Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 333 / 2

Assume a steady, small-Rossby-number interior in which drag is negligible and the nonlinear advection of relative vorticity,is small compared with advection of planetary vorticity. The stretching term proportional to $\psi$ has zero Jacobian with $\psi$. On the beta plane,Hence the forced quasi-geostrophic equation reduces to Sverdrup balanceFor the zonal wind stress $\mathbf{\tau }=\tau (y)\hat{\mathbf{x}}$,so

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The eastern wall has no normal flow, so the quasi-geostrophic streamfunction is constant there. Choose $\psi (L,y)=0$. Integrating the Sverdrup relation givesSince geostrophic balance gives $\psi =g\eta /f_0$,If this interior solution were extended to both walls, the west-minus-east height difference would be

The northward volume transport per unit meridional distance isThusThis is the basin-integrated form of Sverdrup balance.

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In a narrow western layer, $x$ derivatives dominate $y$ derivatives. Subtracting the forced interior balance leavesIf the layer has width $\delta$, then ${\psi }_x=O(\psi /\delta )$ and ${\psi }_{xx}=O(\psi /{\delta }^2)$. Balancing the beta effect with linear bottom drag givesand therefore the Stommel boundary layer width isThis scaling assumes $\delta \ll L$ and that along-boundary variations occur on the basin scale.

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No normal flow makes $\psi$ constant on the connected basin boundary; set that constant to zero. PutThe interior solution is $A(L-x)$. The decaying boundary correction that enforces $\psi (0,y)=0$ isThus, to leading order in $\delta /L$,

If $\delta$ denotes the e-folding width literally, thenSince a boundary-layer width is only defined up to an order-one factor, matching at any point satisfying $\delta \ll x\ll L$ gives the convention-independent leading jumpIt is the height difference needed to return the broad Sverdrup transport in a narrow western boundary current.

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Because $v={\psi }_x=(g/f_0){\eta }_x$, the mean boundary-current velocity is exactly related to the height change byFor the e-folding convention used in part d,Its robust narrow-layer scaling isThus weaker drag makes the current proportionally narrower and faster. Their product is independent of $\gamma$ to leading order:Mass conservation requires the narrow return transport to cancel the broad Sverdrup balance transport. The meridional gradient of planetary potential vorticity makes a frictional closure possible on the western side and produces western intensification; bottom drag supplies the vorticity sink that permits fluid parcels to cross potential-vorticity contours there.

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