Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 73 / 3 / i

Take a beta-plane approximation, $f=f_0+\beta y$ with $\beta >0$, and define the depth transport $\mathbf{T}=(U,V)={\int }_{-H}^0{\mathbf{u}}_{h}dz$. For steady flow with no net flux through surface and bed, depth-integrated continuity gives $U_x+V_y=0$. Vertical integration of momentum givesIn the interior, small depth-to-horizontal aspect ratio makes vertical stress divergence the leading viscous contribution; horizontal viscous terms are neglected there. Taking the vertical component of curl eliminates pressure. Sinceneglecting bottom stress gives the Sverdrup balance,Thus the wind-stress curl sets the interior meridional depth transport, not the local surface meridional velocity. A constant-$f$ plane would have no $\beta V$ term; the use of $\beta$ in this question requires latitude-dependent $f$. For a transport streamfunction with $(U,V)=(-{\bar{\psi }}_y,{\bar{\psi }}_x)$, the interior relation is $\beta {\bar{\psi }}_x=W/{\rho }_0$.