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In dimensionless equatorial wave units with gravity wave speed and beta plane coefficient one, a Fourier mode $e^{i(kx-\omega t)}$ obeys $\omega u-k\eta =iyv$ and $\omega \eta -ku=-i{v}^{\prime }$. Inverting this two-by-two linear system gives the displayed wave polarization, provided ${\omega }^2\ne k^2$. Inserting it in the meridional momentum equation gives the Hermite differential equation after a Gaussian factor is removed. At singular values the original equations, rather than these quotient formulae, determine whether a trapped mode exists.

Use irrotational flow to identify where a velocity potential can describe the velocity field locally. An irrotational flow has $\mathbf{\omega }=\nabla \times \mathbf{u}=0$ throughout the fluid region. For a continuously differentiable velocity field, the Poincaré lemma supplies a velocity potential on sufficiently small neighborhoods. A global single-valued velocity potential requires zero circulation around every closed curve; a simply connected domain is a sufficient domain assumption.

The potential line vortex $\mathbf{u}=K{\mathbf{e}}_{\theta }/r$ is an irrotational flow on the punctured plane, or on an annulus $a<r<b$ with $a>0$, even though its pathlines are circular. Every sufficiently small loop bounding a disk in that fluid has zero circulation, by Stokes theorem. A loop winding once counterclockwise around the excluded origin has circulation $2\pi K$. Thus local irrotational flow and nonzero circulation around a hole are compatible: the enclosing disk passes outside the domain of the velocity field.

An irrotational flow is a kinematic condition on a velocity field, independent of whether a particular model neglects viscosity or assumes incompressible flow. Adding incompressible flow makes the local velocity potential satisfy the Laplace equation. This reduces the number of unknowns and makes potential flow a powerful starting point for flow around bodies and for gravity waves. ^[ref]

Potential-vorticity inversion is the diagnostic recovery of the flow from its advected vortical scalar, using the balance relation and specified boundary conditions. Its usefulness is a separation of tasks: transport the scalar by the current velocity, invert to find the new velocity, and repeat. The inverse is spatially nonlocal even though the transport law is local. The four models differ chiefly in the operator being inverted and the information needed at boundaries.

In two-dimensional vortex dynamics, planar incompressibility permits $\mathbf{u}=(-{\psi }_y,{\psi }_x)$, and vertical vorticity is $\zeta ={\nabla }^2\psi$. The curl of inviscid momentum givesThere is no vortex stretching in strictly planar motion, so vorticity is materially conserved. The two-dimensional potential-vorticity inversion is the Poisson equation ${\nabla }^2\psi =\zeta$. On the plane, with suitable integrability and far-field behavior, its Green's function is $(2\pi {)}^{-1}\log |\mathbf{x}-{\mathbf{x}}^{\prime }|$, givingDifferentiation gives the planar Biot-Savart kernel. Here ${\psi }_h$ is a harmonic function fixed by boundary, circulation and far-field data. A rigid impermeable wall is a streamline, so $\psi$ is constant on it; constants or circulations around distinct holes require specification. The vorticity alone cannot distinguish two flows differing by a permitted harmonic contribution, such as a uniform current on an unbounded plane. A constant added to $\psi$ has no physical effect.

In nonrotating layerwise-two-dimensional vortex dynamics, strong stable density stratification constrains motion to almost horizontal layers. In the ideal independent-layer limit, vertical velocity vanishes at leading order, $(u,v)=(-{\psi }_y,{\psi }_x)$ andhold separately at each fixed $z$. Height is a parameter: the horizontal inverse Laplacian contains no vertical derivative. With background buoyancy $N^{2}z$, the leading Ertel potential vorticity is $Q=\mathbf{\omega }\cdot \nabla b\simeq N^2\zeta$, so the materially advected normalized scalar is $Q/N^2\simeq \zeta$. The same horizontal Poisson equation is inverted layer by layer, using each layer's boundary data. This is an asymptotic constrained model, not arbitrary three-dimensional Euler motion with vertical velocity set to zero: its validity requires small Froude number and controlled vertical scales. At finite buoyancy frequency, the weak vertical motion and tilted isopycnals restore coupling. The inversion may be independent by layer while the reconstructed fields vary with height. The horizontal Euler equations for an inviscid fluid determine the pressure gradient, while hydrostatic balance relates its vertical derivative to buoyancy.

