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The spatial gradient of buoyancy determines the buoyancy anomaly produced by a small parcel displacement: $b^{\prime }=-\mathbf{\xi }\cdot \nabla \bar{b}$. Its vertical component is the squared buoyancy frequency in a stably stratified Boussinesq approximation. A horizontal component enters thermal wind and the propagation of a Boundary Rossby wave.

For a semi-infinite fluid above a rigid horizontal boundary, constant buoyancy frequency $N$, and thermal wind $U=\Lambda z$, a zero-interior-potential vorticity perturbation has vertical dependence $e^{-\mu z}$, where $\mu =NK/|f_0|$ and $K=(k^2+l^2{)}^{1/2}$. Material conservation of boundary buoyancy gives $c{\varphi }^{\prime }(0)+\Lambda \varphi (0)=0$. Thus its phase velocity relative to the boundary flow is $c=\Lambda /\mu$. The wave's time dependence resides in its boundary condition; the interior field follows by potential-vorticity inversion. MIT's Eady-edge-wave notes develop this boundary-wave interpretation.

With uniform thermal wind $U=\Lambda z$ above a plane boundary, write the zero-interior-potential vorticity response as ${\psi }^{\prime }=\mathrm{Re}\{A{e}^{-\mu z+i(kx+ly)}\}$, where $\mu =N\sqrt{k^2+l^2}/f$ and $f>0$. Boundary vertical velocity $ϵ\cos (kx+ly-{\omega }_{0}t)$ gives $C=N^2ϵ/(f\mu )$ and natural angular frequency $\omega =\Lambda k/\mu$. An initially zero response has $A=C(e^{-i{\omega }_{0}t}-e^{-i\omega t})/[i(\omega -{\omega }_0)]$. Its continuous resonant limit is $Ct{e}^{-i\omega t}$: forcing in phase with the boundary mode supplies energy every cycle, and linear undamped theory predicts secular growth.

The basic quasi-geostrophic streamfunction $\bar{\psi }=-\Lambda yz$ gives $\bar{u}=\Lambda z$, $\bar{v}=0$ and constant quasi-geostrophic potential vorticity $\bar{Q}=f$. DefineBecause the background has no potential-vorticity gradient, the linear potential-vorticity equation is simplyIts characteristics have constant $y,z,x-\Lambda zt$, so the general solution is $q^{\prime }=\mathcal{F}(y,z,x-\Lambda zt)$. In particular, $q^{\prime }=F(y,z)G(x-\Lambda zt)$ is the requested arbitrary separated family; one product is not the most general initial field.

Linearizing the vertical velocity requires retaining advection of ${\bar{\psi }}_z=-\Lambda y$ by $v^{\prime }={\psi }_x^{\prime }$. ThereforeThe impermeable boundary at $z=0$ imposesThis is conservation of boundary buoyancy, not merely a condition that ${\psi }_z^{\prime }$ vanish.

For zero interior potential vorticity, write ${\psi }^{\prime }=\mathrm{Re}\{\varphi (z,t)e^{i(kx+ly)}\}$. With $K_h=\sqrt{k^2+l^2}>0$, stratified quasi-geostrophic inversion givesThe decaying solution isNo time dependence has been assumed to obtain this vertical structure. Applying the boundary condition gives $-\mu \dot{A}-i\Lambda kA=0$, henceThus the Eady edge wave travels eastward with horizontal phase velocity $\omega /k=\Lambda /\mu$ for $k\ne 0$.

The propagation mechanism can be made explicit using boundary parcels. Thermal wind gives the background buoyancy gradient ${\bar{b}}_y=f{\bar{\psi }}_{zy}=-f\Lambda$. A small meridional parcel displacement $d$ produces boundary buoyancy perturbation $b^{\prime }=-d{\bar{b}}_y=f\Lambda d$. Meanwhile the decaying potential-vorticity inversion gives $b^{\prime }=f{\psi }_z^{\prime }=-f\mu {\psi }^{\prime }$ at the boundary. Consequently ${\psi }^{\prime }=-\Lambda d/\mu$ there, and the parcel equation $d_t=v^{\prime }={\psi }_x^{\prime }$ becomesA displaced buoyancy contour therefore induces a velocity field that advances that displacement eastward. No interior potential-vorticity gradient is needed because the boundary buoyancy supplies the invertibility data.

