Wind-driven circulation over parabolic basin topography ($H=ax(L_x-x)$)

With $f=f_0+\beta y$ and parabolic depth, the topographic potential-vorticity pseudovelocity is $(-\beta /H,-f{H}_x/H^2)$. For positive $f,\beta$, it points west and turns south on the western slope, north on the eastern slope. Its characteristics are the parabolas $y=Qax(L_x-x)/\beta -f_0/\beta$, where $Q=f/H$. Wind-driven ocean gyres are steered by these depth gradients; drag and boundary data determine the actual ocean transport streamfunction. The zero-depth shoreline is a singular limit that requires a separate physical regularization.