In shallow-water quasi-geostrophic inversion, begin with the exact shallow-water potential vorticity $(f+\zeta )/h$. Linearizing its anomaly about depth $H$ gives, after multiplying by $H$, $\zeta -f_0\eta /H$. For small Rossby number, geostrophic balance gives $\psi =g\eta /f_0$, while $\zeta ={\nabla }_h^2\psi$. Keeping the beta plane background gradient givesThus $q-\beta y$ is inverted with the modified Helmholtz equation $({\nabla }_h^2-L_D^{-2})\psi =q-\beta y$, under the specified boundary and far-field conditions. The velocity follows from $\psi$ and surface elevation from $\eta =f_0\psi /g$. A horizontal Fourier mode of the anomaly satisfiesThe Rossby deformation radius therefore limits the reach of the balanced response. Layer stretching, represented by $-\psi /L_D^2$, is absent from ordinary planar vorticity inversion. Unlike the pure Poisson equation, the finite-deformation operator also responds to the constant part of $\psi$, which specifies mean elevation rather than an arbitrary streamfunction offset. Specified bottom topography adds a known source to $q$; it is subtracted along with the background before inversion.

In stratified quasi-geostrophic inversion, continuously varying vertical displacement replaces the shallow layer's single stretching term. For positive $N^2(z)$ and constant nonzero $f_0$, the three-dimensional quasi-geostrophic potential vorticity isGeostrophic balance and hydrostatic balance give velocity $(-{\psi }_y,{\psi }_x)$ and buoyancy perturbation $b=f_0{\psi }_z$. Invert the anisotropic elliptic boundary value problem using interior $q$, prescribed top and bottom buoyancy as Neumann boundary conditions, and the lateral, far-field and global-flow data. Boundary buoyancy evolves by its own material conservation when the boundary is impermeable. It cannot be discarded: the Eady edge wave in the preceding solution has zero interior potential vorticity anomaly but nonzero boundary buoyancy and a nonzero induced flow.

For constant $N,f_0$, the stretched coordinates for quasi-geostrophic inversion $Z=Nz/|f_0|$ turn the inversion operator into the three-dimensional Laplacian. Its infinite-space Green's function is $-1/(4\pi |\mathbf{X}-{\mathbf{X}}^{\prime }|)$ in $(x,y,Z)$ coordinates. Hence a horizontal disturbance of scale $L$ communicates vertically over scale $|f_0|L/N$, explicitly coupling layers. In a bounded domain with homogeneous difference data, multiplying the difference of two inversions by its streamfunction and applying integration by parts givesThis proves uniqueness up to a constant when the relevant boundary terms vanish; a mean or pressure reference fixes the remaining constant. Compatible boundary fluxes are needed for a purely Neumann boundary-value problem.

Across all four models, inversion provides the induced flow, while scalar transport supplies time evolution. The independent horizontal inverse in the first two models becomes a finite-deformation inverse in shallow water and a vertically coupled elliptic inverse in continuous stratification. The principle applies within the stated balance or constrained model. In the unrestricted Boussinesq equations or shallow water equations, gravity waves and divergent motion can share the same potential vorticity, so the vortical scalar alone does not reconstruct every possible flow.

Constant density in each layer and negligible ambient acceleration far from the front giveThis primitive momentum conservation equation remains valid with slowly varying $B$. In conservative form it iswhere the right side is the longitudinal component of the hydrostatic pressure force on the sloping sidewalls. Omitting that term while retaining variable $B$ would give the wrong acceleration.

