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A Rossby wave is a displacement of a materially conserved potential vorticity distribution whose induced balanced velocity propagates that displacement. In a quasi-geostrophic approximation interior, take a locally uniform zonal current $U$ and meridional background PV gradient $B={\overline{Q}}_y$. A small meridional parcel displacement $\eta$ produces $q^{\prime }=-B\eta$. The linear potential-vorticity equation isFor a shallow-water vertical mode, potential-vorticity inversion is $q^{\prime }=({\nabla }_H^2-L_D^{-2}){\psi }^{\prime }$ and $v^{\prime }={\psi }_x^{\prime }$. Substitution of $e^{i(kx+ly-\omega t)}$ givesThe same structure holds for other vertical modes, with their own deformation lengths, and for barotropic waves with $L_D^{-1}=0$. A topographic or shear-induced PV gradient can replace the planetary one: locally $B=\beta -U^{\prime \prime }$ in a barotropic zonal current, with the corresponding bottom contribution when present.

The restoring process can be seen directly by choosing $\eta ={\eta }_0\cos (kx-\omega t)$ in one horizontal dimension with no mean flow. Then $q^{\prime }=-B\eta$ and ${\psi }^{\prime }=B\eta /(k^2+L_D^{-2})$. The resulting $v^{\prime }={\psi }_x^{\prime }$ is a quarter wavelength out of phase with the displacement and requires $\omega =-Bk/(k^2+L_D^{-2})$. A PV anomaly is therefore not merely carried passively: its nonlocal balanced velocity moves the displaced contours.

For $B>0$, the intrinsic zonal phase velocity isChanging the sign of $k$ gives the conjugate representation of the same real pattern, not an eastward intrinsic phase branch. More generally the intrinsic phase-velocity vector has nonpositive zonal component $-B{k}^2/[|\mathbf{k}{|}^2(k^2+l^2+L_D^{-2})]$. This one-way phase property contrasts with acoustic and internal gravity waves, whose frequency equations have two independent propagation branches. The direction reverses when the relevant PV gradient reverses; a sufficiently strong current can make the laboratory phase move eastward. Thus one-way refers to propagation relative to the mean flow and a specified gradient, not an absolute geographic direction in every problem.

Scale dependence is supplied by potential-vorticity inversion. At short horizontal scales, a fixed PV anomaly creates ${\psi }^{\prime }\sim q^{\prime }/|\mathbf{k}{|}^2$, so the intrinsic phase speed is small, of order $B/|\mathbf{k}{|}^2$. At long scales with finite $L_D$, it approaches $-B{L}_D^2$. The group velocity need not have the phase direction:In particular, short nearly zonal waves can carry a packet eastward relative to the current while their crests move westward. A group-velocity statement requires a slowly varying, propagating packet and a smooth dispersion branch; turning regions and critical levels require further analysis.

Interior PV gradients are not the only support. In a stratified quasi-geostrophic approximation fluid the inversion operator includes ${\partial }_z[(f^2/N^2){\psi }_z^{\prime }]$. The boundary buoyancy $b^{\prime }=f{\psi }_z^{\prime }$ also supplies inversion data. A meridional boundary-buoyancy gradient is equivalent, in the inversion problem, to a boundary PV sheet. Consequently Boundary Rossby waves can exist even if the interior PV gradient vanishes. Their perturbations decay away from the boundary over a horizontal-wavenumber-dependent Rossby height; opposite boundaries can support oppositely directed intrinsic waves because their effective sheet gradients have opposite signs.

The Eady model makes this concrete. For $U=\Lambda z$, constant $N$ and horizontal wave-vector magnitude $k_h$, put $K=N{k}_h/|f|$. Zero interior PV gives ${\psi }_{zz}^{\prime }-K^2{\psi }^{\prime }=0$, and each rigid boundary has the buoyancy conditionA lower isolated boundary has ${\psi }^{\prime }\propto e^{-Kz}$, so $c_{\mathrm{lower}}=U(0)+\Lambda /K$. An upper isolated boundary has ${\psi }^{\prime }\propto e^{K(z-H)}$, so $c_{\mathrm{upper}}=U(H)-\Lambda /K$. For positive shear, the lower wave is intrinsically eastward and the upper wave intrinsically westward, entirely consistently with their opposite effective PV gradients.

