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Exchange two equal fluid elements of volume $V$ without thermal diffusion in a Boussinesq approximation. Each retains its buoyancy. If the separation is $(\delta y,\delta z)$ and the background buoyancy difference is $\Delta b=B\delta y+N^2\delta z$, the pair's gravitational potential energy changes by ${\rho }_{0}V\delta z\Delta b$. For $B<0$, $N^2>0$ and poleward-rising slope $s=\delta z/\delta y$, this is negative exactly when $0<s<-B/N^2$. The minimum at fixed $\delta y$ is attained at half the isopycnal slope. This energetic test alone is not sufficient for dynamical baroclinic instability: rotation and constraints on parcel motion must also be considered.

Mixing across surfaces of constant density, or isopycnals. In the deep ocean, breaking internal waves can supply the mechanical energy needed to mix against stable stratification. The divergence of the turbulent buoyancy flux determines local water-mass transformation.

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.

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.

Southern Ocean westerly wind stress drives northward surface Ekman transport in the Southern Hemisphere. Its divergence and the circumpolar geometry allow deep water to rise towards the surface along tilted isopycnals, providing an important return route for the deep branch of the Atlantic meridional overturning circulation. Surface heat fluxes and freshwater fluxes then transform the upwelled water before it enters shallower return pathways. The winds supply mechanical energy; the overturning cannot be represented entirely by a local northern density difference. Eddy-induced overturning opposes part of the wind-driven contribution, so the net water-mass transformation and transport need not increase in direct proportion to the wind stress.^[ref]

Deep currents and tides moving across rough topography, including the Mid-Atlantic Ridge, generate internal waves. Their breaking produces turbulent mixing across isopycnals, called diapycnal mixing. This transfers buoyancy between deep and overlying waters, transforms dense water into lighter water, and contributes to the upward return needed to close the overturning. The required mechanical energy ultimately comes from winds and tides; roughness is a mechanism for converting this input into mixing, rather than an independent source of energy. Mixing is spatially concentrated, and local vertical motion depends on the divergence of the turbulent buoyancy flux, so enhanced mixing does not imply uniform local upwelling everywhere over a ridge.^[ref]

Potential-vorticity inversion is the diagnostic recovery of the flow from its advected vortical scalar, using the balance relation and specified boundary conditions. Its usefulness is a separation of tasks: transport the scalar by the current velocity, invert to find the new velocity, and repeat. The inverse is spatially nonlocal even though the transport law is local. The four models differ chiefly in the operator being inverted and the information needed at boundaries.

In two-dimensional vortex dynamics, planar incompressibility permits $\mathbf{u}=(-{\psi }_y,{\psi }_x)$, and vertical vorticity is $\zeta ={\nabla }^2\psi$. The curl of inviscid momentum givesThere is no vortex stretching in strictly planar motion, so vorticity is materially conserved. The two-dimensional potential-vorticity inversion is the Poisson equation ${\nabla }^2\psi =\zeta$. On the plane, with suitable integrability and far-field behavior, its Green's function is $(2\pi {)}^{-1}\log |\mathbf{x}-{\mathbf{x}}^{\prime }|$, givingDifferentiation gives the planar Biot-Savart kernel. Here ${\psi }_h$ is a harmonic function fixed by boundary, circulation and far-field data. A rigid impermeable wall is a streamline, so $\psi$ is constant on it; constants or circulations around distinct holes require specification. The vorticity alone cannot distinguish two flows differing by a permitted harmonic contribution, such as a uniform current on an unbounded plane. A constant added to $\psi$ has no physical effect.

In nonrotating layerwise-two-dimensional vortex dynamics, strong stable density stratification constrains motion to almost horizontal layers. In the ideal independent-layer limit, vertical velocity vanishes at leading order, $(u,v)=(-{\psi }_y,{\psi }_x)$ andhold separately at each fixed $z$. Height is a parameter: the horizontal inverse Laplacian contains no vertical derivative. With background buoyancy $N^{2}z$, the leading Ertel potential vorticity is $Q=\mathbf{\omega }\cdot \nabla b\simeq N^2\zeta$, so the materially advected normalized scalar is $Q/N^2\simeq \zeta$. The same horizontal Poisson equation is inverted layer by layer, using each layer's boundary data. This is an asymptotic constrained model, not arbitrary three-dimensional Euler motion with vertical velocity set to zero: its validity requires small Froude number and controlled vertical scales. At finite buoyancy frequency, the weak vertical motion and tilted isopycnals restore coupling. The inversion may be independent by layer while the reconstructed fields vary with height. The horizontal Euler equations for an inviscid fluid determine the pressure gradient, while hydrostatic balance relates its vertical derivative to buoyancy.