Together with volume conservation, the principal coefficient matrix in $(h,u)$ isIts real eigenvalues and characteristic curves areThese are downstream and upstream gravity waves relative to the moving fluid. The factor $1/2$ is geometric: the hydraulic depth is $A/b(h)=h/2$. Both waves move downstream in a sufficiently fast current, whereas one can carry information upstream in a slower current.

For $B=1$, differentiate $c^2=g^{\prime }h/2$ and combine the two equations. The Riemann invariants areFor variable $B$ they instead obeywhich also checks that the invariant claim depends on a prismatic channel.

Assume $\beta >0$, $c=\sqrt{gH}$, and use complex normal mode amplitudes. The zonal momentum and continuity equations giveWhen ${\omega }^2\ne k^2{c}^2$, solving these two linear equations gives

For $\hat{v}=0$, a nonzero solution requires $\omega =\pm kc$ and $\hat{\eta }=\pm c\hat{u}/g$. The remaining momentum equation gives ${\hat{u}}^{\prime }=\mp \beta y\hat{u}/c$. Only the positive sign decays at both infinities, so the equatorial Kelvin wave isIts phase and group velocities are both eastward with speed $c$.

For nonzero meridional velocity, substitute the eliminated amplitudes into $-i\omega \hat{v}+\beta y\hat{u}=-g{\hat{\eta }}^{\prime }$. The result isSet $Y=y\sqrt{\beta /c}$. The decaying Hermite polynomial modes have $\hat{v}=V_n{H}_n(Y)e^{-Y^2/2}$, up to normalization, and their eigenvalue gives the equatorial shallow-water dispersion relation

For $n=0$, the quadratic in $k$ factors asThe admissible Yanai wave root is $k=\omega /c-\beta /\omega$. With $\hat{v}(0)=1$, substitution and cancellation giveThese expressions satisfy the original equations even at the isolated point where the two algebraic roots coincide and the earlier elimination formula has a zero denominator. The other root is not an independent admissible branch: its generic singular reconstruction fails the trapping conditions.

In the zonal momentum equation, the acceleration term is $-i\omega \hat{u}=({\omega }^{2}y/c)\hat{v}$, the Coriolis term is $-\beta y\hat{v}$, and the pressure term is $-ikg\hat{\eta }=k\omega y\hat{v}$. When ${\omega }^2\ll \beta c$, acceleration is small and the leading balance is between Coriolis acceleration and the pressure gradient. When ${\omega }^2\gg \beta c$, acceleration balances pressure and the Coriolis term is small; the branch tends to an eastward gravity wave.

For $n\ge 1$, the fixed-frequency zonal roots areThe radicand is nonnegative exactly when $s={\omega }^2/(\beta c)$ satisfies $s^2-(2n+1)s+1/4\ge 0$. This gives the frequency gap for higher equatorial wave modes:

For the forcing frequency $\omega =\frac{1}{2}\sqrt{\beta c}$, we have $s=1/4$. Already the $n=1$ low-frequency upper bound is $3/2-\sqrt{2}<1/4$, and that bound decreases with $n$, while the high-frequency bound increases. Hence no $n\ge 1$ mode propagates. The available real-wavenumber branches areFor the equatorial Kelvin wave, $c_{g,K}=c$. For the Yanai wave, differentiating $k=\omega /c-\beta /\omega$ givesThus both propagating responses are detected east of the localized forcing, provided it has nonzero projection onto their meridional structures. The Yanai phase travels westward at this frequency, but its wave packet and energy travel eastward. Higher modes contribute only an evanescent response near the forcing.

