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The quasi-geostrophic approximation isolates slowly evolving, nearly balanced motions in a rotating, stably stratified fluid. Ocean eddies and mid-latitude atmospheric disturbances often have this character: a leading Coriolis force–pressure balance suppresses rapid horizontal acceleration, while hydrostatic balance suppresses rapid vertical acceleration. Small ageostrophic flow nevertheless supplies the vertical motion that changes the leading vorticity. The approximation filters fast inertia-gravity waves while retaining nonlinear transport, vertical shear and Rossby waves.

Use a reference density ${\rho }_0$ and remove the background hydrostatic pressure. Let $\pi =p^{\prime }/{\rho }_0$ and let the upward buoyancy perturbation be $b=-g{\rho }^{\prime }/{\rho }_0$. A vertically varying reference density stratification has $N^2(z)>0$. Under the Boussinesq approximation, the inviscid and adiabatic equations areHere $D/Dt={\partial }_t+{\mathbf{u}}_H\cdot {\nabla }_H+w{\partial }_z$. Density variations are small compared with ${\rho }_0$ but retained in the vertical buoyancy force. This is a useful local, incompressible description; a compressible atmosphere needs the corresponding weighted or pressure-coordinate version.

Let horizontal and vertical lengths be $L,H$, speed be $U$, and take the slow time $L/U$. Take $f_0>0$ for definiteness; use $|f_0|$ in the scale estimates in the opposite hemisphere. The principal small parameter is the Rossby number $\mathrm{Ro}=U/(f_{0}L)\ll 1$ at a latitude where $f_0\ne 0$. For the hydrostatic, strongly stratified regime, $H/L\ll 1$ and $f_0/N\ll 1$. Retaining both horizontal vorticity and vortex stretching gives the Burger numberThe geostrophic balance pressure scale is $f_{0}UL$ and the buoyancy perturbation scale is $B=f_{0}UL/H$. Hence $B/(N^{2}H)=\mathrm{Ro}/\mathrm{Bu}\ll 1$: buoyancy perturbations produce small displacements relative to the background stratification scale. The ageostrophic flow is $O(\mathrm{Ro}U)$ and incompressibility then gives $w=O(\mathrm{Ro}UH/L)$. Vertical acceleration is smaller than hydrostatic terms by order ${\mathrm{Ro}}^2(f_0/N{)}^2$ in this scaling. Weak forcing or diffusion can be added at the slow order, but neither is present here.

Define the quasi-geostrophic streamfunction by $\psi =\pi /f_0$. Leading geostrophic balance and hydrostatic balance implyDifferentiating horizontally and vertically gives the thermal-wind balanceThus vertical shear is determined by horizontal buoyancy gradients. The homogeneous Taylor–Proudman theorem is recovered only when those gradients vanish. Stable stratification allows sloping density surfaces and different geostrophic flow at different levels, without abandoning leading balance.

The Prandtl ratio of scales makes that modification quantitative. Horizontal and vertical terms in balanced potential-vorticity inversion have sizes $\psi /L^2$ and $(f_0^2/N^2)\psi /H^2$. Equality givesFor constant $N$, $Z=(N/|f_0|)z$ makes the inversion operator an ordinary three-dimensional Laplacian. Equivalently, an interior balanced disturbance of horizontal size $L$ naturally penetrates a depth of order $|f_0|L/N$. With $N\gg |f_0|$ this is a thin, vertically sheared structure, rather than the arbitrarily tall homogeneous columns suggested by strict Taylor–Proudman theorem behaviour. This is an aspect-ratio statement, not the diffusivity Prandtl number.

To allow a weak meridional variation of the Coriolis parameter, use the beta plane $f=f_0+\beta y$, with $\beta L/f_0=O(\mathrm{Ro})$, or equivalently $\beta L^2/U=O(1)$. The small correction to the leading balance is dynamically important on the slow time. Let $\xi ={\nabla }_H^2\psi$ be vertical relative vorticity and let $D_g={\partial }_t+{\mathbf{u}}_g\cdot {\nabla }_H$. At the first nontrivial order, the vertical vorticity equation and buoyancy equation reduce toRelative-vorticity stretching and tilting are higher order here; planetary-vorticity stretching $f_0{w}_z$ must remain. Eliminate $w$ to expose a materially conserved balanced scalar. Because ${\mathbf{u}}_{g,z}=(-b_y/f_0,b_x/f_0)$, the commutator term ${\mathbf{u}}_{g,z}\cdot {\nabla }_{H}b$ is exactly zero. For $N=N(z)$ this givesAdding the two equations proves the quasi-geostrophic potential-vorticity equationAn additive constant such as $f_0$ changes no dynamics. The horizontal streamfunction advection bracket can express this as $Q_t+{\psi }_x{Q}_y-{\psi }_y{Q}_x=0$. Although small departures from balance generated this evolution equation, the leading geostrophic flow itself advects $Q$ nonlinearly. The stretching term couples horizontal motion to vertical displacement of density surfaces.

