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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 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.

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.

Take $x$ eastward, $y$ northward and $z$ upward. In a rotating reference frame, the horizontal Cauchy momentum equation isHere ${\mathbf{F}}_h$ represents other accelerations, such as friction, and $f$ is the Coriolis parameter. When the Rossby number $U/(|f|L)$ and the ratio $1/(|f|\tau )$ are small, and friction is weak, both advective and local acceleration can be neglected. The Coriolis force then balances the pressure gradient force: this is geostrophic balance, with $fv=p_x/\rho$ and $fu=-p_y/\rho$. The geostrophic wind runs along atmospheric isobars, with low pressure on its left in the northern hemisphere.

A rapid change of pressure need not bring the corresponding geostrophic wind into existence at once. An initially weak wind therefore leaves an inward pressure gradient force around the new low. For a smoothed anomaly of horizontal scale $L=5\times 10^5\mathrm{m}$, geostrophic balance would requireFor the illustrative midlatitude value $|f|=10^{-4}{\mathrm{s}}^{-1}$, the implied Rossby number is already about $0.67$, so curved-flow acceleration matters. With a small tropical value $|f|=10^{-5}{\mathrm{s}}^{-1}$, the inferred speed is $333\mathrm{m}{\mathrm{s}}^{-1}$ and the Rossby number about $67$: the assumed geostrophic balance is untenable. Geostrophic adjustment redistributes the anomaly and emits internal gravity waves, leaving a weaker balanced component.

For a target speed $U_{\ast }=10\mathrm{m}{\mathrm{s}}^{-1}$ on the same scale, the required scale estimate isThus $6\mathrm{hPa}$, or $0.30$ of the initial magnitude, is the midlatitude estimate; at the illustrative tropical $|f|=10^{-5}{\mathrm{s}}^{-1}$ it is $0.6\mathrm{hPa}$, or $0.030$ of the initial magnitude. The paper gives no latitude, so there is no unique numerical reduction. At the equator itself one must use equatorial dynamics rather than divide by $f=0$. Even the tropical $10\mathrm{m}{\mathrm{s}}^{-1}$ estimate has Rossby number $2$, so it describes the scale permitted by the stated balance, rather than a controlled tropical geostrophic balance.

There are two qualifications to “unstable”. A literal top-hat pressure anomaly has a discontinuous edge, where no finite classical geostrophic wind can balance its distributional pressure gradient. The $500\mathrm{km}$ estimate assumes a smoothed edge; a thinner edge requires stronger winds. More generally a large low is not, by its amplitude alone, proved to have an exponentially growing instability. For a concrete counterexample to the claim that such an amplitude cannot be stationary, set $p\prime (s)=-P\exp (-s^2/a^2)$, with cylindrical radius $s$, constant $\rho$ and $f>0$. The cyclonic gradient windis real and satisfies $v_{\theta }^2/s+f{v}_{\theta }=p_s/\rho$ for every $P>0$, with a regular limit at the centre. This is a stationary axisymmetric flow; no claim about its perturbation stability is needed. Here the physically justified conclusion is that a rapidly imposed anomaly without its balancing wind is unbalanced.

Small tropical Coriolis parameter allows only weak horizontal pressure gradients to coexist with ordinary large-scale winds. Hydrostatic balance connects the vertical variation of these pressure gradients to horizontal temperature gradients; equivalently the approximate thermal wind relation gives $f{u}_z\simeq -(g/T)T_y$. For comparable vertical wind shear, small $|f|$ means smaller $|T_y|$. Internal gravity waves can spread heating-induced buoyancy anomalies efficiently before rotation arrests the adjustment. This motivates the weak temperature gradient approximation for the tropical free troposphere; it does not require the land surface or atmospheric boundary layer to have uniform temperature.