Potential temperature and a parcel. For a dry ideal gas of fixed composition, the first law of thermodynamics per unit mass isA reversible adiabatic process has $\delta q=0$. Taking $c_p,R_d$ constant gives $d\ln T=\kappa d\ln p$, where $\kappa =R_d/c_p$. Integrating to a fixed reference pressure $p_0$ defines the potential temperatureIt is the temperature a parcel would have after reversible dry adiabatic process taking it to $p_0$. Equivalently $ds=c_{p}d\ln \Theta$ under these assumptions, making potential temperature a thermodynamic entropy label useful for diagnosing stratification and air-mass motion. Latent heating, radiation, mixing and variable composition require modifying the conservation statement.

An air parcel is a small material volume followed with the flow. The parcel model treats it as internally approximately uniform, large compared with molecular scales and small compared with environmental variation scales. Rapid acoustic wave adjustment makes its pressure approximately equal to the surrounding pressure, while slower heat and mass exchange can be neglected during an idealized displacement. It is not a rigid container or an isolated molecule.

At the parcel's height, environmental hydrostatic pressure supplies an upward pressure force ${\rho }_{e}g$ per unit volume. Dividing by parcel density gives the upward accelerationThe last steps use equal pressure, equal gas constants and the same reference pressure. Moisture-dependent density requires virtual temperature or its potential-temperature counterpart. Nonhydrostatic pressure perturbations and entrainment are excluded from this elementary buoyancy expression.

The convective available potential energy is the positive buoyancy work per unit mass between the level of free convection and the equilibrium level:One sums positive-buoyancy layers when there is more than one. Negative work below free convection contributes convective inhibition rather than positive CAPE. In a lossless parcel estimate, conversion of CAPE to vertical kinetic energy gives $w_{\max }\simeq \sqrt{2\mathrm{CAPE}}$; pressure work beyond this model, mixing and precipitation can change that speed.

Linear stability. Let $\xi$ denote displacement from a neutrally buoyant reference altitude. Dry motion through an adiabatic process retains ${\Theta }_p={\Theta }_e(0)$, while ${\Theta }_e(\xi )={\Theta }_e(0)+\xi {\Theta }_e^{\prime }(0)+O({\xi }^2)$. The linear vertical momentum equation is thereforeFor $N_0^2>0$, $\xi =C\cos N_{0}t+D\sin N_{0}t$: stable buoyancy oscillations with period $2\pi /N_0$. For $N_0^2=0$, $\xi =C+Dt$: neutral stratification with no restoring force. For $N_0^2<0$, growing and decaying exponentials occur, indicating convective instability. The growing solution is generic unless the initial data select only the decaying branch. These are local small-displacement results with an approximately constant environmental gradient.

Lapse-rate estimate. Define $\Gamma =-dT/dz$. Hydrostatic balance and the ideal gas law imply $p^{\prime }/p=-g/(R_{d}T)$, soIn the free-parcel-oscillator interpretation, identifying the ridge encounter interval with the buoyancy period gives $P=10^4/20=500\mathrm{s}$ and $N_0=2\pi /P=0.01257{\mathrm{s}}^{-1}$. With $T=293.15\mathrm{K}$, $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$ and $c_p\simeq 1004\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$,The frequency identification is an extra modeling assumption. A stationary mountain gravity wave generally obeys ${\hat{\omega }}^2=N_0^2{k}_h^2/(k_h^2+m_z^2)$ without rotation, with $\hat{\omega }=U{k}_h$. Ridge spacing fixes $\hat{\omega }$, not the vertical wavenumber $m_z$. For example, $m_z=k_h$ allows the same spacing and wind with $N_0=\sqrt{2}\hat{\omega }$, and a lapse rate about $0.3\mathrm{K}\mathrm{k}{\mathrm{m}}^{-1}$. Thus $5.1$ is the requested parcel-period estimate, not a uniquely identifiable lapse rate for every possible mountain-wave field; a resolved wave field needs its vertical structure as well.

