 Cosine-ridge shallow-water geostrophic response

For a uniform upstream current whose free-surface and bottom slopes compensate, steady quasi-geostrophic approximation flow with uniform incoming potential vorticity obeys the displayed inversion equation. Let $\hat{b}=ϵ\cos (x/L)$ on $|x|<a=\pi L/2$ and zero outside. With $C=(fϵ/h_0)/(L^{-2}+L_R^{-2})$, the decaying even response is $\varphi =C[\cos (x/L)+(L_R/L)e^{-a/L_R}\cosh (x/L_R)]$ inside and $\varphi =C(L_R/L)e^{-a/L_R}\cosh (a/L_R)e^{-(|x|-a)/L_R}$ outside. Continuity of $\varphi$ and ${\varphi }^{\prime }$ fixes the coefficients. Its streamlines satisfy $y=y_{\infty }+\varphi (x)/U$, so the response extends beyond the compact ridge through exponentially decaying balanced tails.