Vortex stretching is what sets the induced flow's vertical reach. In the interior, the zero-potential vorticity condition requiresBoundary isopycnal displacement decreases with height as $e^{-\mu z}$, so fluid columns are stretched or compressed over depth ${\mu }^{-1}=f/(N{K}_h)$. Conservation of potential vorticity makes this stretching balance the relative vorticity. The resulting inversion factor $b^{\prime }=-f\mu {\psi }^{\prime }$ gives the propagation speed: stronger density stratification reduces penetration and speed, while larger $f$ increases them. Zero interior potential vorticity therefore does not imply zero interior flow.

For imposed boundary vertical velocity, retain the same decaying structure and writeThe boundary relation becomes the forced Eady edge wave equationThe integrating factor $e^{i\omega t}$ gives, for ${\omega }_0\ne \omega$,The arbitrary homogeneous coefficient $B$ sets the initial disturbance. With $A(0)=0$, this is $A=C(e^{-i{\omega }_{0}t}-e^{-i\omega t})/[i(\omega -{\omega }_0)]$.

At resonance, ${\omega }_0=\omega$, integration instead yieldsAn initially undisturbed fluid has ${\psi }^{\prime }=Ct{e}^{-\mu z}\cos (kx+ly-\omega t)$. The imposed pumping reinforces the same boundary normal mode every cycle, with no dissipative loss, so the linear amplitude grows secularly. The divergence describes resonance of the undamped linear model; sufficiently large wave amplitude invalidates the assumed small-disturbance approximation.

Use the meridional overturning streamfunction conventionDifferentiating geostrophic and hydrostatic balance gives the thermal wind relation $f_0{\overline{u}}_z=(g/{\rho }_0){\overline{\rho }}_y$. Differentiate it in time and use the mean momentum and density equations. The resulting diagnostic elliptic boundary value problem isFor the transformed circulation it becomes the Eliassen equation for residual circulation, with signs appropriate to the convention above:At the rigid meridional walls, impermeability requires each streamfunction to be constant along each wall. Set those constants to zero, choosing the solution with no imposed net vertical throughflow. Since the prescribed density flux vanishes at both walls, the same conditions apply to ${\mathcal{X}}^{\ast }$.

There is a genuine data limitation: prescribing $F$ alone does not prescribe $R$. The term $-f_0{R}_{yz}$ remains in both circulation equations. To give the usual explicit step-flux circulation, assume vanishing meridional momentum-flux divergence, $R_y=0$, with no independently imposed circulation. The formulas below are conditional on that assumption, not a unique consequence of the density flux alone.

Let $l=\pi /L$, $\kappa =N_{0}l/|f_0|$ and $A_0=-1/S>0$, using the flux unit in which its lower plateau is $-1$. The bounded solution has $\mathcal{X}=\sin (ly)\chi (z)$. Its vertical equation isAt vertical infinity require bounded Eulerian flow and vanishing residual flow. Requiring the Eulerian vertical velocity to vanish also at lower infinity would be inconsistent with the continuing lower-region flux divergence. At $z=0$, continuity of $\chi$ and ${\chi }^{\prime }$ follows because the forcing has a step but no delta function. The step-flux residual circulation in a stratified channel is thereforeHere ${\mathcal{X}}^{\ast }=\sin (ly){\chi }^{\ast }$, because $F/S=A_0\sin (ly)$ below zero and vanishes above it. The residual streamfunction decays at both infinities, whereas the Eulerian streamfunction tends to $-A_0\sin (ly)$ below and to zero above.

For $z\ne 0$, the Eulerian velocity isThe flow descends near the $y=0$ wall and rises near $y=L$, with a negative meridional return flow concentrated near $z=0$. Below the dissipation level its vertical flow balances the continuing divergence of eddy density transport. There is no independently added throughflow, but opposing vertical transports at lower infinity are required by this idealized forcing.

Away from the discontinuity the residual meridional velocity equals the Eulerian value, whileThus the residual mean circulation forms two oppositely rotating cells, confined within vertical distance $O(1/\kappa )$ of the termination of the wave flux. Its streamfunction jumps by $-A_0\sin (ly)$ at zero. Distributionally this supplies the horizontal return transportThe Dirac delta sheet is a consequence of the discontinuous imposed flux, not an extra boundary condition. Smoothing the flux termination replaces it by a thin finite return current. Streamlines must not be drawn as continuously crossing the jump without that transport.

The physical Eulerian mean flow is continuous. In the transformed momentum equation, the sheet contribution to $f_0{v}^{\ast }$ cancels the sheet in the flux divergence, leaving the smooth mean accelerationThus termination of the vertical Eliassen–Palm flux produces a finite column-integrated wave drag, redistributed over depth by the residual mean circulation.

