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For constant Coriolis parameter $f_0$ and buoyancy frequency $N$, thermal-wind balance and transformed mean-flow equations givewhere ${\chi }^{\ast }$ is the residual-circulation streamfunction and $\mathbf{F}$ is the Eliassen–Palm flux.

For a sinusoidal Eliassen–Palm flux whose vertical divergence is constant in $-D<z<D$ and zero elsewhere, the residual mean circulation is determined by a forced one-dimensional modified Helmholtz equation. With $v^{\ast }=C^{\prime }(z)\sin (\pi y/L)$ and $w^{\ast }=-(\pi /L)C(z)\cos (\pi y/L)$, a forcing proportional to $-G_0\sin (\pi y/L)/(2D)$ givesThe response is uniquely selected by decay of the residual circulation at both vertical infinities. An Eulerian mean vertical flow can persist in the undamped wave region even where this residual response vanishes.

Steady conservative small-amplitude waves have divergence-free Eliassen–Palm flux by wave activity conservation. In a balanced mean state with homogeneous impermeable boundaries and no independently forced residual mean circulation, this gives zero residual circulation and zero mean acceleration. The Eulerian mean circulation can nevertheless be nonzero because the residual circulation includes an eddy-induced correction. Wave dissipation, time-dependent wave activity or boundary forcing can invalidate the conclusion.

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.

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.

The geostrophic material derivative isIt advects horizontally with the geostrophic flow. The exact material derivative additionally includes horizontal ageostrophic flow and vertical advection $w{\partial }_z$. The leading balances are geostrophic balance and the hydrostatic approximation:The conventional quasi-geostrophic streamfunction is $p=f_0\psi$, after fixing the horizontally uniform reference pressure consistently with the basic buoyancy profile. ThereforeHere and below pressure is divided by reference density, and $\sigma$ is the anomaly about the prescribed background buoyancy.

For the zonal average assume periodicity or vanishing boundary terms, so that averages of $x$ derivatives vanish and averaging commutes with $y,z$ derivatives. Put $S(z)=f_0^2/N^2(z)$. Since $\overline{{\psi }_x^{\prime }}=0$, the disturbance quasi-geostrophic potential vorticity is $Q^{\prime }={\psi }_{xx}^{\prime }+{\psi }_{yy}^{\prime }+{\partial }_z(S{\psi }_z^{\prime })$. The zonal part of $\overline{{\psi }_x^{\prime }{Q}^{\prime }}$ vanishes as $\frac{1}{2}\overline{{\partial }_x({\psi }_x^{\prime }{)}^2}$. For the other parts, integration by parts givesAs $u^{\prime }=-{\psi }_y^{\prime }$, $v^{\prime }={\psi }_x^{\prime }$, and ${\sigma }^{\prime }=f_0{\psi }_z^{\prime }$, the Eliassen–Palm flux isWe have therefore proved the exact Taylor identity for quasi-geostrophic flux,without imposing small amplitude. This identity is algebraic within the quasi-geostrophic approximation; it does not assert validity of that approximation for arbitrary physical flows.

Now linearize about a steady zonal basic flow $\bar{u}=U(y,z)$, with ${\bar{Q}}_y\ne 0$ and no mean meridional velocity. The linear quasi-geostrophic potential-vorticity equation is $Q_t^{\prime }+U{Q}_x^{\prime }+v^{\prime }{\bar{Q}}_y=0$. Multiply by $Q^{\prime }/{\bar{Q}}_y$ and average. The advection term is a zonal derivative and vanishes, givingThis is quasi-geostrophic wave-activity conservation law. The basic-state denominator is time-independent at the retained order; its change caused by the waves is higher order. The wave activity is signed if ${\bar{Q}}_y<0$, and the formula cannot be used where that gradient vanishes. The original PDF includes the zonal average over the squared disturbance in this definition.