In shallow-water quasi-geostrophic inversion, begin with the exact shallow-water potential vorticity $(f+\zeta )/h$. Linearizing its anomaly about depth $H$ gives, after multiplying by $H$, $\zeta -f_0\eta /H$. For small Rossby number, geostrophic balance gives $\psi =g\eta /f_0$, while $\zeta ={\nabla }_h^2\psi$. Keeping the beta plane background gradient givesThus $q-\beta y$ is inverted with the modified Helmholtz equation $({\nabla }_h^2-L_D^{-2})\psi =q-\beta y$, under the specified boundary and far-field conditions. The velocity follows from $\psi$ and surface elevation from $\eta =f_0\psi /g$. A horizontal Fourier mode of the anomaly satisfiesThe Rossby deformation radius therefore limits the reach of the balanced response. Layer stretching, represented by $-\psi /L_D^2$, is absent from ordinary planar vorticity inversion. Unlike the pure Poisson equation, the finite-deformation operator also responds to the constant part of $\psi$, which specifies mean elevation rather than an arbitrary streamfunction offset. Specified bottom topography adds a known source to $q$; it is subtracted along with the background before inversion.

In stratified quasi-geostrophic inversion, continuously varying vertical displacement replaces the shallow layer's single stretching term. For positive $N^2(z)$ and constant nonzero $f_0$, the three-dimensional quasi-geostrophic potential vorticity isGeostrophic balance and hydrostatic balance give velocity $(-{\psi }_y,{\psi }_x)$ and buoyancy perturbation $b=f_0{\psi }_z$. Invert the anisotropic elliptic boundary value problem using interior $q$, prescribed top and bottom buoyancy as Neumann boundary conditions, and the lateral, far-field and global-flow data. Boundary buoyancy evolves by its own material conservation when the boundary is impermeable. It cannot be discarded: the Eady edge wave in the preceding solution has zero interior potential vorticity anomaly but nonzero boundary buoyancy and a nonzero induced flow.

For constant $N,f_0$, the stretched coordinates for quasi-geostrophic inversion $Z=Nz/|f_0|$ turn the inversion operator into the three-dimensional Laplacian. Its infinite-space Green's function is $-1/(4\pi |\mathbf{X}-{\mathbf{X}}^{\prime }|)$ in $(x,y,Z)$ coordinates. Hence a horizontal disturbance of scale $L$ communicates vertically over scale $|f_0|L/N$, explicitly coupling layers. In a bounded domain with homogeneous difference data, multiplying the difference of two inversions by its streamfunction and applying integration by parts givesThis proves uniqueness up to a constant when the relevant boundary terms vanish; a mean or pressure reference fixes the remaining constant. Compatible boundary fluxes are needed for a purely Neumann boundary-value problem.

Across all four models, inversion provides the induced flow, while scalar transport supplies time evolution. The independent horizontal inverse in the first two models becomes a finite-deformation inverse in shallow water and a vertically coupled elliptic inverse in continuous stratification. The principle applies within the stated balance or constrained model. In the unrestricted Boussinesq equations or shallow water equations, gravity waves and divergent motion can share the same potential vorticity, so the vortical scalar alone does not reconstruct every possible flow.

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.

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.

Now let the basic buoyancy beThermal-wind balance requires the basic along-front velocity to have vertical shear $V_z=M^2/f$, soThe basic absolute vorticity and buoyancy gradient areand hence

For the prescribed disturbance the buoyancy perturbation vanishes, so the linearized buoyancy equation and incompressibility giveThe wavevector must therefore satisfythe disturbance velocity lies along a basic isopycnal. The along-front momentum equation becomesProjecting the remaining momentum equations onto the divergence-free direction eliminates the pressure and yieldsThus this isopycnal disturbance grows if and only if $fQ<0$. Geometrically, $Q$ is the component of the absolute vorticity along the buoyancy gradient, multiplied by $|\nabla B|$; the horizontal buoyancy gradient reduces that component through the term $-M^4/f$. When $M=0$, the result reduces to the most unstable, nearly horizontal-wavevector limit of part i.