Potential-vorticity dynamics. The quasi-geostrophic potential-vorticity equation expresses material conservation of a balanced potential vorticity by a horizontally nondivergent geostrophic flow. Its terms represent relative vorticity, planetary vorticity and the layer-thickness/free-surface contribution. The Rossby deformation radius $L_d$ measures the scale at which that thickness contribution competes with the horizontal-vorticity term. The quasi-geostrophic approximation filters fast acoustic waves and gravity waves, making the slower balanced evolution accessible to early digital forecasts. The 1950 barotropic-vorticity forecasting study demonstrated practical twenty-four-hour forecasts from observed mid-tropospheric fields, while identifying discretization and omitted baroclinicity as error sources.^[ref]

Near the north pole put $\delta =\pi /2-\theta$ and $r\simeq a\delta$. ThenUsing projected rather than arc distance gives the same leading polar expansion. It is a north-polar expression; the sign structure is changed consistently at the opposite pole.

In cylindrical coordinates, horizontal advection is $u_r{\partial }_{r}Q+(u_{\varphi }/r){\partial }_{\varphi }Q$. Substituting the specified streamfunction givesThe factors of $1/r$ are required by cylindrical geometry.

An exact traveling eigenfunction. This is a polar gamma-plane Rossby wave. The radial equation implies ${\nabla }_h^2\psi -\psi /L_d^2=-{\alpha }^2\psi$, so $Q=f-{\alpha }^2\psi$. Since $J(\psi ,\psi )=0$, its nonlinear self-advection vanishes exactly, regardless of amplitude. Also $f_r=-\gamma r$, giving $J(\psi ,f)=\gamma {\psi }_{\varphi }$. HenceFor $\psi =F(r)\cos (\omega t-n\varphi )$, this yieldsThe wave is retrograde relative to the balanced mean flow, since $\omega /n<0$ in this convention.

Multiplying the radial equation by $r^2$ givesFor $K^2={\alpha }^2-L_d^{-2}>0$, setting $y=Kr$ gives the standard Bessel equation with derivatives with respect to $y$. ThusRegularity at the pole excludes $Y_n$ for the usual nonnegative integers. If $K^2<0$, the corresponding general solutions are modified Bessel functions; at $K=0$ they are $r^{\pm n}$, with a constant and $\ln r$ for $n=0$. The printed Bessel hint mixes its independent-variable labels; using $y=Kr$ and derivatives in $y$ resolves that notation.

What the hexagon fixes. With the north-polar approximation and the supplied rotation period,The intrinsic phase speed at that radius is $c_{\varphi }=r_0\omega /n=-\gamma r_0/{\alpha }^2$. Therefore the measured $-4\mathrm{m}{\mathrm{s}}^{-1}$ fixesThe polar-wave deformation-radius identifiability issue is that a unique $L_d$ additionally requires a radial wavenumber or boundary condition. The quoted peak-to-peak meander is an amplitude, not a radial wavelength: multiplying the exact eigenfunction by a constant changes its amplitude without changing $K,\alpha$ or $\omega$. Thus it cannot be silently converted into $K=2\pi /\Delta r$.

A transparent order-of-magnitude closure is a slowly varying radial envelope near $r_0$, so $F^{\prime \prime }+F^{\prime }/r$ is small relative to $n^{2}F/r_0^2$. Then $K^2\simeq n^2/r_0^2$ andThis is a conditional estimate, not a uniquely determined observation. The general real-$K$ bound is $L_d\ge {\alpha }^{-1}\simeq 1.7\times 10^6$ m. Different radial structures give different values: for example $K=3.0\times 10^{-7}$ and $5.0\times 10^{-7}{\mathrm{m}}^{-1}$ yield about $2.0\times 10^6$ and $3.2\times 10^6$ m, respectively, at the same measured phase speed. Their amplitudes remain free. The given $5^{\circ }$ meander corresponds to a radial half-amplitude about $2.5\times 10^6$ m and helps characterize the pattern's size, but alone removes none of this eigenvalue ambiguity.

