Geostrophic adjustment of a finite-width height ramp ($\alpha =L/R_d$)

For an initially resting shallow water layer whose height rises linearly from $-h$ to $h$ across $(-L,L)$, let $R_d=\sqrt{gH}/|f|$ and $\alpha =L/R_d$. Conserved linear shallow-water potential vorticity gives $R_d^2{\eta }_s^{\prime \prime }-{\eta }_s=-{\eta }_0$. Boundedness, oddness, and continuity of height and slope give ${\eta }_s/h=x/L-e^{-\alpha }\sinh (x/R_d)/\alpha$ inside the ramp, and ${\eta }_s/h=\mathrm{sgn}(x)[1-(\sinh \alpha /\alpha )e^{-|x|/R_d}]$ outside. The balanced velocity is $u_s=0$, $v_s=g{\eta }_s^{\prime }/f$. Narrow ramps spread over the Rossby deformation radius; wide ramps retain their initial height except for narrow matching layers.