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The background potential energy is the gravitational potential energy of the statically stable state obtained by adiabatically sorting a fluid's parcels. The difference $A=P-P_b$ is available potential energy. Pure rearrangement changes $A$ but not $P_b$; irreversible density mixing can increase $P_b$. Molecular exchange with internal energy must be separated when defining an irreversible mixing rate.

Depth-weighted decomposition gives reduced depth $H_r=H_1{H}_2/(H_1+H_2)$ and internal radius $R_d^2=g\prime H_r/f_0^2$. On scale $l$, the ratio of interface available potential energy to baroclinic kinetic energy is of order $(l/R_d{)}^2$. Comparable depths recover $l^2{f}_0^2/(g\prime H_i)$; for strongly unequal depths the thinner layer sets the scale.

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.

Cumulative mixing efficiency is the ratio of total irreversible density-mixing conversion to total irreversible mechanical-energy loss over an interval. It is a loss-rate-weighted average of instantaneous mixing efficiency, not an unweighted time average. Stored kinetic energy, changes in available potential energy and boundary work must be accounted for before replacing its denominator by integrated forcing.

For a plane inertia-gravity wave independent of one horizontal coordinate, period-mean kinetic and available potential energy obey $⟨K⟩-⟨A⟩=2⟨K_y⟩$. The transverse rotating velocity is in quadrature with the in-plane velocity. The modified oscillator balance is $⟨K_{xz}⟩=⟨A⟩+⟨K_y⟩$; ordinary kinetic/potential equipartition occurs only when the transverse amplitude vanishes.

A surface on which geopotential is constant. Effective gravity is perpendicular to it. In a local constant-gravity approximation these surfaces are horizontal; comparing their slopes with isopycnals identifies the geometry permitting an adiabatic parcel exchange to release available potential energy.

Instantaneous mixing efficiency is the fraction of instantaneous irreversible mechanical-energy loss that raises background potential energy, after subtracting internal-to-potential exchange. The ratio $\Gamma =M/ϵ$ is a different quantity, related by $\eta =\Gamma /(1+\Gamma )$. Replacing $M$ by buoyancy exchange $B$ additionally assumes negligible change of available potential energy.

Choose $y$ northward and $z$ upward in the Northern Hemisphere. In a locally linear Boussinesq approximation, write the background buoyancy as $\bar{b}=b_{\ast }+By+N^{2}z$, with horizontal buoyancy gradient $B<0$ and positive squared buoyancy frequency $N^2$. The stable density stratification resists a purely vertical displacement, but the isopycnals slope upward poleward:The geopotential surfaces are horizontal in this local approximation. Their inclination differs from that of the isopycnals, which is the relevant baroclinicity.

Exchange two equal small fluid elements, each of volume $V$, between positions separated by $(\delta y,\delta z)$. The adiabatic parcel exchange has no thermal diffusion, so each parcel retains its buoyancy. The background buoyancy difference is $\Delta \bar{b}=B\delta y+N^2\delta z$. Because the variable-density contribution to gravitational potential energy is $-{\rho }_{0}Vbz$, the change in the pair's gravitational potential energy isFor a poleward-rising exchange, set $\delta z=s\delta y$ with $s>0$. ThenThe trajectory lies between a geopotential surface and an isopycnal. The poleward-rising parcel starts more buoyant than the parcel it replaces, and the equatorward-sinking parcel starts less buoyant: the buoyancy does positive work on both. This releases available potential energy into kinetic energy, allowing eddies to amplify if the full rotating dynamics supports the displacement and dissipation does not remove the gain.

A trajectory steeper than the isopycnals, $s>s_{\rho }$, instead costs gravitational potential energy; an equatorward-rising trajectory also costs it. Motion exactly along a geopotential surface or an isopycnal gives zero change in this ideal exchange. At fixed $\delta y$, the largest release occurs at $s=s_{\rho }/2$, since $N^{2}s(s-s_{\rho })$ is minimized there.

This is an energetic possibility argument for baroclinic instability, not a sufficient normal mode instability criterion. In particular, unrestricted parcel motion must not be assumed in a rotating fluid: conservation of angular momentum can impose an additional restoring effect. Symmetric instability has a separate Ertel potential vorticity criterion; ordinary Eady instability can occur without violating that criterion.

