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The Burger number compares the square of a relevant deformation radius with the square of the horizontal flow scale: $\mathrm{Bu}=R_d^2/l^2$. In a two-layer geostrophic model $R_d^2=g\prime H_1{H}_2/[f_0^2(H_1+H_2)]$. Its inverse measures the scale ratio of baroclinic available potential energy to kinetic energy. Keeping it of order unity retains both interface and relative-vorticity contributions in QG scaling.

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.

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.

Full equations and the balanced scaling. Let $H=h_{00}$, free-surface displacement $\eta$ and layer thickness $h=H+\eta -b$. The inviscid rotating shallow water equations areHere gradients are horizontal and $b$ is a fixed bottom height. For horizontal scales $L$, velocities $V$ and time scale $L/V$, acceleration is $V^2/L$, whereas the Coriolis term is $|f|V$. A small Rossby number $\mathrm{Ro}=V/(|f|L)$ therefore gives leading geostrophic balance. DefineThe last length is the Rossby deformation radius. Geostrophy gives $\eta \sim |f|VL/g$ and consequentlyThe usual distinguished quasi-geostrophic limit takes the Burger number $L_R^2/L^2$ of order one, small surface displacement, and bottom amplitude $b/H=O(\mathrm{Ro})$. The ageostrophic velocity is smaller than the geostrophic velocity by a factor of order $\mathrm{Ro}$, but its small divergence supplies the first nonzero thickness tendency. Motions are slow compared with $|f{|}^{-1}$, and the shallow-layer approximation also requires depth small compared with horizontal scale. Other deformation-radius limits are possible only if the small-height condition is retained.

Taking the vertical curl and combining it with thickness continuity gives exact shallow-water potential vorticity conservation,Expand to first order in relative vorticity, displacement and relief; multiplication by $H$ removes the irrelevant constant depth factor. To quasi-geostrophic accuracy,Potential-vorticity conservation transports this scalar with the geostrophic flow. Potential-vorticity inversion solves the elliptic problemwith the required boundary or far-field conditions, then differentiates $\psi$ to recover velocity and surface height. The deformation term suppresses the far-field influence over distances comparable with $L_R$; at scales much shorter than $L_R$ the inversion approaches the two-dimensional Poisson problem.

The full shallow-water system evolves two velocity components and layer thickness independently, with three first-order time equations. The quasi-geostrophic system evolves one scalar; balanced velocity and height are diagnostic consequences of inversion. In particular it filters the two fast inertia–gravity branches, whose resting-layer dispersion is ${\omega }^2=f^2+gH|\mathbf{k}{|}^2$, and does not describe their fast geostrophic adjustment. Reducing the time order restricts admissible initial data; it does not turn an arbitrary full shallow-water state into a quasi-geostrophic state automatically.

The intended Gaussian inversion, and its required background qualification. Set $r^2=x^2+y^2$ and ${\psi }^{\prime }=A{e}^{-r^2/a^2}$. Direct differentiation givesConsequently, in an inversion with uniform PV and its background gradient explicitly compensated, the printed relief is balanced byThis is the Gaussian topographic response with compensated uniform potential vorticity. In that model the perturbation PV is identically zero, and any steady velocity field advects this uniform PV trivially.

There is a genuine uniform-current obstruction to constant shallow-water QG potential vorticity in the literal constant-$f$, finite-$L_R$ formulation. A geostrophic background current $(U,0)$ has total streamfunction $\Psi =-Uy+{\psi }^{\prime }$ and a background free-surface slope. Over flat bottom its PV is $\overline{Q}=f+Uy/L_R^2$, not uniform. For the displayed Gaussian and printed relief, the two disturbance terms cancel in inversion but leaveFor $Uϵ\ne 0$ and finite $L_R$ this residual is nonzero; for example, it is nonzero at $x=a/\sqrt{2},y=0$. Thus the claimed Gaussian is not a steady solution of the standard unforced constant-$f$ equations as literally stated. Omitting the mean contribution to the stretching term is not a legitimate gauge choice.

A precise intended repair is to supply a background PV gradient $-Uy/L_R^2$ from an explicitly compensating weak bottom slope or beta-plane background, so the current really does have uniform PV. Another repair is the rigid-lid limit $L_R\to \infty$. In the uncompensated finite-$L_R$ current model, a Gaussian can instead solve steady equations if the $a^2/L_R^2$ term is removed from the relief: then ${\nabla }^2{\psi }^{\prime }+fb/H=0$ and $Q=f-\Psi /L_R^2$, so $J(\Psi ,Q)=0$. Its PV is not uniform. These repairs distinguish the intended calculation from the false literal claim.