Both descriptions are depth-averaged hydrostatic shallow-layer models. In a one-layer free-surface shallow water equations, a homogeneous layer overlies a rigid bottom and its upper surface moves. Surface elevation changes give a restoring acceleration based on full gravity $g$, with external-wave speed $\sqrt{g{H}_0}$. Barotropic coastal storm surges, long surface gravity waves or a basin seiche are examples.

In the rigid-lid reduced gravity model, the movable boundary is a density interface beneath the active upper layer, while the much deeper lower layer remains at rest. The internal interface mode instead has speed $\sqrt{g^{\prime }{h}_0}$ and represents, for example, thermocline displacements and an upper-ocean baroclinic eddy. The mathematical shallow-water form is retained with $g$ replaced by $g^{\prime }$ and with the appropriate interface-depth convention. A freely moving two-layer surface would support both external and internal modes; the rigid lid filters out the fast external mode, and the resting abyss removes its independent dynamics.

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

Let $K=(k^2+l^2{)}^{1/2}$ and $c=\sqrt{g^{\prime }{h}_0}$. The three branches areFor nonzero $K$ and $f$, the zero-frequency mode is stationary geostrophic flow, with $f\mathbf{k}\times \mathbf{u}=-g^{\prime }\nabla \eta$ and zero horizontal divergence. It carries the linear potential-vorticity disturbance. There is no Rossby-wave propagation on a constant-$f$ plane; a varying Coriolis parameter would be needed for that.

The other two branches are oppositely phased inertia-gravity waves. Their phase-speed magnitude and radial group velocity are $|\omega |/K=\sqrt{c^2+f^2/K^2}$ and $|\partial \omega /\partial K|=c^{2}K/\sqrt{f^2+c^2{K}^2}$. When $K{L}_R\gg 1$, gravity dominates: $|\omega |\simeq cK$ and the waves are approximately nondispersive, with both speeds approaching $c$. When $K{L}_R\ll 1$, rotation dominates: $|\omega |\simeq |f|+c^2{K}^2/(2|f|)$, the frequency approaches the inertial frequency and the group velocity tends to zero. The large long-wave phase speed is not a large signal speed.

Without rotation the propagating branches become nondispersive gravity waves, $\omega =\pm cK$, and the stationary branch is a nondivergent vortical velocity disturbance rather than a finite-pressure geostrophic balance. At $K=0$, the modes reduce to constant interface height and spatially uniform inertial oscillations. The wave branches have zero linear potential-vorticity perturbation; it is the stationary branch that retains the initial potential vorticity.

Work with the explicitly corrected reduced-pressure system in b. Write $d=1/(2H)$ and use the complex wave amplitude, taking real parts at the end. The vertical factor satisfies $({\partial }_z^2-H^{-1}{\partial }_z)w=-(m^2+d^2)w$, so the wave equation givesFor a Kelvin wave, continuity, zonal momentum and the buoyancy equation then determine all polarizations:In particular $u,w,b,\Phi$ have amplitude proportional to $e^{z/(2H)}$, whereas pressure $p^{\prime }$ has amplitude proportional to $e^{-z/(2H)}$. Directly substituting these expressions into vertical momentum gives ${\Phi }_z=-i\omega (m^2+d^2)w/k^2=i(\omega -N^2/\omega )w$, verifying both the sign and the dispersion relation.

Meridional momentum is $\beta yu+{\Phi }_y=0$. Since $\Phi =cu$, it integrates to a Gaussian $\exp [-\beta y^2/(2c)]$. ThusThe eastward intrinsic branch $c>0$ is selected by decay away from the equator; choosing $c<0$ would make the meridional amplitude grow without bound. Although the squared dispersion relation has both frequency signs, a meridionally trapped Kelvin wave has this one phase-propagation direction.