Take eastward $x$, upward $z$, positive wavenumbers $k,K$, and a real instantaneous amplitude $A$. A common growing factor does not affect the phase geometry. Put $C=\cosh (Kz)$ and $S=\sinh (Kz)$, using the hyperbolic cosine and hyperbolic sine. The prescribed streamfunction iswhereFollowing a fixed phase givesTherefore the streamfunction pattern tilts westward with height. The lower boundary leads the upper boundary eastward.

In the conventional quasi-geostrophic streamfunction normalization, hydrostatic balance gives $b^{\prime }=f_0{\psi }_z^{\prime }$. ThereforeFor the phase calculation, multiplication by the constant $f_0$ is immaterial. Define a continuous phase ${\delta }_b=atan2(\alpha C,S)$; this avoids a spurious discontinuity at $z=0$. ThenThe buoyancy perturbation pattern tilts eastward with height, opposite to the streamfunction pattern. At $z=0$, ${\psi }^{\prime }=A\cos (kx)$ while $b^{\prime }=-f_{0}AK\alpha \sin (kx)$, so the two fields are a quarter wavelength apart there.

Both perturbation amplitudes are largest at the boundaries. Indeed,which increase with $|z|$. The physical reason is particularly clear through stratified quasi-geostrophic inversion: the Eady model has no interior background potential-vorticity gradient, and a growing normal mode has zero interior potential vorticity anomaly. The independent thermal anomalies reside at the rigid upper and lower boundaries, where meridional motion advects the basic horizontal buoyancy gradient. Each Eady edge wave induces an interior balanced flow that decays away from its boundary on scale $K^{-1}$. Their interacting, phase-locked combination produces the growing baroclinic instability. Boundary confinement of the thermal sources, rather than a rigid-wall requirement ${\psi }^{\prime }=0$, explains the amplitude pattern.

For a sketch with a quarter-wavelength difference between the boundary streamfunction patterns, chooseAn upper positive streamfunction crest is then ${\lambda }_x/4$ west of its lower counterpart, where ${\lambda }_x=2\pi /k$. The buoyancy perturbation crests slope the other way. The quarter-wavelength boundary assumption is used for the streamfunction; differentiating in $z$ fixes the buoyancy perturbation and does not generally give it the same boundary phase separation.

The connection to poleward heat transport can be checked without relying on the sketch. The meridional geostrophic flow and buoyancy perturbation areAverage over one zonal wavelength, using equal averages $1/2$ for squared sine and cosine and zero for their product:In the Northern Hemisphere this is positive: warm anomalies preferentially move northward and cold anomalies southward. For the same eastward-increasing basic thermal wind in the Southern Hemisphere, $f_0<0$ and the basic buoyancy gradient reverses; the negative northward eddy buoyancy flux is again poleward. For thermal expansion coefficient ${\alpha }_T>0$ and specific heat capacity at constant pressure $c_p$, the corresponding sensible heat flux in a thermal Boussinesq approximation is ${\rho }_0{c}_p\overline{v^{\prime }{T}^{\prime }}={\rho }_0{c}_p\overline{v^{\prime }{b}^{\prime }}/(g{\alpha }_T)$. The growing amplitude makes this flux grow as $A^2$. The tilt supplies the correlation absent in an untilted disturbance and reduces the background available potential energy.

Let $P$ denote total gravitational potential energy, $P_b$ the background potential energy of the adiabatically sorted density field, and $A=P-P_b$ the available potential energy. Define $B$ positive for reversible transfer from kinetic energy to $P$, $ϵ$ as viscous kinetic-energy loss, $F$ as imposed mechanical work, $W$ as molecular transfer from internal energy to potential energy, and $M$ as the irreversible conversion of $A$ to $P_b$. With boundary contributions removed, the stratified mixing energy budget isFor an unforced flow set $F=0$. In an alternative notation, if ${\Phi }_d={\dot{P}}_b$ is the full diffusive increase of background energy and ${\Phi }_i=W$, then $M={\Phi }_d-{\Phi }_i$ and $\dot{A}=B+{\Phi }_i-{\Phi }_d$. This distinguishes irreversible density mixing from reversible buoyancy exchange. The internal reservoir obeys $\dot{I}=ϵ-W$, so summing the kinetic, potential and internal budgets gives $d(K+P+I)/dt=F$ as required by conservation of energy.