In the finite-depth Eady dispersion relation, $\gamma =KH/2$. For $\gamma \gg 1$, $\tanh \gamma$ and $\coth \gamma$ both tend to one. The square root therefore tends to $\gamma -1$, givingThese are the two uncoupled Eady edge waves, not a pair of opposite branches on a single boundary. Their overlap is exponentially small in $KH$. For smaller separation, the interaction can phase-lock the counterpropagating boundary waves; the square root becomes imaginary for $0<KH<2.399\dots$, yielding Eady instability. The short-wave limit is stable and illustrates why interior-PV-only reasoning misses boundary wave propagation and baroclinic instability.

The governing balance requires a small Rossby number, weak fractional height or density perturbations, stable stratification where relevant, and frequencies slow relative to $|f|$. The local gradient must change little across a wavelength; on a planetary beta-plane $\beta \ell /|f|\ll 1$ is the associated small parameter. A hydrostatic stratified example also requires a small vertical-to-horizontal aspect ratio and $N$ large compared with its slow frequency. For reference, midlatitude $|f|\sim 10^{-4}{\mathrm{s}}^{-1}$ and $\beta \sim 2\times 10^{-11}{\mathrm{m}}^{-1}{\mathrm{s}}^{-1}$. Atmospheric deformation lengths of order $10^6\mathrm{m}$ give long-wave speeds of order $\beta L_D^2\sim 20{\mathrm{m}\mathrm{s}}^{-1}$ and synoptic-to-planetary times of days to weeks. An oceanic baroclinic deformation length of order $3\times 10^4\mathrm{m}$ instead gives about $0.02{\mathrm{m}\mathrm{s}}^{-1}$ and much longer basin-crossing times. Structural constants and background advection matter; these are scale estimates, not universal measured speeds.

Finite-amplitude Rossby waves can overturn or strongly filament PV contours, especially near a critical layer where their phase speed approaches the local current. Smooth inviscid motion still conserves each parcel's PV. Apparent mixing comes from the resulting fine structure under coarse-graining or from weak diffusion; it can flatten a coarse-grained background PV gradient and sharpen neighboring jet-edge gradients. A simplest description of Rossby-wave PV mixing and zonal momentum is a downgradient fluxFor barotropic zonal averages with zero mean meridional transport, $u^{\prime }=-{\psi }_y^{\prime }$, $v^{\prime }={\psi }_x^{\prime }$ and $q^{\prime }={\nabla }_H^2{\psi }^{\prime }$. Zonal integration by parts gives the Taylor identity for quasi-geostrophic fluxThus mixing down a positive PV gradient produces westward acceleration in the mixing region: wave-generated Reynolds stress redistributes zonal momentum. The inversion relation ${\overline{Q}}_y=\beta -{\overline{U}}^{\prime \prime }$ then relates PV homogenization to a reshaped current. Total zonal momentum cannot simply disappear in an isolated domain; integrated stress convergence is a boundary flux, so compensating momentum changes or boundary forcing must accompany localized drag.

In small conservative waves with $B>0$, the quadratic wave activity is $\mathcal{A}=\overline{q^{\prime 2}}/(2B)$. Multiplying the linear PV equation by $q^{\prime }/B$ gives ${\mathcal{A}}_t=-\overline{v^{\prime }{q}^{\prime }}$ for a zonal average on a locally uniform current. Together with the momentum equation this yields ${\partial }_t(\overline{U}+\mathcal{A})=0$ during conservative growth or decay. In a commonly used momentum convention the Rossby-wave pseudomomentum is $-\mathcal{A}$; signs must be stated rather than inferred from the word activity. Breaking and absorption invalidate the reversible small-wave description and deposit the wave's momentum through its flux convergence. A steady undamped conservative wave cannot provide sustained local drag without a flux divergence or boundary exchange, as expressed by the non-acceleration theorem for quasi-geostrophic waves.

