 Gaussian topographic response with compensated uniform potential vorticity

In a shallow-water potential-vorticity inversion with the background gradient explicitly compensated, uniform PV gives $({\nabla }^2-L_R^{-2}){\psi }^{\prime }=-fb/H$. The displayed Gaussian function solves it for $b=ϵ[4(1-r^2/a^2)+a^2/L_R^2]e^{-r^2/a^2}$. This is not, without that compensation, a steady solution of the standard finite-depth constant-f current problem: its residual is $U{\psi }_x^{\prime }/L_R^2$. Closed streamlines can occur while Rossby number and surface displacement remain small.