Geostrophic adjustment of a Gaussian height ridge ($(1-R_D^2{\partial }_x^2){\zeta }_s={\zeta }_0$)

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,For each Fourier transform component, $\hat{\zeta }(t)={\hat{\zeta }}_s+({\hat{\zeta }}_0-{\hat{\zeta }}_s)\cos (\sqrt{f^2+gh{k}^2}t)$. 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.