For the resonant solution it is helpful to avoid squaring an imaginary coefficient as though it were real. Writewhere $B_0$ is real. Assume $f_0\ne 0$, $\beta \ne 0$, and $k\ne 0$. Define $c(z)=\cos (mz)$ and $r(z)=(z-H)\sin (mz)$. The complex coefficient of the physical streamfunction disturbance isThe original PDF contains $Bt\cos (mz)$: the converted TeX's missing $t$ would incorrectly remove the resonance and fails the differential equations.

For the vertical potential-vorticity inversion operator, $c^{\prime \prime }=-m^{2}c$ and $r^{\prime \prime }=-m^{2}r+2mc$. ConsequentlySubstitution in the linear potential-vorticity equation givessince ${\kappa }^2{B}_0=2k{f}_{0}U$. Thus this is a genuine resonant topographic quasi-geostrophic wave, with linear amplitude growth rather than an assumed stationary wave.

At the flat upper boundary $h_z(H,t)=0$, so $w^{\prime }(H)=0$. At the lower boundary, linearization of the impermeability condition gives $w^{\prime }(0)=U{\partial }_x{h}_b$, where $h_b=\mathrm{Re}\{\epsilon H{e}^{ikx}\sin (\ell y)\}$ is the bottom corrugation. Here $h_z(0,t)=-CHm$ is time-independent, so the specified vertical velocity givesusing $f_{0}Cm/N^2=1$. Both boundary conditions are verified. The linear solution is valid while its growing displacement and velocity remain small; it is not a large-time nonlinear solution.

To evaluate all fluxes, use $\overline{\mathrm{Re}(a{e}^{ikx})\mathrm{Re}(b{e}^{ikx})}=\frac{1}{2}\mathrm{Re}(a{b}^{\ast })$. The coefficients of $u^{\prime },v^{\prime },{\sigma }^{\prime }$ are respectively $-\epsilon \ell \cos (\ell y)h$, $i\epsilon k\sin (\ell y)h$, and $\epsilon f_0\sin (\ell y)h_z$. HenceThe products of the $B_{0}t$ terms alone and the $C$ terms alone are imaginary in the vertical flux and contribute nothing. Only the cross terms remain. Sincewe obtainDifferentiating $R$ gives $R^{\prime }=m(1+\cos (2mz))=2m{\cos }^2(mz)$. As $SmC=f_0$,For an independent evaluation of the right side of the Taylor identity for quasi-geostrophic flux, write $q=(2f_{0}c-{\kappa }^{2}Cr)-i{\kappa }^2{B}_{0}tc$. Multiplication of $ikh$ by $q^{\ast }$ gives real part $-2k{f}_0{B}_{0}t{c}^2$, because the remaining real cross terms cancel. Thereforewhich verifies the flux identity pointwise in $y,z$.

The basic potential-vorticity gradient is ${\bar{Q}}_y=\beta$, so the wave activity and its derivative areHere ${\kappa }^4{B}_0^2/(2\beta )=k{f}_0{B}_0$ follows from $U=\beta /{\kappa }^2$. Thus ${\mathcal{A}}_t+F_y+G_z=0$. The growth term involves $B_0^2=|B{|}^2=-B^2$; the distinction matters because the printed complex coefficient $B$ is imaginary. These formulas evaluate both flux components, their divergence, the buoyancy and PV covariances, and wave activity. In particular $G(H)=0$ and $G(0)={\epsilon }^{2}k{f}_0{B}_{0}Ht{\sin }^2(\ell y)/2$, so for $\beta >0$ the growing activity is supplied by the positive flux from the lower boundary.