For completeness, the general bounded response can be calculated without setting $R_y$ to zero. Expand $R_{yz}$ in wall-compatible sine modes, with coefficients $B_n(z)$, $l_n=n\pi /L$ and ${\kappa }_n=N_0{l}_n/|f_0|$. Add to either displayed streamfunction the same correctionfor decaying momentum-flux forcing. The Green function sign follows from $({\partial }_z^2-{\kappa }_n^2)e^{-{\kappa }_n|z-z^{\prime }|}=-2{\kappa }_n\delta (z-z^{\prime })$.

An explicit counterexample to uniqueness is the additional momentum flux $R=ϵ[1-\cos (2ly)]e^{-a|z|}$, with $a>0$. It vanishes at both walls and leaves the prescribed density flux unchanged. For $a\ne 2\kappa$ it addsto both circulations. This nonzero bounded, wall-impermeable solution verifies that the original mean equations need an additional momentum-flux specification to select a unique sketch.

For the Boussinesq approximation, subtract the resting background buoyancy $N^{2}z$ and denote its perturbation by $b$. With constant nonzero $f$ and constant $N>0$, the leading geostrophic flow and hydrostatic balance define the quasi-geostrophic streamfunctionThe leading horizontal flow is nondivergent. At the next order, its relative vertical vorticity $q_g={\nabla }_H^2\psi$ is stretched by the small vertical velocity, while the background buoyancy is advected vertically. Thus the curl and buoyancy equations giveThese follow from $D{q}_a/Dt=-q_a{\nabla }_H\cdot \mathbf{u}$ with $q_a\simeq f$ and ${\nabla }_H\cdot {\mathbf{u}}_a=-w_z$, retaining the geostrophic advection of the first-order anomalies. Vertical advection of those anomalies is of higher quasi-geostrophic order.

Differentiate the buoyancy equation in $z$. It is important not to assume without justification that ${\partial }_z$ commutes with $D_g$. Their commutator on $b$ isby hydrostatic balance and thermal wind. Hence $D_g{b}_z=-N^2{w}_z$. Adding $f/N^2$ times this equation to the vorticity equation eliminates $w_z$, yieldingThis is the quasi-geostrophic potential-vorticity equation on an $f$-plane. The additive $f$ makes $Q$ equal to the first-order normalized Ertel PV of total buoyancy $N^{2}z+b$; using a PV anomaly instead removes that constant and does not change the dynamics.

For stretched coordinates for quasi-geostrophic inversion, putThen $(f^2/N^2){\partial }_z^2={\partial }_Z^2$, and the potential-vorticity relation becomesStable stratification makes this operator elliptic. Uniform $N$ and nonzero $f$ are essential to this simple constant-coefficient stretching.

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.

Put $S=M^2/f_0$. The thermal wind is $\bar{U}(z)=-Sz$, and the basic potential vorticity is spatially constant: $\bar{q}=2f_0$ when the full reference buoyancy is included, or $f_0$ when that reference contribution has been subtracted. Either way ${\nabla }_h\bar{q}=0$.

For a smooth normal mode, the linear quasi-geostrophic potential-vorticity equation becomesTake $k\ne 0$ and $K>0$, as required for a nontrivial wave with the printed phase-speed parameterization. Except possibly at an isolated critical level of a shear-flow wave, this implies $\hat{q}=0$; regularity then extends that condition through the isolated level. HenceDecay at infinity excludes the growing exponential, giving the decaying vertical structure of a semi-infinite Eady edge waveThe amplitude $C$ is arbitrary. The semi-infinite Eady model supports a wave trapped at its lower boundary with penetration depth $m^{-1}$. The zero-horizontal-wavenumber case has no nonzero decaying solution of this homogeneous vertical equation. Singular neutral interior potential vorticity sheets belong to a different continuous-spectrum class; they are not the smooth decaying edge-wave eigenfunction requested here.

The PDF contains the request for this vertical structure and uses the phase factor $e^{i(kx+ly-kct)}$. The TeX aid omits that request and also inserts a duplicate factor without $c$. The solution uses the original PDF form throughout.

