= Geostrophic adjustment of a surface-height jump
{title2=$\eta_s=\eta_0\operatorname{sgn}(x)(1-e^{-|x|/R_D})$}
A constant-depth layer initially at rest with a step in its <free surface> adjusts by radiating <inertia-gravity waves>. Conservation of linearized <shallow-water potential vorticity> leaves the bounded balanced profile $\eta_s=\eta_0\operatorname{sgn}(x)(1-e^{-|x|/R_D})$, where $R_D=\sqrt{gH}/|f|$ is the <Rossby deformation radius>. The released <potential energy> per unit transverse length is $3\rho g\eta_0^2R_D/2$; the final <kinetic energy> is $\rho g\eta_0^2R_D/2$, leaving $\rho g\eta_0^2R_D$ in radiated waves.
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