Balances and density gradients. The first two relations describe horizontal geostrophic balance between pressure gradient and Coriolis force accelerations. The vertical relation is hydrostatic balance between the vertical pressure force and gravity. They describe a slowly varying large-scale interior circulation, excluding the frictional surface/bottom layers and large ageostrophic accelerations.

Differentiate the horizontal balances vertically and use $p_z=-\rho g$, treating $f,{\rho }_0$ as vertically constant:These are the oceanic thermal wind relations. With no motion at $z=-H$, the exact surface relations involve the depth-integrated density gradients. If the measured density is representative of the vertical mean, or if those gradients are approximately uniform with depth,Equivalently these hold exactly for the horizontal gradients of $\bar{\rho }=H^{-1}{\int }_{-H}^0\rho dz$ in this fixed-depth reference model. A single arbitrary surface-density gradient does not otherwise determine a vertical integral. In the Northern Hemisphere the inferred surface current runs along horizontal isopycnals with denser water on its left when the no-motion reference and this vertical-gradient approximation apply.

For a steady density pattern and a ship whose actual track velocity is $v_s(\cos \theta ,\sin \theta )$,If the field changes during the survey an Eulerian time derivative must also be included. Here $H=\Delta z>0$ is the upward separation from reference depth to surface, so its sign must not be reversed.

Inversion and velocity estimate. Let $C=f{\rho }_0{v}_s/(gH)$ and denote the two ship measurements by $y_{\theta },y_0$, with the east-going ship assigned the subscript zero. The observation system and its inverse areIt is invertible for $\sin \theta \ne 0$. For two measurements and two unknowns this inverse is also the stated least-squares estimate; the nearly parallel-track geometry explains its poor conditioning at small angle.

The supplied parameters give $C=5.654\times 10^{-5}$ in units converting velocity to the measured density tendency. With $y_{\theta }=1.9\times 10^{-5}$ and $y_0=1.8\times 10^{-5}\mathrm{kg}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$,The current therefore has eastward and southward components. Signs follow from the positive-upward, east–north coordinate convention and the specified no-motion reference.

Uncertainty and track design. The two-track thermal-wind covariance exposes the geometrical limitation. The stated error already applies to the difference of successive density observations. Dividing by the sampling interval, rather than adding an extra factor of $\sqrt{2}$, gives ${\sigma }_y=7.2\times 10^{-4}/3600=2.0\times 10^{-7}\mathrm{kg}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$. Assume independent errors of that size for the two ships, as in the supplied covariance model. ThenAt $5^{\circ }$,The correlation is positive,Both estimates depend on the east-going ship; its positive measurement error lowers $v_g$ and also lowers $u_g$, explaining the sign of the covariance. If the other ship goes north instead, $\theta =90^{\circ }$ gives orthogonal observations: $u_g=y_{90}/C$ and $v_g=-y_0/C$. Both standard errors become $0.00354\mathrm{m}{\mathrm{s}}^{-1}$ and their covariance vanishes. The north-going track improves the poorly constrained zonal component by about a factor of sixteen, while leaving the meridional precision unchanged under the same noise assumptions.

For a semi-infinite stratified fluid with thermal wind $U=\Lambda z$ above a gently sloping boundary $z=\alpha y$, a zero-interior-potential vorticity perturbation decays as $e^{-\mu z}$, where $\mu =N|k|/|f_0|$. The boundary buoyancy perturbation obeys a single oscillatory equation with phase velocity $c=(\Lambda -\alpha N^2/f_0)/\mu$. The topographic and thermal-wind contributions cancel when the boundary follows a background isopycnal.

A change in the amount of ocean water belonging to a specified density class, driven by surface heat flux, freshwater exchange or interior mixing. Advection along an isopycnal transports a water mass without necessarily transforming it; diapycnal mixing can transfer water between density classes.