This gives the practical advection-and-inversion description. Start with interior three-dimensional quasi-geostrophic potential vorticity, advect it with ${\mathbf{u}}_g$, and recover $\psi$ fromThe stratified quasi-geostrophic inversion is elliptic for positive $N^2$. It must be supplied with lateral and vertical boundary conditions and any circulation or mean-flow constraints; interior potential vorticity alone is not sufficient. At rigid horizontal lids, $w=0$ makes boundary buoyancy satisfy $D_{g}b=0$, and $b=f_0{\psi }_z$ supplies the vertical derivative data for inversion. Boundary buoyancy perturbations can therefore support flow even when the interior anomaly is zero. Once $\psi$ is found, its horizontal derivatives give velocity, its vertical derivative gives buoyancy, and the thermodynamic equation diagnoses $w$. This balanced evolution excludes the independent fast-wave initial data present in the full equations.

Finally, a gradient in $f$ supplies a restoring mechanism even in an otherwise uniform fluid. Linearizing about rest with constant $N$, the quasi-geostrophic potential-vorticity equation becomesSubstitution of $\psi \propto e^{i(kx+ly+mz-\omega t)}$ gives the Rossby wave dispersionFor $\beta >0$, its zonal phase speed is westward relative to a resting background. A meridionally displaced parcel retains potential vorticity, so it acquires a relative-vorticity anomaly that induces the velocity tending to return the disturbance. Background shear and stratification modify the full potential-vorticity gradient; they can alter propagation or permit instability. On a constant f-plane the planetary contribution vanishes, but gradients of relative vorticity, boundary buoyancy or topography can still support balanced wave motion. Close enough to the equator that $f_0$ vanishes, this mid-latitude quasi-geostrophic approximation and its scaling must be replaced.

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.

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.

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.

Stable stratification is a constraint on motion, not an isolated parcel spring. A light-over-heavy density profile has $N^2=-(g/{\rho }_0){\bar{\rho }}_z>0$. An adiabatically displaced parcel has buoyancy anomaly $b=-N^2{\xi }_z$ relative to its surroundings. If one neglects the pressure response and movement of surrounding fluid, the vertical equation would be ${\ddot{\xi }}_z=-N^2{\xi }_z$. A finite spherical blob cannot satisfy that isolated-oscillator picture: incompressible flow requires surrounding fluid motion, and the pressure perturbation communicates its acceleration to the rest of the fluid.

This can be shown directly with the linear Boussinesq equations. For a uniform background, take a Fourier wavevector $({\mathbf{k}}_h,m)$. Pressure removes the component of the buoyancy force parallel to this wavevector. Taking the divergence of momentum to determine pressure and projecting the vertical component givessoThis pressure projection in a displaced stratified blob produces frequencies less than $N$ except for wavevectors with $m=0$. A localized spherical perturbation contains a range of directions and therefore a range of frequencies. It deforms and emits internal gravity waves, rather than remaining one sphere oscillating coherently at $N$. The added mass of a rigid sphere is a useful inertia analogy, but its numerical correction does not define an exact oscillation frequency for a deformable material blob in stratified fluid.

Internal waves and energy transport. The restoring force supports anisotropic internal gravity waves: their frequency depends on the direction, not just the length, of the wavevector. For a two-dimensional positive-frequency branch, $\omega =Nk/\sqrt{k^2+m^2}$ with $k>0$. Differentiation givesThe vertical group and phase velocities have opposite signs. Wave energy therefore travels along a direction different from the direction of phase propagation, a point essential in selecting radiation conditions for wave-generating obstacles.

For constant $N$, the linear perturbation energy density per reference mass isDot momentum with velocity and multiply buoyancy evolution by $b/N^2$; the $bw$ terms cancel, leaving ${\partial }_t\mathcal{E}+\nabla \cdot (\Pi \mathbf{u})=0$. The second term is the available potential energy stored by displacement against the stable density profile. In an ideal fluid, that energy is exchanged with kinetic energy or transported away, not irreversibly dissipated by buoyancy diffusion.

Steady flows, layers and topography. Stable stratification penalizes vertical displacement and favors flows with small vertical motion when the stratified Froude number $U/(NH)$ is small. The natural vertical displacement scale associated with kinetic energy $U^2$ is of order $U/N$; obstacles requiring much larger displacements can cause blocking and diversion instead of simple overpassing. In the ideal model fluid retains its buoyancy label, so motion tends to follow density surfaces. This inhibits vertical mixing but does not forbid horizontal rearrangement or turbulence powered by a shear flow.

