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Baroclinic instability converts potential energy of a stratified sheared basic flow into growing disturbances. In the Eady model, coupled Boundary Rossby waves provide a useful mechanism: their induced velocities displace boundary buoyancy anomalies, and suitable wavelengths permit phase locking. A single neutral boundary wave does not by itself imply exponential instability.

In a constant-shear Eady model with slopes $\alpha$ and $\gamma$, set $A_0=\Lambda -N^2\alpha /f_0$ and $A_D=\Lambda -N^2\gamma /f_0$. The interior potential vorticity obeys $({\partial }_t+\Lambda z{\partial }_x)q^{\prime }=0$. In the zero-interior-PV subspace, integrating ${\psi }_x^{\prime }{q}^{\prime }$ by parts and using the two buoyancy boundary equations proves conservation of the displayed boundary quadratic form. Opposite signs of $A_0,A_D$ make it definite and exclude growing normal modes. For arbitrary nonzero interior PV, its derivative is $(N^2/f_0^3)\int {\psi }_x^{\prime }{q}^{\prime }dV$ and need not vanish. The restriction to zero interior PV is automatic for a growing smooth mode with complex phase speed.

A boundary Rossby wave is supported by a buoyancy perturbation or potential-temperature anomaly on a rigid horizontal boundary. Its intrinsic propagation depends on the boundary orientation and basic buoyancy gradient; the upper and lower boundary waves of the Eady model propagate oppositely relative to their local background flows.

A gently sloping lower boundary modifies the lower Boundary Rossby wave of the Eady model, while a horizontal upper boundary retains the usual upper wave. With $M=N|k|D/|f_0|$, $a=\alpha N^2/(f_0\Lambda )$ and $C=c/(\Lambda D)$, the two-boundary dispersion relation isFor $a>1$ both speeds are real for every nonzero wavenumber. For every $a<1$ there is a band of exponentially growing modes; this includes negative slopes. The case $a=1$ has no exponential instability but can have a defective neutral mode at the crossing of the two uncoupled speeds.

When both rigid boundaries of a constant-shear Eady model have the same small slope $\alpha$, define $a=N^2\alpha /(f_0\Lambda )-1$, $M=N|k|D/|f_0|$ and $C=c/(\Lambda D)$. Zero-interior-PV modes have vertical structure $A\cosh (Mz/D)+B\sinh (Mz/D)$. The boundary conditions $(z/D-C){\hat{\psi }}_{z/D}+a\hat{\psi }=0$ at both endpoints give the displayed dispersion relation. For $a\ge 0$ its square root is real and there is no exponential baroclinic instability. The same-sign boundary-pseudomomentum criterion is necessary, not sufficient.

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 hydrostatic buoyancy anomaly is $b^{\prime }=f_0{\psi }_z$, and its leading adiabatic evolution is $D_g{b}^{\prime }/Dt+N_0^{2}w=0$. The small vertical velocity advects the background stratification; vertical advection of the perturbation buoyancy is higher order. ConsequentlyThe impermeability condition at a stationary lower boundary $z=\alpha y$ is $w=\alpha v$. Using $v_g={\psi }_x$ gives the topographic buoyancy boundary condition for quasi-geostrophic flowFor application at the reference plane, the actual small-height condition is $|\alpha |L/D\ll 1$. Slopes scaled by $D/L$ are of Rossby-number order, so this topographic contribution is retained alongside the small ageostrophic vertical velocity. The dimensional slope should not be compared to the Rossby number without also specifying the aspect ratio.

The basic quasi-geostrophic streamfunction is $\overline{\psi }=-\Lambda zy$. It gives $U=\Lambda z$, basic buoyancy gradient ${\overline{b}}_y=-f_0\Lambda$, and spatially constant interior quasi-geostrophic potential vorticity. Linearizing its conservation law therefore givesThe printed equation omits the shear-advection term. This cannot be omitted for general disturbances: an interior anomaly proportional to $\cos [k(x-\Lambda zt)]$ is materially conserved but not independent of time at a fixed location. The intended zero-interior-PV Eady model modes do have $q^{\prime }=0$ and satisfy both expressions; the ensuing mode calculation is valid in that subspace.

At either boundary, linearizing the geostrophic material derivative includes $v^{\prime }{\overline{\psi }}_{zy}=-\Lambda {\psi }_x^{\prime }$. LetThe two boundary equations areThese are exactly the respective bottom and top slope corrections to the rigid-boundary buoyancy condition for quasi-geostrophic waves.