Smallness and closed streamlines of the intended response. Its maximum disturbance speed isIts relative vorticity is of order $|fϵ|/H$, and its surface displacement has maximum $|{\eta }^{\prime }|=|ϵ|a^2/L_R^2$. The relief satisfies $|b|\le |ϵ|(4+a^2/L_R^2)$. Sufficient conditions for quasi-geostrophic consistency are thereforeThe second condition limits the background geostrophic height change across the disturbance scale. The last controls disturbance Rossby number, surface height and bottom relief, including positive layer depth. For $a$ comparable with $L_R$ it reduces to $|ϵ|\ll H$. It does not require disturbance velocity to be small relative to $U$.

Take $U>0$ without loss of horizontal orientation. The total streamfunction is $\Psi =-Uy+A{e}^{-r^2/a^2}$, withIf $\sqrt{2/e}|A|/a<U$, then $u>0$ everywhere and no streamline can close. In terms of the relief amplitude,is the strict no-closed-cell condition. Above this closed-streamline threshold for a Gaussian disturbance in uniform flow, stagnation points occur at $x=0$ on the side opposing the current. The equation $|y|e^{-y^2/a^2}=|U|a^2/(2|A|)$ has two roots: the inner one is a center because both Hessian eigenvalues of $\Psi$ have the same sign; the outer one is a saddle. Closed contours surround the center and are bounded by a separatrix. Equality gives a degenerate onset, with no finite-area closed cell yet. If $U=0$, every nontrivial Gaussian response instead has circular closed streamlines.

Closed streamlines are compatible with small Rossby number: their threshold can be reached with $|ϵ|/H$ of order the already small background Rossby number. They do not alone invalidate quasi-geostrophy. They do, however, disconnect the cell from upstream PV data. If only the upstream PV is specified, the PV in such a cell depends on its formation history and cannot generally be assigned from the upstream value alone. Globally uniform initial PV would remain uniform in the ideal compensated model, but mixing, dissipation and forcing history can select a different physical trapped circulation. These qualifications are separate from the constant-$f$ inconsistency demonstrated above.

For the shallow-water approximation, let the vertical scale be much smaller than the horizontal scale, so vertical acceleration is negligible and pressure is hydrostatic. Assume constant density, a flat impermeable bottom, a free surface under constant atmospheric pressure, and a horizontally moving column with no leading-order vertical shear. Neglect friction. Taking the bottom at $z=0$, hydrostatic pressure is $p=p_a+\rho g(h-z)$, so its horizontal gradient is independent of $z$. The horizontal momentum equations are thereforeIntegrating incompressibility from the bottom to the free surface, using zero bottom vertical velocity and the surface kinematic condition $w=h_t+u{h}_x+v{h}_y$, givesTogether these are the nonrotating shallow water equations.

Rotation with angular velocity $f\hat{\mathbf{z}}/2$ adds Coriolis force terms $-fv$ and $+fu$ to the left sides of the $u$ and $v$ equations, respectively. The centrifugal potential is absorbed into the background pressure and effective gravity. ThusWrite $\zeta =v_x-u_y$ and $d=u_x+v_y$. Taking the horizontal curl, including the derivatives of the advecting velocity, yieldsFor example the curl of the Coriolis force contribution is $fd$, and the curl of nonlinear advection supplies $\zeta d$. Since $q_a=f+\zeta$ and $D_{H}h=-hd$, division givesThis is exact shallow-water potential vorticity conservation within the rotating shallow water equations.

For the quasi-geostrophic approximation, take a horizontal scale $L$, speed $U$, mean depth $H$, and slow time $L/U$. Require small Rossby number $\mathrm{Ro}=U/(|f|L)\ll 1$ and small surface departure $\eta =h-H$. Introduce the Rossby deformation radius $L_R=\sqrt{gH}/|f|$ and Burger number $\mathrm{Bu}=L_R^2/L^2=O(1)$. Geostrophic balance gives $g\eta /L\sim |f|U$, hence $\eta /H\sim \mathrm{Ro}/\mathrm{Bu}\ll 1$. The Froude number is $U/\sqrt{gH}=\mathrm{Ro}/\sqrt{\mathrm{Bu}}\ll 1$. These assumptions select slow balanced motion, with fast gravity-wave transients absent at leading order.

Define the quasi-geostrophic streamfunction by $\psi =g\eta /f$. The leading geostrophic flow is $(u_g,v_g)=(-{\psi }_y,{\psi }_x)$, which is horizontally nondivergent. Expanding the exact shallow-water potential vorticity givesThe correction is smaller than $f/H$ by $O(\mathrm{Ro})$. Advecting it by the leading geostrophic flow, while the constant leading term has zero gradient, givesHere potential-vorticity conservation means that $Q$ is carried unchanged by the leading horizontal trajectories. Potential-vorticity inversion means solving the elliptic equation $({\nabla }_H^2-L_R^{-2})\psi =Q-f$, with appropriate boundary or decay conditions, to recover $\psi$ and thus the velocity. Conservation evolves the scalar field; inversion obtains its velocity field.