With the free-surface elevation denoted by $\zeta$, the rotating shallow water equations over a fixed bottom areThe momentum gradient is that of the surface elevation, not depth alone; bottom slope enters through the depth and continuity equation. Taking ${\partial }_x$ of meridional momentum minus ${\partial }_y$ of zonal momentum gives the vorticity equationThe terms quadratic in velocity gradients combine into ${\omega }_r{\nabla }_H\cdot {\mathbf{u}}_H$; the constant Coriolis parameter contributes $f{\nabla }_H\cdot {\mathbf{u}}_H$. Continuity gives $D_{H}h/Dt=-h{\nabla }_H\cdot {\mathbf{u}}_H$. Dividing the two equations proves the exact shallow-water potential vorticity invariant

For the quasi-geostrophic approximation, let $U_s$ and $\ell$ be representative velocity and horizontal length scales. Require the Rossby number $\mathrm{Ro}=U_s/(|f|\ell )\ll 1$, a shallow layer, and slow evolution on time $\ell /U_s$. Put $L_R=\sqrt{g{h}_{00}}/|f|$ and assume $\mathrm{Bu}=(L_R/\ell {)}^2$ is order one. Leading momentum then gives geostrophic balance, and a quasi-geostrophic streamfunction can be chosen asIts surface-height scale satisfies $|\zeta |/h_{00}=O(\mathrm{Ro}/\mathrm{Bu})\ll 1$. Take the bottom-height ratio of the same small order, with gentle slopes, so $\eta =\zeta -b$ obeys $|\eta |/h_{00}\ll 1$. Relative vorticity is order $\mathrm{Ro}f$, and expanding the exact potential vorticity givesBecause the leading geostrophic velocity is horizontally nondivergent, ageostrophic advection of the anomaly is one order smaller. At the first evolving order the shallow-water quasi-geostrophic potential vorticity equation isThe constant reference $f/h_{00}$ has zero gradient and drops out. These assumptions also require all topographically induced velocities to retain a small Rossby number; the small bottom-height condition alone is not a full justification of quasi-geostrophic balance.

The uniform eastward current has ${\psi }_0=-Uy$ up to an additive constant, so ${\zeta }_0=-(fU/g)y$. With $b_0=-(fU/g)y$, surface and bottom slopes are equal and $h=h_{00}$. Its relative vorticity vanishes and $Q_0=0$. This sloping background is used locally where its surface and bottom departures remain within the small-height regime.

Add the compact ridge and set $\psi =-Uy+\varphi (x)$. Then $u=U$, $v={\varphi }^{\prime }(x)$, and $Q$ depends only on $x$. The steady equation becomes $U{Q}_x=0$. Uniform incoming potential vorticity fixes that constant to zero, giving the cosine-ridge shallow-water geostrophic responseLet $a=\pi L/2$, $\lambda =L_R^{-1}$ andThe forcing is even, so the decaying response is even. A cosine particular solution inside the ridge leaves an even homogeneous term; outside, retain only the decaying exponential. Since the forcing is bounded and contains no delta function, both $\varphi$ and ${\varphi }^{\prime }$ are continuous at the edges. Write $\varphi =A\cos (x/L)+D\cosh (x/L_R)$ inside and $\varphi =B{e}^{-(|x|-a)/L_R}$ outside. Matching at $x=a$ givesSolving yields $D=A(L_R/L)e^{-a/L_R}$, henceA globally decaying homogeneous solution that is continuously differentiable is zero, so this matched solution is unique.

A steady streamline is a constant-$\psi$ curve. If its upstream ordinate is $y_{\infty }$, it therefore hasFor $f>0$, $\varphi$ is positive: the flow first turns toward increasing $y$, passes the crest at its largest deflection, then returns to its upstream ordinate. Reversing the sign of $f$ reverses the deflection. For $L=3L_R$, the center deflection is $A[1+e^{-3\pi /2}/3]/U$ and each tail decays on scale $L_R$. The figure shows several streamlines and the ridge support; the deflection extends beyond the ridge.

One compatible realization of the initial condition has $h(x)=h_{00}-\hat{b}(x)$ and $\zeta =b_0(y)$. Its stated zonal velocity then satisfies $u(x)h(x)=U{h}_{00}$, so the initial mass flux is constant and $v=0$. The data given only far upstream do not uniquely determine the initial height everywhere; this is the natural mass-flux-compatible choice. It is not fully balanced over the ridge: $fu(x)$ differs from $-g{\zeta }_y=fU$. The resulting transverse acceleration initiates geostrophic adjustment and launches fast inertia-gravity waves.

