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

Sverdrup balance applies to a steady, large-scale, small-Rossby number, hydrostatic and nearly geostrophic ocean interior on a beta plane. The flow is depth-integrated, relative-vorticity advection and interior friction are negligible, density is treated as constant for the barotropic balance, and the principal vorticity source is the curl of wind stress. The balancesays that wind input of vertical vorticity is balanced by meridional advection of planetary vorticity, or equivalently by the stretching needed to conserve potential vorticity as parcels move across latitude circles.

Use $u=-{\psi }_y$, $v={\psi }_x$. Since $v=W(y)/\beta$ and the eastern boundary may be chosen as the zero streamline,The lower half-basin has southward interior flow and a northward western return current, giving a clockwise ocean gyre. The upper half has northward interior flow and a southward western return current, giving a counterclockwise gyre. The interior solution cannot satisfy no slip at every wall, so viscous layers complete the circulation.

With constant lateral viscosity $\nu$, the vorticity equation contains $\nu {\nabla }^4\psi$. At the western and eastern walls, balancing $\beta {\psi }_x$ against four $x$ derivatives gives the Munk boundary layer scaleAt the southern and northern walls, $x$ variation remains on the basin scale $L_X$ while four $y$ derivatives occur across the layer, so

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.

For both $y>0$ and $y<0$, the velocity $v=-W_{0}y/(2f_0)$ carries parcels toward $y=0$, where $H=y^2$ decreases. Their leading potential vorticity $f_0/H$ therefore increases along the path, exactly as required by the positive source $W_0/H$. The explicit interior velocity has zero relative vorticity, so the leading change comes from vortex-column stretching rather than from $\zeta /H$. Relative vorticity and friction become important only where boundary layers close the circulation or where the singular depth profile near $y=0$ invalidates the approximation.