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

Put $m=\pi /b$. With $r=0$, Sverdrup balance givesAt an impermeable eastern wall, $u(L,y)=0$. Continuity impliesEquivalently,satisfies the impermeability condition with the boundary streamfunction set to zero.

At a second wall $x=0$, this solution has $u(0,y)=-m{F}_{0}L\sin (my)/\beta$, generally nonzero. A useful integral proof of the obstruction isfor two impermeable walls, whereas the proposed interior gives ${\int }_0^{L}vdx=L{F}_0\cos (my)/\beta$, which varies with $y$. Friction or another departure from the inviscid model is needed for a return flow that closes the circulation.