Initially the relative vorticity is zero, but exact shallow-water potential vorticity is $f/[h_{00}-\hat{b}(x)]$, not the uniform upstream value. That localized anomaly is materially transported downstream, approximately at $U$ in the leading quasi-geostrophic approximation; it is not annihilated by adjustment. Fresh incoming parcels of uniform PV subsequently supply the ridge region, whose balanced potential-vorticity inversion approaches the steady solution above. The fast waves can carry unbalanced energy away on the inertia-gravity time scale, short compared with the advective time at small Rossby number. In an open domain this can leave the local balanced response; an ideal closed or reflecting domain need not settle monotonically, because undamped waves can return. Radiation, advection of the initial PV anomaly, and any weak damping must be distinguished from irreversible loss of PV.

Write $c=\sqrt{gH}$ and $R_d=c/|f|$, with $f\ne 0$. For the profiles below take $f>0$, so $\alpha =L/R_d$ agrees with the stated parameter; changing the sign of $f$ reverses the balanced meridional velocity but not the height profile. The linearized shallow water equations areTaking the horizontal curl gives $(v_x-u_y{)}_t=-f(u_x+v_y)=f{\eta }_t/H$. Hence the signed linear shallow-water potential vorticity anomaly is conserved:The initial data are independent of $y$, so the solution remains independent of $y$ and $v_x=f(\eta -{\eta }_0)/H$. Differentiating continuity in time and using the zonal momentum equation gives

For a horizontal Fourier mode with wavenumber $k$, put ${\Omega }_k^2=f^2+c^2{k}^2$. The solution splits into a balanced component and inertia-gravity waves:These waves have frequencies $|\omega |\ge |f|$, phase velocity $\omega /k$, and group velocity $c^{2}k/\omega$, whose magnitude is less than $c$. They have zero perturbation potential vorticity and carry the unbalanced part away. On the unbounded line, the localized difference between the initial and balanced profiles disperses, leaving a local steady limit. No damping is needed for this local geostrophic adjustment; in a closed periodic domain, undamped waves would persist and a literal pointwise steady limit would generally not exist.

The bounded steady height satisfies the modified Helmholtz equationOddness removes the even homogeneous solution in the central region. The exterior solution is bounded and tends to the appropriate plateau. Matching both height and its first derivative at $x=\pm L$ gives the geostrophic adjustment of a finite-width height ramp:For example, writing the interior solution as $hx/L+B\sinh (x/R_d)$, matching its slope to the decaying exterior at $L$ gives $B=-h{e}^{-\alpha }/\alpha$.

The steady meridional momentum equation gives $u_s=0$, and the zonal equation gives the geostrophic balance $f{v}_s=g{\eta }_s^{\prime }$. ThusThe height is odd and monotone, and the meridional current is even, with maximum $v_s(0)=gh(1-e^{-\alpha })/(fL)$ for $h>0$.

For $\alpha \ll 1$, the initial ramp is narrow compared with the Rossby deformation radius. To leading order,Near the origin the height rises with slope $h/R_d$, rather than $h/L$; the current has width $R_d=L/\alpha$ and peak $gh\alpha /(fL)$. For $\alpha \gg 1$, the central height remains nearly $hx/L$ and the central velocity nearly $gh/(fL)$. Around each end of the ramp there is a matching layer of width $R_d=L/\alpha$, with height correction of order $h/\alpha$; the exterior current decays over the same width. At $x=\pm L$ the current is approximately half its central value. In both limits the zonal current is zero. These are the broad rotational adjustment of a narrow disturbance and the weak edge adjustment of a broad disturbance, respectively.

Use the energy normalization specified for the height energy: the linear total energy density is $(u^2+v^2)/2+g{\eta }^2/(2H)$. The linear equations implyFor $\alpha \ll 1$, let the potential energy loss mean the positive difference between the initial and final values, per unit transverse length. Although both absolute potential energies diverge on the infinite line, their difference is finite. The leading step approximation givesThe finite ramp changes this leading expression by a relative $O(\alpha )$ correction. The initial kinetic energy is zero and the final gain isThe remainder $\Delta V-\Delta T\simeq g{h}^2{R}_d/H$ is energy carried away by the outgoing inertia-gravity waves. This explains why the ratio is less than one without invoking dissipation: conservation of energy includes the waves, not only the final balanced flow.

