= Geostrophic adjustment of a Gaussian height ridge
{c}
{title2=$(1-R_D^2\partial_x^2)\zeta_s=\zeta_0$}
Consider an initially resting layer governed by the linear <shallow water equations>, with mean depth $h$, constant nonzero <Coriolis parameter> $f$, and a localized along-ridge-uniform height disturbance $\zeta_0(x)$. Define $c=\sqrt{gh}$ and the positive <Rossby deformation radius> $R_D=c/|f|$. The <linearized shallow-water potential-vorticity anomaly> is $Q=v_x-u_y-f\zeta/h$, initially $Q_0=-f\zeta_0/h$.
The stationary <geostrophic balance> has $u_s=0$ and $v_s=g\zeta_s'/f$. Substituting in conserved $Q$ gives
$$
(1-R_D^2\partial_x^2)\zeta_s=\zeta_0.
$$
The decaying <Green function> of this <modified Helmholtz equation> gives
$$
\zeta_s(x)=\frac1{2R_D}\int_{\mathbb R}
e^{-|x-x'|/R_D}\zeta_0(x')\,dx'.
$$
Its kernel is positive, has integral one and has width $R_D$: the balanced surface is a smoothed version of the initial ridge with the same signed excess <volume> per unit ridge <length>. For a positive <Gaussian function> $\zeta_0(x)=A e^{-x^2/(2\sigma^2)}$, the final height is positive, even and lower at its central peak; the balanced along-ridge <velocity> is odd. Reversing $f$ changes the sign of this <velocity> without changing the final height or its decay <length>. Using $c/f$ as a positive kernel width would instead require the explicit assumption $f>0$.
For each spatial <Fourier transform> component, initial rest fixes
$$
\widehat\zeta_s(k)=\frac{\widehat\zeta_0(k)}{1+R_D^2k^2},\qquad
\widehat\zeta(k,t)=\widehat\zeta_s(k)
+[\widehat\zeta_0(k)-\widehat\zeta_s(k)]
\cos\bigl(\sqrt{f^2+ghk^2}\,t\bigr).
$$
The second term consists of <inertia-gravity waves>. For localized data on the infinite line they spread away and leave the steady component locally. This local <geostrophic adjustment> does not remove wave <energy> globally; a finite reflecting basin can retain oscillations.
Back to article page