With negligible internal-to-potential exchange and negligible change in available potential energy, the budgets give $M=B$. The model supplies $B=\sqrt{K}{U}_0^2/L_{\rho }$ and $ϵ=K^{3/2}/L_v$, so for $K(t)>0$ the instantaneous mixing efficiency isFor a positive initial seed and $L_f<L_{\rho }$, both loss rates approach positive constants. Their time integrals grow linearly at those rates, implyingIf mixing efficiency instead denotes the flux coefficient, then $\Gamma (t)=L_v{U}_0^2/[L_{\rho }K(t)]$ and ${\Gamma }_{c,\infty }=L_f/(L_{\rho }-L_f)$. During the transient, $\int (B+ϵ)dt=\int Fdt+K_0-K(t)$; equating irreversible loss with the instantaneous forcing would omit changing kinetic-energy storage. A permanently resting zero-seed branch has zero numerator and denominator, so it has no defined loss-based efficiency.

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.

Let the vertical-velocity amplitude $W$ be real. A convenient inertia-gravity wave polarization, avoiding pressure denominators, isThe physical velocities and buoyancy are their real parts. The period-mean kinetic energy density and available potential energy density areThe latter follows from $A={\rho }_0{b}^2/(2N^2)$, or from $A={\rho }_0{N}^2{\xi }_z^2/2$ for vertical fluid displacement. The factor $1/4$ includes both the energy definition and the mean of a squared harmonic.

Using the dispersion relation,This energy partition of rotating internal waves shows that the printed request for ordinary kinetic/potential equipartition in the rotating case is false in general. An explicit counterexample is $f=1$, $N=2$, $k=m=1$, $W=1$, giving ${\omega }^2=5/2$, $⟨K⟩=3{\rho }_0/5$ and $⟨A⟩=2{\rho }_0/5$.

The valid modified oscillator balance isThe transverse rotational velocity is in quadrature with the in-plane motion, so it supplies an additional positive energy term on the displacement side of this oscillator balance. It remains physically kinetic energy, not gravitational potential energy. Ordinary kinetic/potential equipartition is recovered when $f=0$ or when the transverse amplitude vanishes. For a real harmonic wave the instantaneous total $K+A$ is constant at a fixed point: the coefficient of ${\cos }^2(kx+mz-\omega t)$ from in-plane motion equals the coefficient of ${\sin }^2$ from transverse motion and buoyancy. This consistency does not imply equality of their separate period means.

Integrating the local two-layer quasi-geostrophic energy conservation law over a horizontal domain and using the specified vanishing boundary flux givesPeriodic boundaries, or suitable fixed streamfunction boundary data eliminating the displayed energy flux, provide examples. The gradient terms are the two layer kinetic energies; the last term is available potential energy, equal to $g^{\prime }{\chi }^2/2$ under the interface-displacement relation.

The baroclinic energy ratio and deformation scale for unequal layer depths uses a decomposition uses the depth-weighted barotropic streamfunction $\Psi =(H_1{\psi }_1+H_2{\psi }_2)/(H_1+H_2)$ and $d=2\tilde{\psi }$. Define $H_r=H_1{H}_2/(H_1+H_2)$. ThenFor variations on horizontal scale $l$, this givesFor comparable layer depths $H_r$ is of order either $H_i$, reproducing the requested scale $l^2{f}_0^2/(g^{\prime }{H}_i)$. If one layer is much thinner, its depth controls this ratio; a depth-independent arithmetic barotropic average would leave unwanted cross terms in the energy decomposition.

Baroclinic potential energy dominates at scales much larger than the two-layer internal deformation radius; baroclinic kinetic energy dominates at much smaller scales. They are comparable near $l\sim R_d$. The independent barotropic kinetic energy has no interface-displacement partner, so this scale comparison refers specifically to the baroclinic component.

Let $b$ denote buoyancy in a stably stratified Boussinesq approximation. Geostrophic balance and hydrostatic balance imply the thermal wind relation $f_0{U}_z=-{\bar{b}}_y$. Consequently the specified vertical shear requiresFor northern-hemisphere $f_0>0$ and $\Lambda >0$, the mean potential temperature decreases northward and increases upward: ${\bar{\Theta }}_y=-({\Theta }_0/g)f_0\Lambda$, ${\bar{\Theta }}_z=({\Theta }_0/g)N^2$. The corresponding mean isentropes slope upward toward the north, with $dz/dy=f_0\Lambda /N^2$. Stable potential-temperature stratification does not require the actual temperature to increase upward; the dry adiabatic lapse also enters that comparison.

In the rigid-lid constant-shear baroclinic model, perturbation quasi-geostrophic potential vorticity isThe mean interior potential-vorticity gradient vanishes on this constant-$f_0$, constant-$N$, linear-shear basic state. Linear advection of that perturbation therefore becomes $({\partial }_t+\Lambda z{\partial }_x)q^{\prime }=0$, the supplied interior equation. It retains slow balanced motion and omits friction, planetary-vorticity gradients and nonlinear disturbance advection.