Rossby propagation is the slow, direction-selective motion produced by PV advection coupled to balanced inversion. Interior gradients and boundary sheets share that mechanism; inversion determines the scale dependence, while wave packets, boundary-wave interaction and breaking govern how energy and momentum reach other regions.

The real-wavenumber dispersion relation has the explicit branchIts frequencies are real. A real-axis Fourier transform propagator multiplies every component by $e^{-i\omega (k)t}$, of modulus one, so a square-integrable localized disturbance has constant $L^2$ norm. For sufficiently smooth localized data with an integrable Fourier transform, its pointwise amplitude also has a time-independent upper bound. In particular, there is no exponential growth along a fixed or moving ray.

The Briggs-Bers criterion makes the same conclusion while enforcing causality. Start the temporal inversion line at $\mathrm{Im}\omega =\sigma >0$ and the spatial inversion contour on the real $k$ axis. The spatial roots can be followed explicitly:Choose the principal logarithm along this upper temporal half-plane. Since $\mathrm{Im}(1-2\omega )<0$, the $j=0$ root lies above the real $k$ axis; the $j=-1$ root lies below it. Successive roots remain separated by $i\pi /a$. As $\sigma$ is lowered, $k_0$ approaches the real axis when $\omega <1/2$ is real. The spatial contour is then indented beneath that root, as required by continuation from positive $\sigma$. No lower root meets it: their separation stays nonzero.

Algebraically,for every finite complex $k$, so there is no finite spatial pinch point. The logarithmic singularity at $\omega =1/2$ sends roots to $\mathrm{Re}k=+\infty$; it is on the real-frequency boundary, not a positive-growth pinch in the upper temporal half-plane. The continuum of neutral poles is the real curve $\omega <1/2$ in the $\omega$ plane. It is reached as a causal boundary value, rather than producing an upper-half-plane singularity.

The figure follows neighboring spatial roots while lowering horizontal temporal contours. It shows why merely finding roots in both spatial half-planes does not establish an instability: those roots never collide across the inversion contour.

These modes have neither absolute wave-packet instability nor convective wave-packet instability; they are temporally neutral. Their positive real group velocity transports wave packets downstream. Nonreal spatial roots describe spatial continuation and evanescence, not temporal amplification. The exponential growth criteria concern the causal impulse response, not an unrestricted choice of a complex root.

Expanding the exponential for $|ka|\ll 1$ gives $e^{-2ka}=1-2ka+2k^2{a}^2+O((ka{)}^3)$. Substitution of the plane wave $A=e^{i(kx-\omega t)}$ in the model giveswhich is exactly the quadratic approximation to the dispersion relation.

For the slowly varying width, put $X=ϵx$ and use the WKB approximationThe derivatives areAt leading order $\Omega =ak-a^2{k}^2$. On the downstream small-wavenumber branch writeso $q$ is independent of $X$. The next order gives the slowly varying triangular-jet envelope transport equationIntegration therefore yieldsIndeed, $q/(1-2q)=\Omega +O({\Omega }^2)$. The requested power is the leading low-frequency envelope, not the exact finite-frequency exponent of the quadratic model. The corresponding leading field is this wave envelope times $\exp (i{\int }_0^{x}q/a(ϵs)ds-i\Omega t)$. Positive slowly varying width and separation from the zero of the local group velocity, $1-2q=0$, are needed for this WKB approximation. Extremely large $|\log [a(X)/a(0)]|$ can also invalidate replacing the exact exponent by $\Omega$.

