Ekman spin-down in a shallow-water layer (${\tau }_k=\frac{H}{\sqrt{f\nu /2}}(1+k^{-2}{R}_D^{-2})$)

In a thin Bottom Ekman layer, slowly evolving geostrophic flow drives upward Ekman pumping into a uniform-depth exterior layer. With $f>0$, $\alpha =\sqrt{f\nu /2}$ and $R_D=\sqrt{gH}/f$, its balanced height satisfies $({\nabla }_h^2-R_D^{-2}){\eta }_t=-(\alpha /H){\nabla }_h^2\eta$. Small-scale Fourier modes decay on $H/\alpha$; large-scale modes obey a diffusion equation with diffusivity $g\alpha /f^2$. The balanced approximation requires $|f|\tau \gg 1$.