Potential-vorticity mixing over a finite bottom slope ($\Delta Q=-Sy{1}_{\{|y|<a\}},S=fϵ/H$)

Homogenizing the potential vorticity over a finite constant bottom slope creates the odd anomaly displayed here while leaving the exterior unchanged. Inverting $({\partial }_y^2-L_R^{-2})\psi =\Delta Q$ with decay at both infinities and matching $\psi ,{\psi }_y$ gives a central return current and flanking jets. The deformation radius sets the smoothing length. For $L_R\ll a$, the central surface follows the bottom slope to leading order, but edge corrections and decaying exterior flow are essential.