Geostrophic adjustment

Geostrophic adjustment is the radiation of inertia-gravity waves from an unbalanced disturbance, leaving a slower flow in geostrophic balance while conserving linearized potential vorticity.

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.

For an initially resting shallow water layer whose height rises linearly from $-h$ to $h$ across $(-L,L)$, let $R_d=\sqrt{gH}/|f|$ and $\alpha =L/R_d$. Conserved linear shallow-water potential vorticity gives $R_d^2{\eta }_s^{\prime \prime }-{\eta }_s=-{\eta }_0$. Boundedness, oddness, and continuity of height and slope give ${\eta }_s/h=x/L-e^{-\alpha }\sinh (x/R_d)/\alpha$ inside the ramp, and ${\eta }_s/h=\mathrm{sgn}(x)[1-(\sinh \alpha /\alpha )e^{-|x|/R_d}]$ outside. The balanced velocity is $u_s=0$, $v_s=g{\eta }_s^{\prime }/f$. Narrow ramps spread over the Rossby deformation radius; wide ramps retain their initial height except for narrow matching layers.

For a resting shallow-water strip $-L<x<0$ adjacent to an impermeable coast, conserved linear potential vorticity gives $a^2{\eta }_{xx}-\eta =-{\eta }_{\mathrm{initial}}$. Decay offshore, matching and conserved volume give $\eta ={\eta }_0{e}^{x/a}\sinh (L/a)$ outside the strip and $\eta ={\eta }_0[1-e^{-L/a}\cosh (x/a)]$ inside it. The adjusted velocity is $v=g{\eta }_x/f$, with zero tangential wall velocity inherited from initial rest and alongshore uniformity.

With constant density divided out, a decaying or periodic geostrophic balance on an f-plane haswhere $q=\zeta -f\eta /H$ is the signed linearized shallow-water potential-vorticity anomaly. The stationary height equation and integration by parts prove the identity. Replacing $q$ by $|q|$ destroys it: an odd height paired with symmetric opposite-sign sheets would then misleadingly give zero.

Consider a layer governed by the linear shallow water equations of mean depth $H$ on an f-plane with nonzero constant Coriolis parameter $f$. Initially the surface is flat, the cross-strip velocity is zero, and the along-strip velocity is $u_0(y)=U{1}_{|y|<a}$. The two edges are opposite vortex sheets. Let $c=\sqrt{gH}$ and use the positive Rossby deformation radius $L_D=c/|f|$.

The conserved linearized shallow-water potential-vorticity anomaly is $q=v_x-u_y-f\eta /H$. Initially it isThe bounded final geostrophic balance has $u_f=-g{\eta }_f^{\prime }/f$ and henceThe decaying Green function of this one-dimensional operator is $-L_D{e}^{-|y|/L_D}/2$. Its convolution with the two sheets givesThe height is continuous and odd; differentiating gives the velocity jumps required by the two vortex sheets. Reversing $f$ reverses this height profile while leaving its positive decay length unchanged. The formula with $c/f$ as a decay length therefore requires the explicit restriction $f>0$.

Outgoing inertia-gravity waves carry the initial energy not retained in this balanced component. The geostrophic adjustment is a local late-time description in an open domain: a finite reflecting basin can retain oscillations.

For an initially uniform strip current of half-width $a$, the final balanced energy fraction isThe fraction tends to $1-r+O(r^2)$ for a narrow strip and to $1/(2r)$ for a wide strip. The difference is outgoing inertia-gravity wave energy, consistent with conservation of energy; the infinite-strip energies are interpreted per unit along-strip length.

A constant-depth layer initially at rest with a step in its free surface adjusts by radiating inertia-gravity waves. Conservation of linearized shallow-water potential vorticity leaves the bounded balanced profile ${\eta }_s={\eta }_0\mathrm{sgn}(x)(1-e^{-|x|/R_D})$, where $R_D=\sqrt{gH}/|f|$ is the Rossby deformation radius. The released potential energy per unit transverse length is $3\rho g{\eta }_0^2{R}_D/2$; the final kinetic energy is $\rho g{\eta }_0^2{R}_D/2$, leaving $\rho g{\eta }_0^2{R}_D$ in radiated waves.

An inertia-gravity wave is restored jointly by buoyancy or free-surface gravity and the Coriolis force.

With downward buoyancy $b=g\rho \prime /{\rho }_0$ in a uniform rotating Boussinesq approximation, $q=\zeta -f{b}_z/N^2$ is time-independent. The linear pressure obeys ${\nabla }^2{p}_{tt}+f^2{p}_{zz}+N^2{\nabla }_H^{2}p={\rho }_{0}f{N}^{2}q$. The initial potential vorticity supplies the steady pressure source; nonzero-frequency inertia-gravity waves have zero perturbation PV.

For a plane inertia-gravity wave on a resting layer, $\eta ={\eta }_0\cos (kx-\omega t)$, the horizontal velocities are $u=U\cos \theta$ and $v=V\sin \theta$, where $U=\omega {\eta }_0/(k{h}_0)$ and $V=f{\eta }_0/(k{h}_0)$. First-order integration at a fixed equilibrium coordinate gives $\xi =-U\sin \theta /\omega$, $\upsilon =V\cos \theta /\omega$. The particle trajectory is an ellipse with axis ratio $|f|/\omega$. For positive $f$, the orbit is clockwise in the $(x,y)$ plane.

An external gravity wave displaces a fluid's free surface and moves nearly uniformly through the depth in the long-wave limit. In a layer of depth $H$, its leading shallow-water approximation speed is $\sqrt{gH}$.

On an f-plane, linear shallow-water plane waves have one zero-frequency geostrophic mode and two inertia-gravity modes with ${\omega }^2=f^2+c^2(k^2+l^2)$. The crossover scale $c/|f|$ is the barotropic deformation radius.

A coastal Kelvin wave is trapped within a Rossby deformation radius of a boundary. It propagates with the boundary on its right in the Northern Hemisphere and on its left in the Southern Hemisphere, and its linear shallow-water potential-vorticity anomaly vanishes.

For a harmonic coastal Kelvin wave, $v=(c/H)\eta$, $c=\sqrt{gH}$ and the offshore decay length is $c/|f|$. Integrating the period-mean pressure-work flux over the offshore direction gives $\mathcal{F}=\rho g^{2}H{A}_0^2/(4|f|)$ for peak coastal amplitude $A_0$. Slow changes with negligible dissipation or scattering conserve this flux, giving $A_0\propto \sqrt{|f|/H}$. The changing mode width explains why depth scaling differs from fixed-width wave shoaling. The approximation is not uniform at the equator.

A slowly varying coastal Kelvin wave can force long Rossby waves at the same absolute frequency and alongshore wavenumber. In the regime $k^2+{\ell }^2\ll L_d^{-2}$, the offshore group velocity is westward with magnitude $\beta L_d^2$. With coastal phase velocity magnitude $c$, matching gives $|k|=c|\ell |/(\beta L_d^2)$. The zonal long-wave condition therefore additionally requires $|\ell |\ll \beta L_d/c$; $|\ell |L_d\ll 1$ alone is insufficient. This mechanism transmits coastal thermocline anomalies into an ocean basin over a much slower timescale than coastal propagation.

Exponentially weighted cross-channel integrals of velocity and pressure isolate the two oppositely propagating boundary Kelvin waves. Their conserved characteristic amplitudes supply the two integral constraints missing from a channel geostrophic-adjustment problem.