Away from forcing, set $F=0$. For a linear sinusoidal disturbance $A=\mathrm{Re}[\hat{A}{e}^{i(kx-\omega t)}]$, the reduced equation givesSince $\sigma ,\beta ,{\alpha }_1>0$, the phase velocity $c_p=\omega /k=\sigma k^2/(\beta {\alpha }_1)$ is positive for every nonzero wavenumber. The group velocity isThus both the linear phase and wave packet travel downstream; the sign of $k$ does not introduce an upstream branch.

Now seek a permanent nonlinear wave with $\xi =x+ct$, for which the propagation velocity is $-c$. Substitution gives $\sigma A^{(3)}-\beta {\alpha }_{1}c{A}^{\prime }-{\alpha }_{2}A{A}^{\prime }=0$. Integration with vanishing far-field data givesMultiplying by $A^{\prime }$ and integrating again, with zero integration constant, givesHere ${\alpha }_2>0$, because ${\gamma }_0>{\gamma }_1>0$ and $R<R+1$. A smooth nonzero wave approaching zero requires $c>0$: for $c<0$ the right-hand side is negative at sufficiently small nonzero $A$. At $c=0$ the allowed positive branch has no nonzero turning point, so it cannot form a smooth localized pulse.

When $c>0$, the nonzero turning point is $A_m=-3\beta {\alpha }_{1}c/{\alpha }_2$. The energy curve is positive between $A_m$ and zero and negative below $A_m$, giving a homoclinic negative pulse. The positive branch has no turning point and cannot return to zero. ThereforeAn explicit solution, also confirming smooth exponential tails, isThis establishes the opposite propagation directions in the annular displacement equation: linear waves move downstream while its localized nonlinear depression waves move upstream.

The full Love wave problem replaces the rigid base by a decaying substrate displacement. With $Y=A\cos (qz)$ in the layer and $Y=B{e}^{-\kappa (z-h)}$ below it,where ${\mu }^{\prime }={\rho }^{\prime }{\beta }^{\prime 2}$ and $\mu =\rho {\beta }^2$. Displacement continuity gives $B=A\cos (qh)$ and traction continuity gives ${\mu }^{\prime }q\sin (qh)=\mu \kappa \cos (qh)$. The rigid base is therefore a good approximation only when $\mu \kappa \gg {\mu }^{\prime }q$: then the substrate displacement is small and $qh$ lies close to ${\chi }_n=(n+1/2)\pi$.

Put $ϵ={\beta }^{\prime }/\beta$, $\eta ={\mu }^{\prime }/\mu$, $y=qh$, $W=\omega h/{\beta }^{\prime }$ and $K=kh$. The complete dispersion branch can be parameterized without root finding:Then $c={\beta }^{\prime }W/K$ and $U={\beta }^{\prime }{W}^{\prime }/K^{\prime }$. At the lower end, $c\to \beta$ and $U\to \beta$. The Love-wave cutoff frequencies are ${\omega }_{c,n}=n\pi {\beta }^{\prime }/[h\sqrt{1-{ϵ}^2}]$, with zero cutoff for the fundamental. Thus the rigid-layer formula cannot describe the fundamental's low-frequency limit or the low-frequency part of an overtone: it would either give no propagating mode or predict $c>\beta$, outside the trapping interval ${\beta }^{\prime }<c<\beta$.

For comparable densities and a fixed mode index, $\eta =({\rho }^{\prime }/\rho ){ϵ}^2\ll 1$. Write $y={\chi }_n-\delta$ near the strong-dispersion transition. In the region controlling the group-velocity minimum of a high-contrast Love wave, $\delta \ll 1$ and $\eta /\delta \gg ϵ$. The leading derivatives giveTheir minimum is atThe frequency estimate follows by inserting ${\delta }_{\min }$ into $W\simeq ({\chi }_n-\delta )[1+{\eta }^2/(2{\delta }^2)+{ϵ}^2/2]$: the two leading terms of size $\delta$ cancel. The real group velocity has a finite minimum and turns upwards near the rigid-layer threshold, whereas the rigid formula has zero group speed at its artificial threshold.

