Past exam of the physics of atmospheres and oceans course of the University of Oxford / 2021 / 6 / c / ii / Solution

Along $y=0$, differentiating the approximate streamfunction givesin the leading small-friction limit. Hence the velocity is strongly southward at the west wall, crosses zero after a few Stommel boundary layer widths, and approaches a much weaker northward interior value $F_0/\beta$. The leading zero is $x\simeq {\delta }_W\ln (L/{\delta }_W)$ when $L/{\delta }_W\gg 1$.

For plotting or finite-friction evaluation, use the exact derivativeIt increases monotonically from its negative western value to its positive eastern value because $v^{\prime }(x)=-A[{\lambda }^2{C}_W{e}^{-\lambda x}+{\gamma }^2{C}_E{e}^{\gamma (x-L)}]>0$. Moreover ${\int }_0^{L}v(x)dx=\Psi (L)-\Psi (0)=0$, as required for a closed basin.