In an adiabatic quasi-geostrophic approximation, hydrostatic approximation and geostrophic balance give $b^{\prime }=f_0{\psi }_z$. The leading buoyancy equation is $D_g{b}^{\prime }+N^{2}w=0$; hence $w=-D_g(f_0{\psi }_z/N^2)$. At the rigid surface $z=\alpha y$, the no-normal-flow condition is $D_g(z-\alpha y)=0$, so $w=\alpha v_g=\alpha {\psi }_x$. The small boundary slope allows the disturbance condition to be evaluated at the flattened boundary $z=0$ to the retained order.

For thermal wind $U=\Lambda z$, choose the basic quasi-geostrophic streamfunction $\overline{\psi }=-\Lambda zy$. Its buoyancy anomaly is $\overline{b}=-f_0\Lambda y$, and its interior quasi-geostrophic potential vorticity is constant. Therefore the linear disturbance equations areThe term $-\Lambda {\psi }_x^{\prime }$ is the advection of the basic meridional buoyancy gradient. Since $q^{\prime }$ is simply advected at each fixed height, $q^{\prime }=0$ initially implies $q^{\prime }=0$ forever. At the lower boundary define $a=\alpha N^2/f_0$; thenThe PDF uses $\alpha$ for the lower-boundary slope throughout; the TeX's isolated $z=cy$ is a transcription error.

For nonzero zonal wavenumber let $\mu =N|k|/|f_0|>0$. The zero-interior-potential vorticity condition gives ${\hat{\psi }}_{zz}-{\mu }^2\hat{\psi }=0$. In the semi-infinite domain choose decay as $z\to \infty$. Writing $B_0(t)={\hat{\psi }}_z(0,t)$, potential-vorticity inversion then givesThis sloping-boundary Eady edge wave has all its time dependence in the boundary buoyancy equation; the interior responds instantaneously through elliptic potential-vorticity inversion. A single boundary wave oscillates without exponential growth. Vertical shear and topography contribute oppositely to its propagation. The contributions cancel when $\alpha =f_0\Lambda /N^2$, the slope of a background isopycnal. For $f_0>0$, positive $\Lambda$ without slope gives eastward propagation, while a positive slope without shear gives westward propagation. The phase velocity varies as $1/|k|$, and the trapping depth is $1/\mu$. Its frequency is independent of $|k|$ on each signed branch, so its zonal group velocity is zero in this ideal semi-infinite model.

With a rigid horizontal upper boundary, put $B_D(t)={\hat{\psi }}_z(D,t)$ and $M=\mu D$. Solving the same second-order inversion problem with both derivative data givesDifferentiation at $0$ and $D$ checks the prescribed values. At the upper boundary $w^{\prime }=0$, so $({\partial }_t+ik\Lambda D)B_D=ik\Lambda \hat{\psi }(D)$. Combining this with the lower-boundary condition givesThus two Boundary Rossby waves interact across the layer. For normal modes proportional to $e^{-ikct}$ their dimensional wave-speed matrix isFor $\Lambda \ne 0$, define $C=c/(\Lambda D)$ and $\tilde{\alpha }=a/\Lambda$. Its characteristic polynomial is the Eady model with a sloping lower boundary dispersion relationThe source writes $\tilde{\mu }=NkD/f_0$; for $k,f_0>0$ this is $M$, and the dispersion relation is unchanged by replacing that signed quantity by its absolute value. Its two roots areIf $\tilde{\alpha }>1$, the second term is positive, so $\Delta >0$ for every $M>0$: all modes are spectrally stable. If $\tilde{\alpha }<1$, the increasing function $M\tanh M$ runs from zero to infinity, so there is a unique $M$ withThere the squared term is zero and $\Delta <0$, giving an exponentially growing member of the complex conjugate pair. By continuity it lies in an unstable band. This proves both the requested instability for $0\le \tilde{\alpha }<1$ and instability for negative slopes in the sloping-boundary Eady model. It does not assert that every wavelength is unstable. Strongly negative slopes move the resonant band to large $M$, where the coupling is exponentially weak.

At $\tilde{\alpha }=1$ the speeds are real, but their crossing at $M=\coth M$ can give a defective neutral mode and algebraic growth; absence of exponential instability is weaker than boundedness of every initial disturbance. The nondimensional slope is undefined when $\Lambda =0$; the dimensional matrix remains valid and then has two real speeds. The instability for $\tilde{\alpha }<1$ is the counterpropagating wave instability mechanism: lower and upper boundary waves can match their laboratory phase speeds and exchange energy with the vertical shear.

