= Geostrophic adjustment of a finite-width height ramp
{title2=$\alpha=L/R_d$}
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 $R_d^2\eta_s''-\eta_s=-\eta_0$. 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 $\eta_s/h=\operatorname{sgn}(x)[1-(\sinh\alpha/\alpha)e^{-|x|/R_d}]$ outside. The balanced <velocity> is $u_s=0$, $v_s=g\eta_s'/f$. Narrow ramps spread over the <Rossby deformation radius>; wide ramps retain their initial height except for narrow matching layers.
Back to article page
