= Geostrophic adjustment of a Gaussian height ridge
{c}
{title2=$(1-R_D^2\partial_x^2)\zeta_s=\zeta_0$}
An initially resting, along-ridge-uniform <shallow water> layer with height disturbance $\zeta_0(x)$ conserves $q-f\zeta/h=-f\zeta_0/h$. Its balanced remainder satisfies the displayed <modified Helmholtz equation>, with $R_D=\sqrt{gh}/f$. On the infinite line,
$$
\zeta_s(x)=\frac1{2R_D}\int_{\mathbb R}e^{-|x-x'|/R_D}\zeta_0(x')\,dx',\qquad u_s=0,\quad v_s=\frac gf\zeta_s'.
$$
For each <Fourier transform> component, $\widehat\zeta(t)=\widehat\zeta_s+(\widehat\zeta_0-\widehat\zeta_s)\cos(\sqrt{f^2+ghk^2}\,t)$. The oscillatory part is <inertia-gravity waves>. For a localized <Gaussian function> on the infinite line those waves spread away, leaving the steady component locally; a reflecting finite basin generally retains oscillations.
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