Finally put $W=\pi /\ell$ and write the height-independent mean velocity change as $\delta U(y)$. Since $\delta {\bar{\psi }}_y=-\delta U$ and $\delta \bar{\psi }$ can be chosen independent of height, the mean potential vorticity becomesTo be uniform in $y$ this requires $\delta U_y=\beta y-K$ for some constant $K$. Integrating and applying both endpoint conditions gives $\delta U(0)=0$, $\delta U(W)=0$, and hence $K=\beta W/2$. ThereforeThe constant is the initial channel-mean potential vorticity, as also follows from ${\int }_0^W\delta Qdy=-[\delta U{]}_0^W=0$. This is the momentum loss from uniform potential-vorticity mixing. Averaging the velocity change across the channel gives the momentum change per unit mass,For positive $\beta$ the mean current is reduced everywhere in the interior. The imposed bottom can exchange momentum with the fluid, so this change is consistent with the complete conservation of momentum budget.

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 $z=0$.

The corresponding Eulerian mean streamfunction isFor $f_0>0$ its bracket is positive and decreases with $z$. Thus the Eulerian meridional flow is southward everywhere; the vertical flow is downward in the southern half and upward in the northern half. Above the wave region it vanishes, but below itThis nonzero remote Eulerian vertical flow is cancelled by the eddy contribution in the residual circulation. Imposing zero Eulerian vertical velocity there would be incompatible with the prescribed undamped wave density flux.

The residual circulation consists of two oppositely directed cells: $C$ is positive above $z=0$ and negative below. Inside the dissipation layer $v_a^{\ast }>0$; outside it $v_a^{\ast }<0$. The upper cell moves northward on its lower flank, upward on the northern side, southward aloft and downward on the southern side; the lower cell has the reverse vertical sense.

Finally, integrate the momentum equation in height. Since $C$ vanishes at both ends and $\mathcal{F}(+\infty )-\mathcal{F}(-\infty )=1$, vertically integrated wave-drag acceleration isPutting $s=mD$ and integrating the interior expression givesThe full vertical integral equals the deposited wave momentum independently of stratification. The fraction within the dissipating layer increases from zero to one: for $s\ll 1$ it is $s+O(s^2)$, while for $s\gg 1$ it is $1-1/(2s)+O(e^{-2s}/s)$. Weak stratification or a thin layer spreads acceleration over the remote vertical scale $1/m$; strong stratification or a thick layer confines most acceleration to the dissipation region.

Begin with the Boussinesq approximation primitive equations on a beta plane, decompose every field into a zonal mean and a disturbance, and average over longitude. The zonal momentum equation then contains the divergence of the eddy momentum flux $\overline{u^{\prime }{v}^{\prime }}$, while the mean density equation contains the divergence of the eddy density flux $\overline{{\rho }^{\prime }{v}^{\prime }}$. At small Rossby number, use geostrophic balance, hydrostatic pressure, thermal-wind balance, and the leading eddy equations to combine those fluxes.

Defineand introduce the residual mean circulationThe added eddy-induced velocity is nondivergent, soIt absorbs the eddy density-flux divergence into advection by the transformed circulation, giving

The mean zonal momentum equation becomeswhere the zonally averaged Eliassen–Palm flux in the meridional-vertical plane isThus the transformed Eulerian mean gathers the wave forcing into one flux divergence and makes density evolve under one residual circulation.

Put $l=\pi /L$. For the specified quasi-geostrophic streamfunction, the complex amplitudes of the disturbance fields areThe product of these two amplitudes is purely imaginary after one is conjugated, so its zonal mean vanishes:Consequently

Geostrophic balance and hydrostatic pressure giveThereforeSince $d{\rho }_s/dz=-{\rho }_0{N}^2/g$, the vertical Eliassen–Palm flux isThus it has the stated formwith

Define the transformed Eulerian mean circulation byThe added terms form a nondivergent eddy-induced circulation, so ${\partial }_y{\overline{v}}_a^{\ast }+{\partial }_z{\overline{w}}_a^{\ast }=0$. The buoyancy equation becomesSubstituting ${\overline{v}}_a={\overline{v}}_a^{\ast }+N^{-2}{\partial }_z\overline{v^{\prime }{\sigma }^{\prime }}$ into the momentum equation giveswhere the components of the Eliassen–Palm flux are