6.5.3 Island integrals for the volume transport

As a final step in the streamfunction solution, it is necessary to formulate a prognostic equation for the volume transports $\mu_{r}(t)$. To do so, reconsider the elliptical problem for the streamfunction

$$\nabla_{h} \wedge \left( \frac{1}{H} \hat{z} \wedge \nabla_{h} \psi_{t} \right) -\frac{1}{\rho_{o}} \nabla_{h} \wedge \left( \overline{\nabla_{h}p_{b}} - \overline{{\bf G}} \, \right).$$ = (6.46)

Now integrate both sides over the area bounding a particular island with label r, and employ Stokes' Theorem. The direction normal to the island's surface is $\hat{z}$, and the area element on the island is $d\Omega_{r}$. The left hand side becomes

$$\int\int \hat{z} \, d\Omega_{r} \; \nabla_{h} \wedge \left( \frac{1}{H} \hat{z} \wedge \nabla_{h} \psi_{t} \right) \int\int \hat{z} \, d\Omega_{r} \; \nabla_{h} \wedge \left( \frac{1}{H} \hat{z} \wedge \nabla_{h} \psi_{r} \right) \, \dot{\mu}_{r}$$ =

$$\dot{\mu}_{r} \; \oint dl \, \hat{t} \cdot \frac{1}{H} \hat{z} \wedge \nabla_{h}\psi_{r}$$ =

$$-\dot{\mu}_{r} \; \oint dl \, \hat{n} \cdot \frac{1}{H} \nabla_{h}\psi_{r},$$ = (6.47)

where $\hat{t} \wedge \hat{z} = -\hat{n}$ was used, and $\hat{n}$represents the outward normal to the island boundary. The right hand side becomes

$$- \int\int \hat{z} \, d\Omega_{r} \; \nabla_{h} \wedge \frac{1}{\... ...la_{h} \wedge \left( \overline{\nabla_{h}p_{b}} - \overline{{\bf G}} \, \right) -\frac{1}{\rho_{o}} \oint dl \, \hat{t} \cdot (\overline{\nabla_{h}p_{b}} - \overline{{\bf G}}).$$ = (6.48)

Equating yields the prognostic equations for the volume transports around an island

$$\dot{\mu}_{r} \frac{1}{\rho_{o}} \; \frac{\oint dl \, \hat{t} \cdot (\overline{... ...erline{{\bf G}}) } {\oint dl \, \hat{n} \cdot \frac{1}{H} \nabla_{h} \psi_{r}}.$$ = (6.49)