For a useful estimate of the validity range, put $\Delta =W-{\chi }_n>0$. The rigid prediction has $K_R^2=W^2-{\chi }_n^2\simeq 2{\chi }_n\Delta$. The finite-load phase correction is $\delta \simeq \eta {\chi }_n/(\kappa h)$; its changes to both $K$ and $dK/dW$ are small ifThe first condition keeps $K_R$ away from the substrate propagation boundary; the second controls the rapidly changing interface phase. Equivalently, the rigid formulas are reliable sufficiently above the actual group-velocity minimum, beyond its narrow transition interval. At fixed $\omega /{\omega }_n>1$ separated from one, they become asymptotically accurate as the contrast grows. They are particularly simple at $\omega \gg {\omega }_n$, where both velocities approach ${\beta }^{\prime }$. Within the transition and below it the full dispersion relation is required.

The small speed ratio alone is insufficient if the layer density grows so strongly that ${\mu }^{\prime }/\mu$ is not small. For such materials the exact load criterion $\mu \kappa \gg {\mu }^{\prime }q$ and its frequency derivatives should be used; the preceding contrast estimates assume a bounded order-one density ratio and fixed $n$.

Let the layer occupy $0<z<h$, with a free surface at $z=0$ and rigid base at $z=h$. Write an SH-wave as $u_y=Y(z)e^{i(kx-\omega t)}$. The equation and boundary conditions giveThus $Y=A\cos (qz)$ and the guided SH modes between rigid and free planes haveThe propagating branch has $\omega >{\omega }_n$ and $k=\sqrt{{\omega }^2-{\omega }_n^2}/{\beta }^{\prime }$. Hence the rigid-base approximation to Love waves predictsAt each threshold $c_n\to \infty$ and $U_n\to 0$. As frequency rises, the phase velocity decreases towards ${\beta }^{\prime }$ and the group velocity increases towards ${\beta }^{\prime }$. Frequencies below ${\omega }_n$ have imaginary horizontal wavenumber and do not give propagating guided modes. The infinite threshold phase speed does not violate causality: the limiting group speed is zero. These quarter-wave thresholds differ from the actual Love-wave cutoff frequencies of an elastic substrate.

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.

Let $S(z)=d{\rho }_s/dz$, $N^2=-gS/{\rho }_0>0$, and define the quasi-geostrophic streamfunction by $\psi =\tilde{p}/({\rho }_0{f}_0)$. Use horizontal scale $L$, vertical scale $D$, velocity scale $U$ and advective time $L/U$. The required quasi-geostrophic approximation has small Rossby number $\epsilon =U/(|f_0|L)$, $\beta L/|f_0|=O(\epsilon )$, and Burger number $N^2{D}^2/(f_0^2{L}^2)=O(1)$. Hydrostatic, thin-layer scaling has $D/L\ll 1$; density variations are small compared with ${\rho }_0$, as required by the Boussinesq approximation. The leading horizontal flow is nondivergent, with vertical velocity $w=O(\epsilon UD/L)$. Retain forcing of the same slow order as density advection, $r=O(U\tilde{\rho }/L)$; stronger forcing need not preserve this balanced scaling.

Leading geostrophic balance and the hydrostatic approximation giveWrite $D_g/Dt={\partial }_t-{\psi }_y{\partial }_x+{\psi }_x{\partial }_y$. The density equation becomesTaking the curl of the horizontal momentum equations at the next order and using continuity gives the stretching equationSubstitute the expression for $w$. Vertical differentiation commutes with the geostrophic material derivative acting on ${\psi }_z$ here: the extra Jacobian is $J({\psi }_z,{\psi }_z)=0$. Since $N$ depends only on height, this yields the diabatically forced quasi-geostrophic potential vorticity equationThe minus sign is appropriate to the specified density forcing; a positive upward buoyancy forcing is $R=-gr/{\rho }_0$ and gives $D_{g}q/Dt=f_0{\partial }_z(R/N^2)$.

