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

Seek $\psi =\Psi (x)\cos (my)$, with $m=\pi /b$. The ordinary differential equation isIts constant particular solution and characteristic exponents areBoth $\lambda ,\gamma$ are positive; the two homogeneous solutions are $e^{-\lambda x}$ and $e^{\gamma x}$. Their product satisfies $\lambda \gamma =m^2$. At small $rm/\beta$,For the rectangular basin, choose zero streamfunction on its connected impermeable boundary. The factor $\cos (my)$ already gives zero at $y=\pm b/2$. Enforcing $\Psi (0)=\Psi (L)=0$ yields the exact resultwhereThe narrow Stommel boundary layer approximation requiresThen $e^{-\lambda L}$ is negligible, $C_W\simeq \gamma L$ and $C_E\simeq 1$, givingIt is an asymptotic formula: at $x=0$ the remaining mismatch is of order $A(\gamma L{)}^2$. Expanding the eastern exponential as well gives the familiar leading Sverdrup balance interior plus western correction,Printed inequality. The original PDF explicitly has ${\lambda }^{-1}\gg L$ as well as ${\gamma }^{-1}\gg L$. The first sign is reversed for the requested small-friction limit. This can also be disproved directly: if $\lambda L=0.02$ and $\gamma L=0.01$, the printed approximate bracket at the western wall is $1-0.01-e^{-0.01}\simeq -4.98\times 10^{-5}$, while at the eastern wall it is $-0.01e^{-0.02}\simeq -0.00980$. The latter is of the same order as the retained basin-scale signal, violating the boundary condition at leading order. The exact expression above remains available without imposing the erroneous inequalities.

The streamlines are contours of $\psi$. Since $A<0$, the exact streamfunction is negative inside the rectangle and zero on its perimeter. It has one interior minimum, with closed anticlockwise streamlines: a broad northward interior flow returns southward along the west wall; flow is westward near the north edge and eastward near the south edge. The western parts of the streamlines are closely packed because the western boundary current is much faster.

The figure uses the exact ordinary differential equation solution, so it also remains consistent with both wall conditions when the asymptotic scale separation is only moderate. It depicts the stipulated dynamical model, not observed winds or a reconstruction of the source illustration.

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.