A uniform current crossing topography generates stationary internal waves when its intrinsic frequency lies in the internal-wave range. For horizontal wavenumber $k$, stationarity gives $m^2=N^2/U^2-k^2$. Height-dependent wind and stratification replace this by the Scorer parameter relation derived in Question 1. A decrease of that parameter can produce a turning level, evanescence and vertical trapping, explaining lee-wave trains confined near a lower boundary. The exact channel superpositions in Question 1 also show that nonlinearity does not automatically destroy every wave solution.

Stability of the background density profile does not imply stability of every flow built on it. Sufficiently large waves can overturn density surfaces, and shear can supply energy to disturbances. Ideal advection can create fine structure, but irreversible homogenization of buoyancy in a real fluid additionally involves diffusion or dissipation. A finite-amplitude ideal solution is consequently distinct from a guaranteed stable or physically realizable state.

How rotation changes the picture. With the vertical Coriolis parameter $f$, the corresponding uniform-background inertia–gravity dispersion becomesWhen $N>|f|$, the propagating branch lies between $|f|$ and $N$. Rotation also allows a slow balanced sector: horizontal pressure and Coriolis forces balance, vertical pressure and buoyancy balance, and the circulation can be described through potential-vorticity inversion. Small Rossby number and small stratified Froude number lead to the quasi-geostrophic equations rather than to the fast internal-wave dynamics. Vertical shear is then tied to horizontal buoyancy gradients through thermal-wind balance.

In this balanced sector, a planetary or topographic PV gradient supports Rossby waves, whose restoring mechanism is conservation of PV under displacement across that gradient. Stable stratification couples horizontal levels through the ${\partial }_z[(f_0^2/N^2){\psi }_z]$ term, and the mountain-wave calculation in Question 3 illustrates its vertical propagation filter. Thus stratification can support rapid buoyancy-restored waves, constrain slowly evolving balanced circulation, and shape interactions with boundaries; which description applies depends on the velocity, length, time and rotation scales.

Use the streamfunction convention $u={\psi }_z$, $w=-{\psi }_x$, and take $v=0$. With $J(A,B)=A_x{B}_z-A_z{B}_x$, the material derivative is $D={\partial }_t-J(\psi ,\cdot )$. This convention differs from the horizontal quasi-geostrophic streamfunction convention used below. The nonlinear Boussinesq equations areThe $y$ component of vorticity is $\zeta =u_z-w_x={\nabla }^2\psi$. Differentiate the first momentum equation in $z$ and subtract the $x$ derivative of the second. The pressure terms cancel; the derivatives of the advecting velocity cancel by $u_x+w_z=0$. Thus $D\zeta =-{\sigma }_x$, orConservation of total buoyancy, with background gradient $N^2(z)$, givesBoth equations retain the nonlinear advection terms.

Write $\psi =\bar{\psi }(z)+\varphi$, with ${\bar{\psi }}_z=U(z)=\bar{u}(z)$, and take the background buoyancy anomaly to be zero. The linear equations for $(\varphi ,\sigma )$ areFor a normal mode $\varphi =\hat{\psi }(z)e^{ik(x-ct)}$, the second equation gives $\hat{\sigma }=N^2\hat{\psi }/(U-c)$. Substitute this into the first to obtain the Taylor–Goldstein equationHere $k\ne 0$ and the calculation is initially away from critical levels $U=c$. The first term is the restoring effect of stable buoyancy on vertically displaced parcels. The second comes from advection of the background vorticity gradient $U^{\prime \prime }$: a displacement creates a vorticity anomaly, whose induced velocity acts back on the displacement. This is the shear-flow counterpart of a Rossby wave restoring mechanism; its sign can also permit instability, so it is not invariably stabilizing.

Rigid boundaries imply $w=-{\varphi }_x=0$, hence $\hat{\psi }(0)=\hat{\psi }(\pi H)=0$. For a real, regular normal mode, multiply the Taylor–Goldstein equation by $\hat{\psi }$, integrate, and integrate its second derivative by parts. The endpoint term vanishes, leavingFor a smooth real dispersion branch, differentiate this identity with respect to $k$. The stated stationarity of $I$ removes the term involving $d\hat{\psi }/dk$, so $2k=I_{c}dc/dk$. Direct differentiation at fixed $\hat{\psi }$ givesSince the frequency is $kc$, the group velocity is $d(kc)/dk$, and thereforeThis is variational group velocity of a Taylor-Goldstein mode. Singular critical levels or a vanishing denominator require separate limiting analysis; no stationarity proof is needed here.