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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.

Balanced variables and PV derivation. Write the disturbance reduced pressure as $p^{\prime }/{\rho }_0=f_0\psi$. Leading geostrophic balance gives $(u_g,v_g)=(-{\psi }_y,{\psi }_x)$, and leading hydrostatic balance gives buoyancy anomaly ${\sigma }^{\prime }=p_z^{\prime }/{\rho }_0=f_0{\psi }_z$. The geostrophic material derivative isIt includes horizontal advection, not an independent prognostic vertical velocity. The appropriate regime has small Rossby number, small stratified Froude number, thin aspect ratio and slow evolution; a common scaling has $[NH/(f_{0}L){]}^2$ of order one and $\beta L/|f_0|\ll 1$.

Only the vertical rotation component is retained in the traditional approximation. In the horizontal momentum balance, the horizontal part of the rotation vector multiplies the small vertical velocity; in vertical momentum, its contribution is small compared with leading hydrostatic pressure and buoyancy in the thin-layer scaling. Away from the equator these corrections are smaller by a factor of order $(H/L)\cot \lambda$. They need not be small in equatorial or strongly nonhydrostatic settings, so the approximation has a definite regime of validity.

Put $A(z)=f_0^2/N^2(z)$. Eliminating $w$ from buoyancy givesThe last step needs justification because $D_g$ and ${\partial }_z$ do not commute on arbitrary fields. Their extra terms here areSubstitution into the supplied vorticity equation givesSince $D_{g}f=\beta v_g=\beta {\psi }_x$,This is three-dimensional quasi-geostrophic potential vorticity conservation; the height-dependent coefficient is retained inside the derivative.

At a small steady lower elevation $b(x,y)$, impermeability gives $w=u_g{b}_x+v_g{b}_y=D_{g}b$, evaluated at $z=0$ to leading order. Substitution into buoyancy and division by $N^2$ gives the topographic buoyancy boundary condition for quasi-geostrophic flow,

Rossby waves at rest. For constant $N$ let $\alpha =f_0^2/N^2$ and $K^2=k^2+\alpha m^2$. Linearization about relative rest gives ${\partial }_t({\psi }_{xx}+\alpha {\psi }_{zz})+\beta {\psi }_x=0$. A Fourier wave givesFor $\beta >0$, the zonal phase speed $\omega /k=-\beta /K^2$ is westward. At fixed altitude the PV perturbation is $q^{\prime }=-K^2{\psi }^{\prime }$; its total field is the background $f_0+\beta y$ with sinusoidally displaced contours, as sketched below.

A northward parcel displacement has $q^{\prime }=-\beta {\xi }_y$: it retains the smaller background PV of its origin. The inverted geostrophic velocity associated with a positive PV crest is northward on its eastern flank and southward on its western flank. Advection of the background PV then decreases the anomaly on the eastern flank and increases it on the western flank, shifting the crest westward. This is a PV-restoring mechanism, not a westward mean current.

Differentiating the dispersion relation gives the group velocityFor nonzero $k,m$, the vertical phase speed is $\omega /m=-\beta k/[m(k^2+\alpha m^2)]$, soThus upward energy propagation accompanies downward vertical phase propagation, and conversely. For $m=0$ the vertical group speed is zero and a vertical phase speed is not defined.

Stationary topographic response. Assume $f_0\ne 0$, $\beta >0$ and nonzero corrugation wavenumber $k$. In a uniform current the laboratory dispersion is ${\omega }_{\mathrm{lab}}=Uk-\beta k/K^2$. For a stationary quasi-geostrophic mountain wave, $K^2=\beta /U$, givingFor $U\ne 0$, the linear lower-boundary condition becomes $U{\partial }_x[(f_0/N^2){\psi }_z^{\prime }+b]=0$. Its forced nonzero-$k$ component therefore hasIf $0<U<\beta /k^2$, the vertical wavenumber is real. Both signs would be bounded aloft, so boundedness alone is insufficient for uniqueness. Impose an upward radiation condition, excluding waves incident from infinity; equivalently switch on the terrain forcing causally with arbitrarily weak absorption aloft. Since $c_{gz}$ has the sign of $km$, chooseThe unique outgoing forced component is thenThe bottom derivative and $w^{\prime }(0)=U{b}_x$ verify the phase and amplitude. In this stationary pattern it is the intrinsic frequency ${\omega }_{\mathrm{int}}=-Uk$ that describes downward vertical phase propagation relative to the current; the laboratory frequency is zero.

If $U<0$ or $U>\beta /k^2$, defineExclude exponential growth at infinity. The unique decaying forced component isAt the excluded value $U=\beta /k^2$, a nonzero prescribed bottom derivative cannot be matched by a bounded constant/linear vertical solution; the steady idealization is singular. At $U=0$ the boundary is not crossed and there is no forced response in the linear nonzero-$k$ sector: ${\psi }^{\prime }=0$, rather than an expression obtained by dividing by $U$. Uniqueness throughout refers to the prescribed forced Fourier component, with the reference state fixed and no independently incident waves or added mean-state changes.

The propagation criterion explains why topographically forced planetary Rossby waves can carry energy into a moderately westerly stratosphere, while easterlies and sufficiently strong westerlies block their vertical propagation. Larger horizontal scales more readily meet the criterion. Real dissipation, height-dependent winds and critical levels modify the simple solution and allow wave–mean-flow interaction; the ideal constant-coefficient calculation establishes the propagation filter, not a complete theory of stratospheric evolution.