Now take constant $N$ and linearize about rest. Put $s=f_0^2/N^2$ and $S_0=g{f}_0\mu r_0/({\rho }_0{N}^2)$. The forced equation isFor the harmonic heating response of quasi-geostrophic flow, represent the two traveling Fourier components with frequencies ${\omega }_{+}={\omega }_0$ and ${\omega }_{-}=-{\omega }_0$:Each component of the forcing has amplitude $S_0/2$, soEquivalently its vertical structure obeysThese are the time-periodic forced responses; unspecified initial data may add free Rossby waves. Selecting no incoming waves describes forcing switched on causally and then its persistent harmonic part.

For $\beta >0$, the eastward component always has ${\lambda }_{+}^2>0$. Let ${\lambda }_{+}>0$. The growing exponential would introduce an unbounded velocity and pressure disturbance at infinity, so discard it. The bottom condition ${\varphi }_{+}(0)=0$ then givesIf $\mu ={\lambda }_{+}$, this expression has the regular limit $-C_{+}z{e}^{-\mu z}/(2\mu )$, rather than a physical singularity.

For ${\omega }_0>\beta /k$, the westward component also has positive ${\lambda }_{-}^2$. Choose ${\lambda }_{-}>0$ and use the same decaying expression with minus subscripts, including the same coincident-exponent limit. Thus both components are vertically evanescent above the heating.

For $0<{\omega }_0<\beta /k$, putBoth homogeneous solutions $e^{\pm imz}$ are bounded, so boundedness is insufficient. A free Rossby wave with phase $kx+mz-\omega t$ hasThe radiation condition demands energy propagation away from the heating, upward into $z>0$. For $\beta ,k>0$ that selects positive $m$, hence $e^{imz}$ in the westward component. The resulting solution isIt satisfies ${\varphi }_{-}(0)=0$ and, away from the decaying forcing, ${\varphi }_{-}^{\prime }\sim im{\varphi }_{-}$. Its far-field amplitude does not decay in this nondissipative model: continuous forcing supplies an outgoing wave train. Thus the low-frequency westward component propagates vertically while the eastward component remains evanescent.

At the threshold ${\omega }_0=\beta /k$, the westward vertical wavenumber is zero. Excluding the linearly growing homogeneous solution and retaining the bottom condition gives ${\varphi }_{-}=C_{-}(e^{-\mu z}-1)/{\mu }^2$. This bounded limiting response tends to a constant and has zero vertical group velocity; demanding both bottom zero and decay to zero at infinity would overdetermine this threshold problem. For $\beta \le 0$, the branch labels and propagation inequality must be reconsidered; the comparison stated here assumes the usual positive beta-plane gradient.

Let $R=\overline{u^{\prime }{v}^{\prime }}$ be the eddy momentum flux, $F=\overline{{\rho }^{\prime }{v}^{\prime }}$ the eddy density flux, and $S=d{\rho }_s/dz<0$. The PDF's hydrostatic equation is $g\overline{\rho }=-{\overline{p}}_z$; the converted TeX mistakenly replaces the pressure derivative by a density derivative. With $N_0^2=-gS/{\rho }_0$, defineThe residual mean circulation is nondivergent because $v_y^{\ast }+w_z^{\ast }={\overline{v}}_{ay}+{\overline{w}}_{az}$. Substituting in the original mean momentum and density equations gives the transformed Eulerian mean equationswith the unchanged geostrophic and hydrostatic balances. Here the Eliassen–Palm flux components areThe eddy buoyancy flux has been absorbed into the residual transport, while both types of wave forcing enter the momentum equation through the Eliassen–Palm flux divergence.

