{1}{H} \hat{z} \wedge \nabla_{h} \psi_{t} \right) -\frac{1}{\rho_{o}} \nabla_{h} \wedge \left( \overline{\nabla_{h}p_{b}} - \overline{{\bf G}} \, \right).$$ = (6.38)

The elliptic boundary value problem

 

$$\nabla_{h} \wedge \left( \frac{1}{H} \hat{z} \wedge \nabla_{h} \psi_{t} \right) -\frac{1}{\rho_{o}} \nabla_{h} \wedge \left( \overline{\nabla_{h}p_{b}} - \overline{{\bf G}} \, \right) \qquad \mbox{interior points}$$ = (6.39)

$$\psi \mu_{r}(t) \qquad \mbox{on island $r$ },$$ = (6.40)

can be separated into two simpler BVPs. The first one is a forced elliptical problem with homogeneous boundary conditions

$$\nabla_{h} \wedge \left( \frac{1}{H} \hat{z} \wedge \nabla_{h} (\psi_{o})_{t} \right) -\frac{1}{\rho_{o}} \nabla_{h} \wedge \left( \overline{\nabla_{h}p_{b}} - \overline{{\bf G}} \, \right) \qquad \mbox{interior points}$$ = (6.41)

$$\psi_{o} 0 \qquad \mbox{on islands}.$$ = (6.42)

This system can be solved for $\psi_{o}$ using some time-stepping scheme, such as leap-frog. The second BVP is a time-independent unforced elliptical problem with constant boundary conditions on the islands

$$\nabla_{h} \wedge \left( \frac{1}{H} \hat{z} \wedge \nabla_{h} \psi_{r} \right) 0 \qquad \mbox{everywhere, except island boundaries}$$ = (6.43)

$$\psi_{r} 1 \qquad \mbox{on islands}.$$ = (6.44)

This BVP can be solved for $\psi_{r}$ using some type of an elliptical solver, such as congugate gradient. The full streamfunction is built from the sum

$$\psi(\lambda,\phi,t) = \psi_{o}(\lambda,\phi,t) + \sum^{R}_{r=1} \mu_{r}(t) \, \psi_{r}(\lambda,\phi),$$ (6.45)

which can be shown to satisfy the original boundary value problem.