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

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 $v_s=g{\zeta }_s^{\prime }/f$. Substituting in conserved $Q$ givesThe decaying Green function of this modified Helmholtz equation givesIts 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 fixesThe 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.