For a mean current $U(y)=-{\bar{\psi }}_y$, substitute $Q=\bar{Q}+Q^{\prime }$ and $\psi =\bar{\psi }+{\psi }^{\prime }$ into potential-vorticity conservation. Retaining first-order terms givesLet $\Delta Q=\bar{Q}(0^{+})-\bar{Q}(0^{-})>0$. Differentiating the mean inversion relation gives ${\bar{Q}}_y=-U^{\prime \prime }+L_R^{-2}U$. Away from $y=0$ this is zero, so decay at both infinities gives $U=U_0{e}^{-|y|/L_R}$. A velocity jump would introduce a derivative of a delta function, absent from ${\bar{Q}}_y$, so $U$ is continuous. Integrating through zero now gives $-[U^{\prime }]=\Delta Q$, and therefore

For the regular interfacial normal mode ${\psi }^{\prime }=\varphi (y)e^{ik(x-ct)}$, the linear equation readsIn each region, choose zero perturbation potential vorticity, giving ${\varphi }^{\prime \prime }-(k^2+L_R^{-2})\varphi =0$. This supplies a regular solution even if $U=c$ at isolated locations away from the interface. Continuity and decay require $\varphi =A{e}^{-\gamma |y|}$, where $\gamma =\sqrt{k^2+L_R^{-2}}$. Integrating the equation across zero givesSince $[{\varphi }^{\prime }]=-2\gamma A$, the nonzero normal mode hasThis potential-vorticity step edge wave is an interfacial Rossby wave. Displacing the potential vorticity jump creates alternating anomalies; their inverted velocity displaces the interface further with an intrinsic propagation opposite to the positive mean current. The mechanism requires a nonzero jump, or more generally a gradient, of the mean potential vorticity. It does not require planetary variation of $f$. Reversing the jump reverses the intrinsic propagation; setting the jump to zero removes this interfacial restoring mechanism.

Define the quasi-geostrophic streamfunction by $p^{\prime }/{\rho }_0=f_0\psi$. Leading horizontal geostrophic balance, $f_0\hat{\mathbf{z}}\times {\mathbf{u}}_g=-{\nabla }_{H}P$, gives $(u_g,v_g)=(-{\psi }_y,{\psi }_x)$. The hydrostatic approximation gives $P_z=\sigma$, henceThus $D_g$ is the material derivative along the leading geostrophic flow, and the vertical derivative of pressure determines the buoyancy perturbation. This also gives thermal wind: $u_{gz}=-{\sigma }_y/f_0$ and $v_{gz}={\sigma }_x/f_0$.

The balanced regime is favored by small Rossby number, a shallow aspect ratio, and a Burger number $N^2{H}^2/(f_0^2{L}^2)$ of order one. Equivalently the stratified Froude number $U/(NH)=\mathrm{Ro}/\sqrt{\mathrm{Bu}}$ is small. On a beta plane, require $\beta L/|f_0|=O(\mathrm{Ro})$, and take slow, initially balanced motion away from the equator. Under the traditional approximation, the horizontal Coriolis force generated by vertical velocity is small because vertical velocity is small, while the vertical rotational force is small compared with the leading hydrostatic pressure and gravity terms. Strong stratification and small aspect ratio justify retaining only the vertical component of Earth's rotation in this regime; this argument is not uniform near the equator.

Put $A(z)=f_0^2/N^2(z)$. The buoyancy equation gives $w=-f_0{D}_g{\psi }_z/N^2$. Substitution into the vertical vorticity equation yieldsThere is an important commutator cancellation. With $J_{xy}(a,b)=a_x{b}_y-a_y{b}_x$,Also $D_{g}f=\beta {\psi }_x$ for $f=f_0+\beta y$. Adding terms therefore proves potential-vorticity conservation:This is the three-dimensional quasi-geostrophic potential vorticity.

At an impermeable stationary lower boundary $z=b(x,y)$, the kinematic condition is $w={\mathbf{u}}_H\cdot {\nabla }_{H}b=D_{g}b$. For small elevation and slope, evaluate this at the reference level $z=0$ to leading order. Combining it with the buoyancy equation gives the rigid-boundary buoyancy condition for quasi-geostrophic wavesThe quantity in parentheses is the boundary displacement plus its buoyancy-equivalent displacement.

