= Ekman spin-down in a shallow-water layer
{c}
{title2=$\tau_k=\frac{H}{\sqrt{f\nu/2}}(1+k^{-2}R_D^{-2})$}
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 $|f|\tau\gg1$.
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