Full equations and the balanced scaling. Let $H=h_{00}$, free-surface displacement $\eta$ and layer thickness $h=H+\eta -b$. The inviscid rotating shallow water equations areHere gradients are horizontal and $b$ is a fixed bottom height. For horizontal scales $L$, velocities $V$ and time scale $L/V$, acceleration is $V^2/L$, whereas the Coriolis term is $|f|V$. A small Rossby number $\mathrm{Ro}=V/(|f|L)$ therefore gives leading geostrophic balance. DefineThe last length is the Rossby deformation radius. Geostrophy gives $\eta \sim |f|VL/g$ and consequentlyThe usual distinguished quasi-geostrophic limit takes the Burger number $L_R^2/L^2$ of order one, small surface displacement, and bottom amplitude $b/H=O(\mathrm{Ro})$. The ageostrophic velocity is smaller than the geostrophic velocity by a factor of order $\mathrm{Ro}$, but its small divergence supplies the first nonzero thickness tendency. Motions are slow compared with $|f{|}^{-1}$, and the shallow-layer approximation also requires depth small compared with horizontal scale. Other deformation-radius limits are possible only if the small-height condition is retained.

Taking the vertical curl and combining it with thickness continuity gives exact shallow-water potential vorticity conservation,Expand to first order in relative vorticity, displacement and relief; multiplication by $H$ removes the irrelevant constant depth factor. To quasi-geostrophic accuracy,Potential-vorticity conservation transports this scalar with the geostrophic flow. Potential-vorticity inversion solves the elliptic problemwith the required boundary or far-field conditions, then differentiates $\psi$ to recover velocity and surface height. The deformation term suppresses the far-field influence over distances comparable with $L_R$; at scales much shorter than $L_R$ the inversion approaches the two-dimensional Poisson problem.

The full shallow-water system evolves two velocity components and layer thickness independently, with three first-order time equations. The quasi-geostrophic system evolves one scalar; balanced velocity and height are diagnostic consequences of inversion. In particular it filters the two fast inertia–gravity branches, whose resting-layer dispersion is ${\omega }^2=f^2+gH|\mathbf{k}{|}^2$, and does not describe their fast geostrophic adjustment. Reducing the time order restricts admissible initial data; it does not turn an arbitrary full shallow-water state into a quasi-geostrophic state automatically.

The intended Gaussian inversion, and its required background qualification. Set $r^2=x^2+y^2$ and ${\psi }^{\prime }=A{e}^{-r^2/a^2}$. Direct differentiation givesConsequently, in an inversion with uniform PV and its background gradient explicitly compensated, the printed relief is balanced byThis is the Gaussian topographic response with compensated uniform potential vorticity. In that model the perturbation PV is identically zero, and any steady velocity field advects this uniform PV trivially.

There is a genuine uniform-current obstruction to constant shallow-water QG potential vorticity in the literal constant-$f$, finite-$L_R$ formulation. A geostrophic background current $(U,0)$ has total streamfunction $\Psi =-Uy+{\psi }^{\prime }$ and a background free-surface slope. Over flat bottom its PV is $\overline{Q}=f+Uy/L_R^2$, not uniform. For the displayed Gaussian and printed relief, the two disturbance terms cancel in inversion but leaveFor $Uϵ\ne 0$ and finite $L_R$ this residual is nonzero; for example, it is nonzero at $x=a/\sqrt{2},y=0$. Thus the claimed Gaussian is not a steady solution of the standard unforced constant-$f$ equations as literally stated. Omitting the mean contribution to the stretching term is not a legitimate gauge choice.

A precise intended repair is to supply a background PV gradient $-Uy/L_R^2$ from an explicitly compensating weak bottom slope or beta-plane background, so the current really does have uniform PV. Another repair is the rigid-lid limit $L_R\to \infty$. In the uncompensated finite-$L_R$ current model, a Gaussian can instead solve steady equations if the $a^2/L_R^2$ term is removed from the relief: then ${\nabla }^2{\psi }^{\prime }+fb/H=0$ and $Q=f-\Psi /L_R^2$, so $J(\Psi ,Q)=0$. Its PV is not uniform. These repairs distinguish the intended calculation from the false literal claim.

Smallness and closed streamlines of the intended response. Its maximum disturbance speed isIts relative vorticity is of order $|fϵ|/H$, and its surface displacement has maximum $|{\eta }^{\prime }|=|ϵ|a^2/L_R^2$. The relief satisfies $|b|\le |ϵ|(4+a^2/L_R^2)$. Sufficient conditions for quasi-geostrophic consistency are thereforeThe second condition limits the background geostrophic height change across the disturbance scale. The last controls disturbance Rossby number, surface height and bottom relief, including positive layer depth. For $a$ comparable with $L_R$ it reduces to $|ϵ|\ll H$. It does not require disturbance velocity to be small relative to $U$.