Now take constant $N,f$ and the mean quasi-geostrophic streamfunction $\bar{\psi }=-(U_0+\Lambda z)y$. Its mean buoyancy perturbation is $-f\Lambda y$, consistent with thermal wind, and its interior potential vorticity is constant. In the boundary condition, the perturbation velocity $v^{\prime }={\psi }_x^{\prime }$ advects the mean boundary buoyancy gradient. Keeping this contribution givesIn particular the $-\Lambda {\psi }_x^{\prime }$ term must not be omitted.

The linear interior potential-vorticity equation is $({\partial }_t+U(z){\partial }_x)Q^{\prime }=0$. Starting with zero interior perturbation potential vorticity, and with forcing only through the boundary, it remains zero. For a steady solution this is also the regular zero-interior-PV choice, continued through any isolated zero of $U$. The inversion equation for a horizontal mode isThe printed convention $K>0$ takes $f>0$; for either hemisphere the decay rate is defined using $|f|$. Boundedness aloft selects the decaying mode. Writewith real part understood. The linear boundary condition becomesFor $K{U}_0+\Lambda \ne 0$, a steady quasi-geostrophic mountain wave is thereforeOn $y=0$, its real meridional velocity isFor the positive-parameter convention $k,ϵ,f,U_0>0$ and $K{U}_0+\Lambda >0$, the velocity minimum lies one quarter wavelength downstream of each topographic maximum; its maximum lies one quarter wavelength downstream of each topographic minimum. If the velocity prefactor changes sign, maxima and minima interchange. The condition $K{U}_0+\Lambda >0$ alone does not fix that prefactor if $U_0$ is allowed to be negative. The upper panel shows the positive-current convention.

At $K{U}_0+\Lambda =0$, the free boundary Eady edge wave has zero frequency in the ground frame. If $kϵU_0\ne 0$, a steady amplitude cannot satisfy the forced amplitude equation. With $A(0)=0$, integration givesConsequentlyThe symbol multiplying $i$ here is the horizontal wavenumber $k$, as in the PDF. Taking the real part and differentiating in $x$ givesFor $U_0=-\Lambda /K>0$, positive $ϵ,f$, and $t>0$, the meridional-velocity minima coincide with topographic maxima, and its maxima with topographic minima. This is the lower panel. The change from quadrature to opposition is the phase of a resonantly forced topographic edge wave.

The stationary relief continuously excites a boundary Rossby wave whose intrinsic propagation exactly cancels mean advection. The forcing therefore remains coherent with that wave; with no damping or radiation loss for this trapped mode, its linear amplitude grows secularly. This is a resonant topographic quasi-geostrophic wave. Unlimited growth is a conclusion of the linear idealization: eventually the small-disturbance approximation fails.

The assertion that resonance must be unsteady needs nonzero forcing. If $k=0$, then $b_x=0$ and the boundary current does not cross the relief; if $U_0=0$, the forcing also vanishes. In either case the zero disturbance is a steady solution even if $K{U}_0+\Lambda =0$. For example, choose $k=0$, $\ell \ne 0$, and $\Lambda =-K{U}_0$; this is a concrete exception to an unconditional resonance claim. When both horizontal wavenumbers vanish, $K=0$ and the stated decaying-wave ansatz itself does not apply.

Use small Rossby number $U/(f_{0}l)\ll 1$, slow evolution on the advective time scale, small interface displacements relative to each layer depth, shallow hydrostatic layers, stable reduced gravity $g^{\prime }>0$, and inviscid unforced flow. On a beta plane, take $\beta l/f_0$ of the same small order as the Rossby number. The internal Burger number is retained at order unity so stratification and relative-vorticity effects can both enter the leading potential-vorticity anomaly.

The rigid-lid pressure in two-layer flow comes from neglecting the free-surface volume displacement in the rigid-lid approximation; it does not permit setting the common horizontal pressure gradient to zero. The small surface displacement multiplied by $g$ retains a finite lid-pressure multiplier. Write $h_1=H_1+\chi$, $h_2=H_2-\chi$, and let $\Pi$ denote this common pressure potential. Leading geostrophic balance giveswith ${\mathbf{u}}_i=(-{\psi }_{iy},{\psi }_{ix})$ at leading order. Literally setting $h_1+h_2$ to a spatial constant in the momentum gradients before taking the rigid-lid limit would suppress the upper-layer pressure field and fail to produce general two-layer QG dynamics.

Taking curl of each shallow-water momentum equation and using layer continuity gives material conservation of $(f+{\zeta }_i)/h_i$. Expanding it to first order and advecting the anomaly by the leading geostrophic velocity yieldsThe two-layer quasi-geostrophic potential vorticity equations areHere the common background $f_0/H_i$ has been removed and the anomaly multiplied by $H_i$. The advection term is retained at the same slow order as the time derivative even though the leading velocity/pressure balance was linear geostrophy. On an $f$ plane simply set $\beta =0$.