Use the corrected PDF continuity equation ${\overline{v}}_{a,y}+{\overline{w}}_{a,z}=0$. Set $S=d{\rho }_s/dz=-{\rho }_0{N}^2/g<0$, $M=\overline{u^{\prime }{v}^{\prime }}$, and $B=\overline{{\rho }^{\prime }{v}^{\prime }}/S$. The transformed Eulerian mean replaces the Eulerian mean meridional circulation byThese substitutions preserve nondivergence, while density evolution becomes ${\overline{\rho }}_t+S{\overline{w}}_a^{\ast }=0$. Substituting ${\overline{v}}_a={\overline{v}}_a^{\ast }+B_z$ into the momentum equation givesThe geostrophic balance and hydrostatic approximation retain their original form. The vector $\mathbf{F}$ is the Eliassen–Palm flux. It gathers both eddy stresses into one mean-flow forcing rather than treating the eddy density flux as direct heating of the mean density.

For small-amplitude Rossby waves, the quasi-geostrophic wave-activity conservation law has the form ${\mathcal{A}}_t+\nabla \cdot \mathbf{F}=\mathcal{D}$, with the consistent signed wave activity convention and $\mathcal{D}$ representing sources or dissipation. Locally for a slowly varying wave packet, its Eliassen–Palm flux follows the group velocity multiplied by the signed wave-activity density. In particular, a wave can have westward phase propagation and upward wave-activity transport. Flux convergence during dissipation transfers wave momentum to the mean flow.

The non-acceleration theorem for quasi-geostrophic waves states that steady conservative waves have $\nabla \cdot \mathbf{F}=0$, and, with no independent boundary forcing or residual circulation, cannot accelerate the balanced mean zonal flow. The residual elliptic problem then has the zero solution under homogeneous impermeable/decaying conditions. The Eulerian mean circulation need not vanish: its eddy correction may be nonzero even when the residual mean circulation is zero. Time-dependent wave activity, dissipation and nonhomogeneous boundary conditions are the ways to evade the theorem. This distinction and the flux convention agree with Haynes's wave–mean-flow discussion, sections 6.2–6.3.

For the specified forcing let $\ell =\pi /L$ and $m=N\ell /|f_0|>0$. ThenIntroduce a streamfunction for the residual flow with the explicit sign conventionThe impermeable sidewalls require $v_a=0$. Since $B$ and $B_z$ vanish there, they also require $v_a^{\ast }=0$, giving this sine dependence. Differentiating geostrophic balance and hydrostatic approximation yields thermal wind, $f_0{\overline{u}}_z=(g/{\rho }_0){\overline{\rho }}_y$. Its time derivative, together with the transformed density equation, gives $f_0{\overline{u}}_{tz}=N^2{\overline{w}}_{a,y}^{\ast }$. The momentum equation consequently reduces to the Eliassen equation for residual circulation:Choose decay of the residual response at both vertical infinities, eliminating independently imposed remote circulation and exponentially growing homogeneous solutions. The wave flux itself need not vanish below the dissipating layer. These are conditions on the residual response, not an assumption that both Eulerian velocities vanish at both infinities.

Because ${\mathcal{F}}^{\prime \prime }=[\delta (z+D)-\delta (z-D)]/(2D)$, the decaying Green function $-e^{-m|z|}/(2m)$ gives the localized wave-drag response in a stratified channel:The derivative coefficient is explicitlyThe streamfunction is continuous; its derivative jumps by $+G_0/(2f_{0}D)$ at $z=-D$ and $-G_0/(2f_{0}D)$ at $z=D$ in accordance with the delta sources. The sharp change in dissipation rate permits the residual velocity jumps. The Eulerian velocity and zonal acceleration remain continuous.

Using the momentum equation givesHere $U(z)<0$ at every finite height, and it tends to zero at both infinities. Dissipation therefore produces westward acceleration throughout the channel, with exponentially decaying tails outside the directly forced layer. The geostrophic balance relation fixes the pressure tendency, up to a spatially uniform gauge:Set that gauge to zero by requiring pressure tendency to decay at both vertical infinities. For $f_0>0$, it is negative on $0<y<L/2$, zero at $L/2$, and positive on $L/2<y<L$.

The density tendency needed for the PDF's sketch follows from the transformed buoyancy equation:For $f_0>0$, it is negative in the upper southern quadrant and positive in the upper northern quadrant; below $z=0$ those signs reverse. It vanishes along $y=L/2$, along $z=0$, and at both vertical infinities. The TeX has misread this density-tendency request as a pressure tendency; the pressure formula above is also provided, but its signs do not reverse across $z=0$.