Take $U>0$ without loss of horizontal orientation. The total streamfunction is $\Psi =-Uy+A{e}^{-r^2/a^2}$, withIf $\sqrt{2/e}|A|/a<U$, then $u>0$ everywhere and no streamline can close. In terms of the relief amplitude,is the strict no-closed-cell condition. Above this closed-streamline threshold for a Gaussian disturbance in uniform flow, stagnation points occur at $x=0$ on the side opposing the current. The equation $|y|e^{-y^2/a^2}=|U|a^2/(2|A|)$ has two roots: the inner one is a center because both Hessian eigenvalues of $\Psi$ have the same sign; the outer one is a saddle. Closed contours surround the center and are bounded by a separatrix. Equality gives a degenerate onset, with no finite-area closed cell yet. If $U=0$, every nontrivial Gaussian response instead has circular closed streamlines.

Closed streamlines are compatible with small Rossby number: their threshold can be reached with $|ϵ|/H$ of order the already small background Rossby number. They do not alone invalidate quasi-geostrophy. They do, however, disconnect the cell from upstream PV data. If only the upstream PV is specified, the PV in such a cell depends on its formation history and cannot generally be assigned from the upstream value alone. Globally uniform initial PV would remain uniform in the ideal compensated model, but mixing, dissipation and forcing history can select a different physical trapped circulation. These qualifications are separate from the constant-$f$ inconsistency demonstrated above.

The vertical component $q_z=\zeta -f\eta /H$ is $H$ times the first-order anomaly in shallow-water potential vorticity about rest:Thus $\mathbf{q}=q_z\hat{\mathbf{z}}$ records the combination of relative vorticity and column stretching that is conserved by the unforced linearized shallow water equations. Indeed their curl and continuity equation give $(q_z{)}_t=0$. It is the persistent forcing of the balanced component during geostrophic adjustment, while inertia-gravity waves carry away the unbalanced component.

Initially $q_z=-f{\eta }_0\mathrm{sgn}(x)/H$. In the final geostrophic flow, $u=0$, $v=g{\eta }_x/f$, and $\zeta =g{\eta }_{xx}/f$. Conservation of the linearized shallow-water potential vorticity therefore givesBoundedness on both sides and continuity of $\eta ,{\eta }_x$ at the origin determine the constants:The solution is the locally attained balanced state after outgoing inertia-gravity waves have departed; the full inviscid system need not lose its globally conserved wave energy. The signed volume flux in the $y$ direction is

For double Kelvin-wave adjustment at a depth step, put $c_{\pm }=\sqrt{g{H}^{\pm }}$ and use $R_D^{\pm }=c_{\pm }/f$ on the corresponding side of the step. The PDF's far-field wording must mean $\eta \to {\eta }_0\mathrm{sgn}(x)$, rather than ${\eta }_0$ for every $x$. In the exponentially trapped outer response, cross-step geostrophic balance gives, at $y=0$,To retain the propagation of this slowly varying trapped disturbance, the along-step acceleration in $u_t-fv=-g{\eta }_x$ must also be retained. It supplies the small cross-step ageostrophic flow. Thus the two cross-step volume fluxes areHere $d\mathrm{sgn}(x)/dx=2\delta (x)$ as a distributional derivative. Equating the fluxes and dividing by $c_{+}+c_{-}$ givesThe printed forcing is missing a factor of two. It cannot follow from the printed sign-function surface profile with the usual Dirac delta distribution normalization. Pure geostrophic balance in both horizontal directions would also omit $A_t$ and could not determine the wave evolution.

The method of characteristics with zero initial amplitude gives the corrected weak solutionBehind the eastward front, $0<x<\Delta ct$, the surface elevation at the step becomes $-{\eta }_0$, matching the upstream side. All the original cross-step transport is diverted into flow along the step once the wave has passed a fixed location: $v=0$ there, with opposing along-step velocities above and below the step. This is the complete barrier in the adjusted region, and in the long-time limit at every fixed $x$. At finite time a front still carries cross-step flux: the formulas above give $H^{\pm }{v}^{\pm }=(2{\eta }_0{c}_{+}{c}_{-}/f)\delta (x-\Delta ct)$. The discontinuous outer representation does not resolve the short-scale fast adjustment at either front.

