= Solution
Write the parabolic depth as $H(x)=a x(L_x-x)$ with $a>0$, and use $f=f_0+\beta y$, $\beta>0$. The <topographic potential-vorticity pseudovelocity> is
$$
\boxed{\widetilde u_x=-\frac\beta{a x(L_x-x)},\qquad \widetilde u_y=-\frac{(f_0+\beta y)(L_x-2x)}{a x^2(L_x-x)^2}.}
$$
It is westward everywhere in the interior. In the Northern Hemisphere it is also southward on the western slope and northward on the eastern slope. Its characteristics are the background <potential vorticity> contours
$$
\boxed{y=\frac{Qa}{\beta}x(L_x-x)-\frac{f_0}{\beta},\qquad Q=f/H,}
$$
which bend towards the south on approaching either shallow side. Along these characteristics forcing accumulates into a transport response and drag spreads it across neighboring characteristics. The actual current follows contours of $\psi$, not contours of this <pseudovelocity> in a forced region.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-333-parabolic-basin.png]
{title=Illustrative wind-driven streamfunction contours and westward topographic pseudovelocity in a basin with parabolic depth and no normal boundary transport}
{height=420}
The sketch uses an explicit illustrative positive drag, a single connected impermeable boundary with $\psi=0$, $H=4x(1-x)$, and $f=1+0.25y$ in normalized coordinates, with small forcing amplitude $F_0=10^{-3}$; linearity makes its value affect amplitudes rather than contour shapes. Solid and dashed contours distinguish the two signs of $\psi$; arrows indicate the <pseudovelocity>, not the real current. The positive lower-half wind curl tends to drive cyclonic circulation and the negative upper-half wind curl anticyclonic circulation; topographic steering bends the gyres, so their dividing contour need not coincide with the forcing's zero line. The plot is an example, not a uniquely specified circulation: the question supplies neither a drag magnitude nor full boundary data.
The westward propagation of transport information explains the connection with <western boundary currents>: a broad wind-driven interior generally needs a narrow western return region to satisfy the impermeable boundary condition. For constant depth the characteristic direction is purely westward and the <Stommel boundary layer> width is $r/\beta$ for this drag normalization. Here strong depth gradients also steer the return along slopes and can make topographic boundary currents important; a conventional flat-bottom western-current profile does not follow unchanged. \b[The equation becomes singular where $H=0$.] The drawn shore limits impose zero transport; a physically resolved shoreline would need a positive-depth cutoff or a separate near-shore model, and the small-<Rossby number> approximation need not remain uniform there.
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