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

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.