Assume periodicity horizontally and first consider $q^{\prime }=0$. Multiply its defining equation by ${\psi }_x^{\prime }$ and integrate over the layer. The two horizontal Laplacian terms vanish by integration by parts; the remaining volume term after one vertical integration is an $x$-derivative. ThereforeFor nonzero $A_0,A_D$, the boundary equations giveThe top advection term integrates to zero. Subtracting proves conservation of the boundary pseudomomentum of a sloping Eady layer:The original PDF uses squared vertical derivatives and the upper slope $\gamma$ in the first denominator; these details are corrupted in the converted TeX. For arbitrary nonzero interior PV the same calculation instead gives $d\mathcal{P}/dt=(f_{0}s{)}^{-1}\int {\psi }_x^{\prime }{q}^{\prime }dV$, which need not vanish. For example, at an instant take ${\psi }^{\prime }=\cos kx+(z/D{)}^2\sin kx$, independent of $y$; the zonal average of ${\psi }_x^{\prime }{q}^{\prime }$ is $-ks/D^2$. Thus the boundary-only conservation statement also needs the zero-interior-PV qualification.

For an exponentially growing smooth normal mode, this qualification is automatic: the interior equation is $ik(\Lambda z-c)\hat{q}=0$, and a phase speed with nonzero imaginary part cannot equal the real basic velocity. Therefore $\hat{q}=0$. If $A_0{A}_D<0$, the conserved boundary quadratic form is definite, so a nonzero exponentially growing boundary amplitude is impossible. Instability consequently requiresfor $\Lambda \ne 0$. When one $A$ vanishes, the corresponding boundary derivative is simply advected and must vanish for a growing mode. The other boundary then yields a real single-edge-wave speed; if both vanish, the zero-interior-PV Neumann problem has no nonzero growing mode. These degenerate cases are neutral and should not be handled by dividing by zero.

For equal slopes, put $\zeta =z/D$, $C=c/(\Lambda D)$, $M=N_0|k|D/|f_0|$, and $a=N_0^2\alpha /(f_0\Lambda )-1$. The normal-mode interior equation is ${\Phi }_{\zeta \zeta }-M^2\Phi =0$ for the zero-interior-PV modes. ThusThe coefficient equations areSetting their determinant to zero, without dividing by $C$ or $a$, givesso the Eady model with parallel sloping boundaries hasFor positive $k,f_0$, $M$ is the printed $\mu$, and $a$ is the printed $\tilde{\alpha }$. The expression is even in the sign of that vertical scale. For $a\ge 0$ the radicand is at least $1/4$, so both speeds are real and these modes have no exponential growth. For equal slopes the earlier necessary condition is merely $a^2>0$, which is satisfied in many stable cases and is not sufficient. At $a=0$ the speeds are $0$ and $\Lambda D$; the invariant with nonzero denominators cannot be used, but the determinant still correctly gives neutral modes. No claim excluding transient amplification is needed.

For an exponentially growing normal mode with $k\ne 0$, $\mathrm{Im}(kc)>0$. Such a complex $c$ cannot equal the real basic velocity anywhere, so the interior equation forces $\hat{q}=0$ at every height. Decay then fixes the unique vertical structure $C{e}^{-mz}$, and the lower-boundary condition fixes $c=-S/m$, which is real. This contradiction proves the absence of exponential instability in the semi-infinite Eady model:For $k=0$, there is no advective frequency in the linear interior equation, so an exponentially growing mode would again have $q^{\prime }=0$. The wall buoyancy equation then forces ${\psi }_z^{\prime }(0)=0$, which the nonzero decaying exponential cannot satisfy. Thus these modes do not provide a missing growing branch. This conclusion rules out exponential eigenmode growth, not every possible transient response or singular neutral potential vorticity disturbance.

With a second rigid boundary at $z=D$, the same uniform-shear flow is the Eady model. Both boundaries carry buoyancy perturbations and support Boundary Rossby waves. Their intrinsic propagation directions oppose one another relative to their respective local basic flows. At suitable wavelengths their interacting fields can phase-lock and release basic-state potential energy, producing baroclinic instability. The single-boundary system lacks that second interacting boundary wave.

For a direct comparison, write $\Lambda ={\bar{U}}_z=-S$ and $m=NK/|f_0|$. The vertical solution between the two boundaries is $A\cosh (mz)+B\sinh (mz)$, and their buoyancy conditions areEliminating $A,B$ gives the finite-depth Eady dispersion relationThe bracket is negative for $0<mD<2.399\dots$, so those modes with $k\ne 0$ have a growing and a decaying member. At short wavelengths the two edge waves interact weakly and the frequencies are real. In the semi-infinite limit, the lower-wave root tends to $c=\Lambda /m=-S/m$, recovering the neutral result above.

A uniformly stratified quasi-geostrophic approximation fluid with constant vertical shear occupies $z>0$, has a flat rigid lower wall, and has decaying perturbations aloft. On an f-plane, the basic interior potential vorticity is constant. Its smooth edge-wave modes have zero interior anomaly and propagate along the lower-boundary buoyancy gradient. The absence of an upper wall removes the pair of interacting edge waves that makes the finite-depth Eady model unstable.