For completeness, solving the equation literally as printed instead gives $A={\eta }_0[\Theta (x)-\Theta (x-\Delta ct)]$. Its amplitude is half that required; $\eta (x,0,t)$ retains a jump ${\eta }_0$ at the origin after passage, so that literal equation does not imply complete blocking. The corrected equation and the literal printed equation are distinguished rather than asserting an inconsistent barrier result.

Initially $P=-f{\varphi }_0\mathrm{sgn}(x)/c^2$. In the final steady state,Equating its potential vorticity to the initial value givesThis is geostrophic adjustment: inertia-gravity waves remove the unbalanced part. Their group speed is at most $c$, so at any finite time only a region whose distance from the initial jump is $O(ct)$ can have reached the steady state.

With no $y$ dependence, the linearized shallow water equations areTheir conserved linear potential-vorticity anomaly isbecause the initial velocity jump has derivative $2v_0\delta (x)$. The final steady state is in geostrophic balance, so $u=0$ and $fv=g{\eta }_x$. HenceTaking $f>0$ for definiteness and requiring decay at infinity gives

Initially, the potential energy is zero and, per unit length in $y$,For $L\gg R_d$, the final kinetic and potential energies are equal:Thus $E_{msteady}=\rho H_0{v}_0^2{R}_d$. During geostrophic adjustment, inertia-gravity waves carry the excess energy out of $|x|<L$; energy in that finite region is therefore not conserved. Their long-wave speed is $c=\sqrt{g{H}_0}$, so the adjustment of the stated region takes

Initially $\zeta =0$, so conservation of the linear potential vorticity givesThe final flow is in geostrophic balance, henceConsequently the adjusted height solves the modified Helmholtz equationwhere $L_R$ is the Rossby deformation radius.

The initial condition is independent of $x$ and odd in $y$. The bounded solution that is continuously differentiable at $y=0$ isIt givesDuring geostrophic adjustment, the part of the initial energy incompatible with the conserved potential-vorticity distribution radiates away as inertia-gravity waves. The remaining current has width $O(L_R)$ and is geostrophically balanced.

Write $h=h_0+\eta$, with small interface displacement $\eta$ and horizontal variation scale $L$. The geostrophic pressure-Coriolis balance givesSubtracting the uniform reference state and the mass-conserving linear potential-energy term, the perturbation energy densities per unit horizontal area areThereforeThe internal deformation radius $L_R$ is the distance an internal gravity wave travels in an inverse-Coriolis time. Balanced broad disturbances $L\gg L_R$ mainly store available potential energy; balanced fine structures $L\ll L_R$ require relatively large kinetic energy. This scale result presupposes small Rossby number $U/(|f|L)$; reducing the length at fixed displacement eventually violates geostrophy. It does not say an initially resting small-scale height perturbation already has that large kinetic energy: such a perturbation undergoes geostrophic adjustment.

A numerical scale estimate requires a reference ocean density and latitude not specified exactly. Taking ${\rho }_0\simeq 1025\mathrm{kg}{\mathrm{m}}^{-3}$, $g\simeq 9.81\mathrm{m}{\mathrm{s}}^{-2}$ and a representative midlatitude $|f|\simeq 10^{-4}{\mathrm{s}}^{-1}$ givesFor a small-amplitude interface displacement of scale $L=1\mathrm{km}\ll L_R$, gravity drives rapid horizontal acceleration and radiates most of the displacement in inertia-gravity waves. The gravity adjustment time is $L/c\sim 8$ minutes, much shorter than $1/|f|\sim 2.8$ hours. A weak balanced remnant remains in an unbounded inviscid adjustment problem; energy is carried away by waves, not dissipated merely by geostrophic adjustment.

For $L=100\mathrm{km}\gg L_R$, rotational adjustment produces a largely retained interface displacement with a geostrophic current. Wave emission and inertial oscillations accompany the adjustment, but the balanced structure is much more important. To quantify that distinction under an initially resting, linear single-Fourier-component assumption, potential-vorticity conservation givesThis is about $0.002$ for the kilometre scale and $0.95$ for the hundred-kilometre scale. These fractions use $L$ as an inverse wavenumber scale, not a full wavelength, and are illustrative rather than unique predictions for an unspecified disturbance or bounded basin.