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The local meridional eddy buoyancy flux is the covariance of disturbance meridional velocity with disturbance buoyancy. In a Boussinesq approximation, $b^{\prime }=-g{\rho }^{\prime }/{\rho }_0$. This is a local transport density, distinct from a plume's cross-section-integrated buoyancy flux. In the transformed Eulerian mean, division by the background buoyancy frequency squared gives an eddy-induced meridional overturning streamfunction.

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 $f(y)$ denote the signed vertical component of the Coriolis parameter; it is negative in the southern hemisphere. In the steady, linear, inviscid interior, depth integration of horizontal momentum giveswhere $\mathbf{U}={\int }_{-H}^0(u,v)dz$ and $\Pi ={\int }_{-H}^{0}pdz/{\rho }_0$. Horizontal curl eliminates the pressure. Its left side is $f{\nabla }_h\cdot \mathbf{U}+\beta V$. Incompressibility and impermeable flat top and bottom imply ${\nabla }_h\cdot \mathbf{U}=w(-H)-w(0)=0$, givingThis is Sverdrup balance. It assumes steady small-amplitude motion, a Boussinesq approximation, negligible nonlinear advection, negligible Rayleigh and bottom/lateral drag in the interior, a flat fixed-depth domain and no normal top/bottom flux. The applied surface wind stress remains a forcing, even though interior friction is neglected. With cooling absent, the steady buoyancy equation additionally requires $B_z=0$, if that equation is imposed; the curl balance itself does not depend on the cooling coefficient. In the closed, $x$-independent channel below, depth-integrated $V=0$, so a nonzero wind-stress curl cannot be balanced by an inviscid interior alone.

Steady forced channel. Now take $r>0,\alpha >0$. Put $a=\pi /L$, $s=(z+h{)}_{+}=\max (z+h,0)$ and write ${\tau }_0$ for the positive wind-stress amplitude. The prescribed profiles giveThe buoyancy jumps across the sharp forcing interface in this simplified model; pressure and the circulation below are continuous. Let $P=p_y/{\rho }_0$, $D=f^2+r^2$, and $b_y^{\mathrm{top}}=-a{B}_0\sin ay/(\alpha h)$. Hydrostatic balance gives $P=C(y)+b_y^{\mathrm{top}}s$. The horizontal momentum equations and their solution areThe no-normal-flow conditions imply ${\int }_{-H}^{0}vdz=0$: this integral is independent of $y$ by incompressibility and is zero at the meridional walls. It determines the otherwise free depth-uniform pressure gradient,DefineA meridional overturning streamfunction satisfying $v=-{\Psi }_z$, $w={\Psi }_y$ isIt vanishes on all four boundaries. Together with the expressions for $b,P,u$, this solves all the steady equations. The formula permits variable $f(y)$; freezing it at a negative $f_0$ gives the simple schematic $w=aA\cos ayG(z)$. No unspecified latitude dependence is silently needed for the sketch.

Since $A>0$, upper-layer meridional transport is northward and the deep return is southward. At constant $f$, the circulation rises on the southern side and sinks on the northern side. The meridional velocity changes sign at $z=-h+h^2/(2H)$, slightly above the forcing interface, not exactly at $-h$. The required schematic follows from this derived field:

Forcing comparison. The buoyancy and wind contributions have the same vertical shape and, with these signs, the same circulation sense. Their ratio isThe printed scaling $f{\tau }_0/{\rho }_0\simeq B_0$ must be understood in magnitude, since $f<0$ and both forcing amplitudes are positive. Surface confinement supplies the small factor $h/L$, making direct buoyancy forcing negligible in this circulation model.

The missing physical mechanism in the stratified Southern Ocean is eddy-induced overturning. The wind-driven mean Deacon cell tilts isopycnals and builds available potential energy; baroclinic instability produces mesoscale eddies whose buoyancy transport largely opposes the mean overturning. The residual circulation depends on the small net balance and the diabatic buoyancy transformation. Wind energy also enters mean kinetic energy; eddies redistribute momentum through form stress and transfer energy toward dissipative scales and bottom friction. The displayed linear model already balances its wind input by imposed drag, but omits this mean-advection/eddy buoyancy budget, so the necessity of an opposing eddy circulation is a physical extension, not another mathematical constraint on the four given equations.