Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 333 / 2 / iii / Solution

At small Rossby number, $q=(\zeta +f_0)/H\simeq f_0/H$. For steady flow with $H=y^2$, the forced potential-vorticity equation becomessoDefine a transport streamfunction by $Hu=-{\psi }_y$ and $Hv={\psi }_x$. Choosing the eastern wall as $\psi =0$ gives the topographic Sverdrup balance solutionand thereforeIn each half-basin, the transport streamlines are the level curves $(L_X-x)y^3=\text{constant}$. They move westward and toward $y=0$. This interior solution treats the two sides of the degenerate line $H=0$ separately and requires boundary layers to enforce solid-wall conditions.