For geostrophic disturbances, the Taylor identity for quasi-geostrophic flux givesFor example, zonal averaging removes $\overline{{\psi }_x^{\prime }{\psi }_{xx}^{\prime }}$ and converts the remaining horizontal and vertical terms in $\overline{{\psi }_x^{\prime }{q}^{\prime }}$ into derivatives of $\overline{{\psi }_x^{\prime }{\psi }_y^{\prime }}$ and $(f_0^2/N_0^2)\overline{{\psi }_x^{\prime }{\psi }_z^{\prime }}$, giving precisely the displayed fluxes.

Locally, for a basic flow with nonzero meridional PV gradient $Q_y$, the quasi-geostrophic wave pseudomomentum is $\mathcal{A}=\overline{q^{\prime 2}}/(2Q_y)$. Multiplying the linear perturbation PV equation by $q^{\prime }/Q_y$ giveswhere $\mathcal{D}$ is a PV source or sink. In a locally uniform conservative Rossby wave packet, $\mathsf{F}=\mathcal{A}{\mathbf{c}}_g$. Thus the flux tracks group velocity rather than the generally different phase velocity, with a signed wave activity if $Q_y<0$.

The non-acceleration theorem for quasi-geostrophic waves states that statistically steady conservative waves, with no wave-activity sources, sinks or independent boundary forcing, produce no balanced mean acceleration. Their flux divergence vanishes. In the present stable, impermeable mean-flow problem the homogeneous residual-circulation equation then has the zero solution, so $v^{\ast }=w^{\ast }=0$ and ${\overline{u}}_t=0$. The Eulerian mean flow can still circulate because its eddy-induced correction need not vanish. Dissipation, absorption at a critical layer, time-dependent wave activity or boundary forcing produces a nonzero flux divergence and invalidates the non-acceleration conclusion.

Use incompressibility, negligible viscosity and surface tension, and an initially irrotational flow. Take the undisturbed free surface at $z=0$ and bed at $z=-H$. The velocity potential $\phi$ satisfies Laplace equation and ${\phi }_z(-H)=0$. For a Fourier mode, writeThe linear kinematic boundary condition and linear Bernoulli equation at the free surface are ${\eta }_t={\phi }_z$ and ${\phi }_t+g\eta =0$. Eliminating $A,{\eta }_a$ gives the surface gravity wave dispersion relationThe rightward phase velocity and group velocity areThe phase velocity transports a crest; the group velocity transports a narrow-band envelope and its leading-order energy. For $kH\gg 1$, $\omega \sim \sqrt{gk}$ and $c_g\sim c_p/2\sim \sqrt{g/k}/2$: deep-water surface gravity waves are dispersive. For $kH\ll 1$, $\omega \sim k\sqrt{gH}$ and $c_g\sim c_p\sim \sqrt{gH}$, the nondispersive shallow-water dispersion relation. The boundary velocity controls the excited amplitude; it does not change the linear dispersion relation.

For a nonrotating inviscid Boussinesq fluid, start from $\nabla \cdot \mathbf{u}=0$, $D\rho /Dt=0$ andThe resting background has ${\bar{p}}_z=-g\bar{\rho }$ and $N^2=-g{\bar{\rho }}_z/{\rho }_0>0$. Write $\rho =\bar{\rho }(z)+{\rho }^{\prime }$, $p=\bar{p}+p^{\prime }$ and $\sigma =-g{\rho }^{\prime }/{\rho }_0$. Keeping first-order terms givesThe PDF has a genuine sign error: its momentum equation uses a minus sign before $\sigma \hat{\mathbf{z}}$ despite defining $\sigma =-g{\rho }^{\prime }/{\rho }_0$. A lighter parcel has $\sigma >0$ and accelerates upward, so the plus sign is necessary. Retaining both printed signs would yield negative squared wave frequency and exponential growth in a supposedly stable fluid; no oscillatory stable-wave dispersion relation follows from that inconsistent system. Alternatively one can define a downward density variable $\tilde{\sigma }=g{\rho }^{\prime }/{\rho }_0$, but then its density equation must have ${\tilde{\sigma }}_t=+N^2{w}^{\prime }$.