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Ageostrophic flow is the part of velocity beyond leading geostrophic flow. Its small horizontal divergence supplies vertical motion and the stretching term in the quasi-geostrophic potential-vorticity equation.

For a Boussinesq approximation, constant buoyancy frequency $N$, and buoyancy heating rate $R$, the quasi-geostrophic potential-vorticity equation becomes $D_{g}q/Dt=(f_0/N^2)R_z$, where $q={\nabla }_h^2\psi +(f_0^2/N^2){\psi }_{zz}+\beta y$. Heating changes the vertical gradient of buoyancy and therefore produces a potential-vorticity tendency proportional to its height derivative.

The quasi-geostrophic potential-vorticity equation is the leading balanced form of potential-vorticity conservation for an inviscid rotating fluid with stable density stratification, small Rossby number and an adiabatic process for each fluid element. Its parent conservation law is Ertel potential vorticity conservation. Horizontal geostrophic balance, vertical hydrostatic balance, and the beta-plane approximation reduce it to advection of one scalar by the leading horizontal geostrophic flow.

One can see its origin by combining the vertical vorticity equation with the thermal equation. Set $D_g={\partial }_t+{\mathbf{u}}_g\cdot {\nabla }_h$, ${\mathbf{u}}_g=(-{\psi }_y,{\psi }_x)$, and ${\zeta }_g={\nabla }_h^2\psi$. The quasi-geostrophic approximation givesHere $b$ is the buoyancy perturbation relative to the reference stable density stratification, $N=N(z)$ its buoyancy frequency, and $w$ the small vertical velocity. The stretching of absolute vorticity in the first equation is offset by the change of stratification represented by the second. More precisely,The last dot product is zero because thermal-wind balance makes ${\partial }_z{\mathbf{u}}_g$ perpendicular to ${\nabla }_{h}b$. Adding therefore yieldsThe constant $f_0$ has no effect on the tendency, but its inclusion makes the connection with absolute vorticity clear. Vertical advection of $q$ is beyond the retained order; the vertical velocity has already supplied the stretching and thermodynamic terms.

For a simple balanced weather forecast, the essential cycle is diagnose, advect, invert:

This is an elliptic boundary value problem, not a local algebraic diagnosis. Its solution supplies the next geostrophic flow and, through $b=f_0{\psi }_z$, the next thermal field. Repeat the cycle to move the balanced weather systems forward.

Interior potential vorticity alone is insufficient: boundary thermal anomalies can induce flow even with zero interior anomaly, as in the Eady model. The method filters fast inertia-gravity waves and is useful for large-scale extratropical weather, while tropical dynamics, convection, moisture, friction and diabatic heating require additional physics. If vertical motion is wanted, the quasi-geostrophic omega equation supplies a further diagnostic calculation.

The quasi-geostrophic approximation isolates slowly evolving, nearly balanced motions in a rotating, stably stratified fluid. Ocean eddies and mid-latitude atmospheric disturbances often have this character: a leading Coriolis force–pressure balance suppresses rapid horizontal acceleration, while hydrostatic balance suppresses rapid vertical acceleration. Small ageostrophic flow nevertheless supplies the vertical motion that changes the leading vorticity. The approximation filters fast inertia-gravity waves while retaining nonlinear transport, vertical shear and Rossby waves.

Use a reference density ${\rho }_0$ and remove the background hydrostatic pressure. Let $\pi =p^{\prime }/{\rho }_0$ and let the upward buoyancy perturbation be $b=-g{\rho }^{\prime }/{\rho }_0$. A vertically varying reference density stratification has $N^2(z)>0$. Under the Boussinesq approximation, the inviscid and adiabatic equations areHere $D/Dt={\partial }_t+{\mathbf{u}}_H\cdot {\nabla }_H+w{\partial }_z$. Density variations are small compared with ${\rho }_0$ but retained in the vertical buoyancy force. This is a useful local, incompressible description; a compressible atmosphere needs the corresponding weighted or pressure-coordinate version.

Let horizontal and vertical lengths be $L,H$, speed be $U$, and take the slow time $L/U$. Take $f_0>0$ for definiteness; use $|f_0|$ in the scale estimates in the opposite hemisphere. The principal small parameter is the Rossby number $\mathrm{Ro}=U/(f_{0}L)\ll 1$ at a latitude where $f_0\ne 0$. For the hydrostatic, strongly stratified regime, $H/L\ll 1$ and $f_0/N\ll 1$. Retaining both horizontal vorticity and vortex stretching gives the Burger numberThe geostrophic balance pressure scale is $f_{0}UL$ and the buoyancy perturbation scale is $B=f_{0}UL/H$. Hence $B/(N^{2}H)=\mathrm{Ro}/\mathrm{Bu}\ll 1$: buoyancy perturbations produce small displacements relative to the background stratification scale. The ageostrophic flow is $O(\mathrm{Ro}U)$ and incompressibility then gives $w=O(\mathrm{Ro}UH/L)$. Vertical acceleration is smaller than hydrostatic terms by order ${\mathrm{Ro}}^2(f_0/N{)}^2$ in this scaling. Weak forcing or diffusion can be added at the slow order, but neither is present here.