Substitution of ${\psi }^{\prime }=\Psi (z)\cos (ly)e^{ik(x-ct)}$ givesA possible branch has zero interior perturbation potential vorticity, soBoth supplied applications have $f_0>0$, so this equals the source's expression. For an unstable mode the nonzero imaginary part of the phase speed prevents a real critical level $\Lambda z=c$; for real speeds the displayed solution remains a valid zero-interior-PV branch away from, and smoothly through, any critical level. Taking the real part gives the physical disturbance.

The rigid lids enforce $w^{\prime }=0$. Linearized adiabatic buoyancy conservation isUse the geostrophic conventions $v^{\prime }={\psi }_x^{\prime }$ and $b^{\prime }=f_0{\psi }_z^{\prime }$. At a lid this gives $({\partial }_t+\Lambda z{\partial }_x){\psi }_z^{\prime }-\Lambda {\psi }_x^{\prime }=0$, henceThese are the no-normal-flow condition expressed through boundary buoyancy advection, not conditions of zero horizontal velocity. Boundary temperature anomalies can interact even though this solution has no interior perturbation PV, allowing baroclinic instability to extract mean available potential energy.

Write the vertical complex amplitude as $\Psi (z)=a(z)e^{i\varphi (z)}$. A constant streamfunction phase has $kx-k{c}_{r}t+\varphi (z)=\mathrm{const}$, hence $dx/dz=-{\varphi }_z/k$. For $k>0$, westward tilt with height implies ${\varphi }_z>0$. With $v^{\prime }={\psi }_x^{\prime }$ and $b^{\prime }=f_0{\psi }_z^{\prime }$, a zonal average over the wave givesfor the northern-hemisphere conventions used here. This is the westward tilt and poleward eddy heat flux relation. Warm anomalies move northward and cold anomalies southward, transporting heat down the mean gradient ${\bar{b}}_y=-f_0\Lambda <0$. The tilt describes the relative phases needed for a nonzero flux, not the literal trajectories of fluid particles along a tilted streamfunction crest.

The flux reduces the mean meridional temperature contrast and converts mean available potential energy into eddy available potential energy. In the perturbation buoyancy-variance budget its production term is $-\overline{v^{\prime }{b}^{\prime }}{\bar{b}}_y/N^2>0$. Correlated rising warm anomalies and sinking cold anomalies have positive $\overline{w^{\prime }{b}^{\prime }}$, converting that eddy potential energy into kinetic energy. The flow is therefore energized by the baroclinic reservoir associated with tilted density/potential-temperature surfaces. It is not simply a claim that the disturbance amplifies while leaving the basic state's thermal structure unchanged.

With boundary fluxes suppressed, let $B$ transfer kinetic energy to potential energy, $M$ transfer available potential energy irreversibly to background potential energy, and $W$ transfer internal energy to potential energy. Then $\dot{P}=B+W$, ${\dot{P}}_b=M+W$, $\dot{A}=B-M$, and $\dot{K}=F-B-ϵ$. The internal reservoir has $\dot{I}=ϵ-W$, conserving total energy apart from forcing $F$. The distinction between reversible buoyancy exchange and irreversible mixing is essential outside a quasi-steady negligible-$A$ regime.

For inviscid unforced two-layer QG dynamics, $H_1{F}_1=H_2{F}_2=f_0^2/g\prime$ makes the coupling symmetric in the layer-depth inner product. Multiplication by $H_i{\psi }_i$ gives the positive energy $E=\frac{1}{2}[{\sum }_i{H}_i|\nabla {\psi }_i{|}^2+(f_0^2/g\prime )({\psi }_1-{\psi }_2{)}^2]$. It is conserved after integration when the lateral energy flux vanishes. The interface term is available potential energy; the gradient terms are kinetic energy.

For a disturbance proportional to $\mathrm{Re}[a(z)e^{i\varphi (z)}{e}^{ik(x-ct)}]$, westward phase tilt means ${\varphi }_z>0$ for $k>0$. With $v^{\prime }={\psi }_x^{\prime }$ and $b^{\prime }=f_0{\psi }_z^{\prime }$, a zonal average gives the displayed positive flux in the northern hemisphere, up to any meridional amplitude and time-growth factors. When the background buoyancy decreases northward, this flux extracts mean available potential energy and supports growing eddies.