For short vertical wavelengths, $m^2\gg k^2+d^2$, so ${\omega }^2\simeq N^2{k}^2/m^2$, $c\simeq N/|m|$ and $L_D^2\simeq N/(\beta |m|)$. The wave is a slowly oscillating, hydrostatic internal gravity wave with strong vertical variation and relatively narrow trapping. Here $u\simeq -(m/k)w$ and $p^{\prime }/{\rho }_0\simeq -(\omega m/k^2)w$; the buoyancy expression above remains valid.

For long vertical wavelengths, $|m|\ll \sqrt{k^2+d^2}$, the exact limiting speed is $c=N/\sqrt{k^2+d^2}$. To obtain the stated height-controlled limit also requires long horizontal wavelength, $|k|H\ll 1$. With both $|m|H\ll 1$ and $|k|H\ll 1$,The finite density scale height prevents the hydrostatic $1/|m|$ phase speed from diverging as $m\to 0$. The polarizations tend to $u\simeq -iw/(2Hk)$, $\Phi \simeq -iNw/k$ and $b\simeq -iNw/(2Hk)$. Long vertical wavelength alone at fixed large $kH$ would not justify $c=2NH$. In the opposite horizontally short limit the frequency approaches $N$ instead. The growth $e^{z/(2H)}$ compensates decreasing reference density in wave-energy density; a radiating wave train is a local modal description, not a finite-total-energy eigenfunction over an infinite vertical domain.

A mountain width is not an exactly specified sinusoidal wavelength. Taking its characteristic horizontal inverse length as $|k|\sim 1/a$ with $a=10\mathrm{km}$ gives $|k|\sim 10^{-4}{\mathrm{m}}^{-1}$. Since $N/U=10^{-3}{\mathrm{m}}^{-1}$, the stationary vertical wavenumber isThis is close to the long-horizontal-scale/hydrostatic estimate $2\pi U/N=6.28\mathrm{km}$. If instead the quoted width is identified with a sinusoidal wavelength ${\lambda }_x=10\mathrm{km}$, then $|k|=2\pi /{\lambda }_x$ and ${\lambda }_z\simeq 8.1\mathrm{km}$. Thus several kilometres, roughly $6$–$8\mathrm{km}$ under these two common width conventions, is the justified estimate; the terrain profile would fix its actual forcing spectrum.

For the inverse-width choice, both $k$ and $m$ are negative for upward energy radiation. The stationary phase lines $kx+mz=\text{constant}$ tilt upstream with increasing height, while energy propagates upward and slightly downstream. Numerically $\widehat{{\mathbf{c}}_g}\simeq (0.10,0.995)\mathrm{m}{\mathrm{s}}^{-1}$, with ray inclination about $84.3^{\circ }$. Intrinsic phase propagation points upstream and downwards; the ground pattern itself remains fixed. The original sketch below uses this explicit convention.

A ground observer may see stationary lenticular or wave clouds where ascent cools sufficiently moist air, with continuing air flow through a cloud whose position remains almost fixed. Descending regions can clear or warm. An aircraft crossing the pattern encounters alternating updraughts and downdraughts; gliders can exploit the lift. Strong amplitudes, shear or rotors may cause turbulence, but a linear gravity wave can also be smooth, and clear-air waves need not produce any visible cloud. Mountain height, humidity and nonlinear stability are not supplied, so neither a vertical-wind amplitude nor a definite atmospheric precipitation forecast follows.

Potential-vorticity inversion recovers a balanced velocity field from potential vorticity and specified boundary and global data. In two-dimensional vortex dynamics this is a Poisson equation; shallow-water quasi-geostrophic inversion and stratified quasi-geostrophic inversion use different elliptic boundary value problems. Advance the materially conserved scalar, then invert again to recover the evolving flow. Boundary buoyancy can induce flow with zero interior anomaly. The balance relation and boundary conditions are essential: unrestricted gravity waves are additional degrees of freedom. McIntyre's potential-vorticity account explains the general invertibility principle.