Define the quasi-geostrophic streamfunction by $\psi =\pi /f_0$. Leading geostrophic balance and hydrostatic balance implyDifferentiating horizontally and vertically gives the thermal-wind balanceThus vertical shear is determined by horizontal buoyancy gradients. The homogeneous Taylor–Proudman theorem is recovered only when those gradients vanish. Stable stratification allows sloping density surfaces and different geostrophic flow at different levels, without abandoning leading balance.

The Prandtl ratio of scales makes that modification quantitative. Horizontal and vertical terms in balanced potential-vorticity inversion have sizes $\psi /L^2$ and $(f_0^2/N^2)\psi /H^2$. Equality givesFor constant $N$, $Z=(N/|f_0|)z$ makes the inversion operator an ordinary three-dimensional Laplacian. Equivalently, an interior balanced disturbance of horizontal size $L$ naturally penetrates a depth of order $|f_0|L/N$. With $N\gg |f_0|$ this is a thin, vertically sheared structure, rather than the arbitrarily tall homogeneous columns suggested by strict Taylor–Proudman theorem behaviour. This is an aspect-ratio statement, not the diffusivity Prandtl number.

To allow a weak meridional variation of the Coriolis parameter, use the beta plane $f=f_0+\beta y$, with $\beta L/f_0=O(\mathrm{Ro})$, or equivalently $\beta L^2/U=O(1)$. The small correction to the leading balance is dynamically important on the slow time. Let $\xi ={\nabla }_H^2\psi$ be vertical relative vorticity and let $D_g={\partial }_t+{\mathbf{u}}_g\cdot {\nabla }_H$. At the first nontrivial order, the vertical vorticity equation and buoyancy equation reduce toRelative-vorticity stretching and tilting are higher order here; planetary-vorticity stretching $f_0{w}_z$ must remain. Eliminate $w$ to expose a materially conserved balanced scalar. Because ${\mathbf{u}}_{g,z}=(-b_y/f_0,b_x/f_0)$, the commutator term ${\mathbf{u}}_{g,z}\cdot {\nabla }_{H}b$ is exactly zero. For $N=N(z)$ this givesAdding the two equations proves the quasi-geostrophic potential-vorticity equationAn additive constant such as $f_0$ changes no dynamics. The horizontal streamfunction advection bracket can express this as $Q_t+{\psi }_x{Q}_y-{\psi }_y{Q}_x=0$. Although small departures from balance generated this evolution equation, the leading geostrophic flow itself advects $Q$ nonlinearly. The stretching term couples horizontal motion to vertical displacement of density surfaces.

This gives the practical advection-and-inversion description. Start with interior three-dimensional quasi-geostrophic potential vorticity, advect it with ${\mathbf{u}}_g$, and recover $\psi$ fromThe stratified quasi-geostrophic inversion is elliptic for positive $N^2$. It must be supplied with lateral and vertical boundary conditions and any circulation or mean-flow constraints; interior potential vorticity alone is not sufficient. At rigid horizontal lids, $w=0$ makes boundary buoyancy satisfy $D_{g}b=0$, and $b=f_0{\psi }_z$ supplies the vertical derivative data for inversion. Boundary buoyancy perturbations can therefore support flow even when the interior anomaly is zero. Once $\psi$ is found, its horizontal derivatives give velocity, its vertical derivative gives buoyancy, and the thermodynamic equation diagnoses $w$. This balanced evolution excludes the independent fast-wave initial data present in the full equations.

Finally, a gradient in $f$ supplies a restoring mechanism even in an otherwise uniform fluid. Linearizing about rest with constant $N$, the quasi-geostrophic potential-vorticity equation becomesSubstitution of $\psi \propto e^{i(kx+ly+mz-\omega t)}$ gives the Rossby wave dispersionFor $\beta >0$, its zonal phase speed is westward relative to a resting background. A meridionally displaced parcel retains potential vorticity, so it acquires a relative-vorticity anomaly that induces the velocity tending to return the disturbance. Background shear and stratification modify the full potential-vorticity gradient; they can alter propagation or permit instability. On a constant f-plane the planetary contribution vanishes, but gradients of relative vorticity, boundary buoyancy or topography can still support balanced wave motion. Close enough to the equator that $f_0$ vanishes, this mid-latitude quasi-geostrophic approximation and its scaling must be replaced.

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.

Potential-vorticity conservation makes the evolving PV distribution a set of parcel-carried labels. In the ideal model it can be rearranged into filaments and complicated geometry, but it is not created or destroyed by advection. Apparent homogenization, such as that assumed for the shallow-water experiment, describes a coarse-grained mixed distribution rather than pointwise destruction of materially conserved PV.

Within a balanced regime, potential-vorticity inversion turns those labels and the boundary data into a velocity field. This makes understanding a flow a coupled geometric problem: determine how PV is rearranged, invert the resulting distribution, and use the recovered velocity to find the next rearrangement. Inversion is nonlocal, so a compact PV anomaly influences fluid well outside its support, as the spherical example explicitly demonstrates.

The quasi-geostrophic potential-vorticity equation has only one time derivative:An initial PV field and the inversion data determine the balanced evolution; an independent initial acceleration is not needed. Pressure, buoyancy and the leading velocity are diagnosed from that field at each instant. This reduced initial-data requirement reflects the removal of freely oscillating inertia-gravity waves, which in the full fluid equations require additional initial information. The principles are consequently powerful for slow large-scale flow, but do not recover every unbalanced motion or remove the need to state boundary conditions and the balance approximation.

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.

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.

Adopt the Boussinesq approximation, hydrostatic approximation, traditional approximation, and inviscid quasi-geostrophic approximation. Assume stable constant $N_0^2>0$, nonzero Coriolis parameter $f_0$, a beta plane, small Rossby number, $\beta L/|f_0|$ of that small order, and small displacement of the background stratification. Vertical advection of the buoyancy perturbation is then higher order. Define the quasi-geostrophic streamfunction byThe buoyancy and vertical vorticity equations at the retained order areDifferentiate the first in $z$. Since $J({\psi }_z,{\psi }_z)=0$, its material-derivative term becomes $f_0{D}_g{\psi }_{zz}$. Elimination of $w_z$ gives diabatically forced quasi-geostrophic potential vorticity:Thus it is the height gradient of heating, rather than heating alone, that produces potential vorticity at this order.

For the steady response linearized about rest, assume $\beta \ne 0$. The quasi-geostrophic potential-vorticity equation reduces to $\beta {\psi }_x=f_0{R}_z/N_0^2$. Choosing the horizontally uniform pressure gauge so that $\psi \to 0$ in the east gives the steady quasi-geostrophic response to localized heatingSetUsing the error function to integrate the Gaussian function givesThese velocities vanish as $x\to +\infty$. A depth-dependent but horizontally uniform addition to $\psi$ is dynamically irrelevant for the requested horizontal velocity. At $z=0$, streamlines are contours of $e^{-y^2/L^2}I(x)$. For $f_0,\beta ,R_0>0$, flow approaches the heating region from the west on its northern side, turns southward, and returns westward on its southern side. The contours are open, not closed gyres: $I$ tends to a nonzero constant in the west.

For $-H<z<0$, $R_z<0$ and the forcing decreases potential vorticity; steady balance requires $\beta v_g=f_0{R}_z/N_0^2<0$, giving southward flow. Below $z=-H$, $R_z>0$ and the meridional flow reverses. A constant heating rate independent of $z$ would have no such interior potential vorticity source.

For the nonlinear diagnostic quasi-geostrophic omega equation, write $\zeta ={\nabla }_h^2\psi$. Apply ${\nabla }_h^2$ to buoyancy evolution and $f_0{\partial }_z$ to vorticity evolution:Subtracting eliminates the pressure tendency and yieldsThe beta term is linear and must remain when the nonlinear terms are neglected. For the steady weak-forcing solution, $f_0\beta {\psi }_{xz}=f_0^2{R}_{zz}/N_0^2$, soThis also follows immediately from the steady linear buoyancy equation. It satisfies $w=0$ at $z=0$, decay at depth and horizontal infinity, and the linear quasi-geostrophic omega equation; with these homogeneous conditions the elliptic problem has no extra decaying homogeneous solution. The rigid-lid interpretation is an additional boundary approximation for this diagnostic response.

At $x=0$, the displayed $v_g$ and $w$ provide the requested meridional/vertical arrows. For positive heating there is upwelling concentrated near $y=0$, maximal in depth at $z=-H$, with northward flow below that depth and southward flow above. The vertical circulation tends to zero far from the forcing. A western zonal wake persists, but its meridional and vertical components vanish as $x\to -\infty$. Horizontal ageostrophic flow supplies the divergence associated with the upwelling; the projected $(v_g,w)$ slice is not itself a two-dimensional incompressible flow and need not have closed streamlines.

The steady calculation fixes an interior circulation, not its attainability from every possible initial and boundary state. For example, a rigid lid with initially uniform boundary buoyancy conserves that buoyancy at linear order because $R(0)=w(0)=0$. The formal steady profile instead generally has nonuniform $b^{\prime }(0)=f_0{\psi }_z(0)$. Such additional initial boundary data require a time-dependent adjustment and cannot simply be imposed on this particular steady solution. No boundary buoyancy data are supplied for the requested steady problem.

Linearization about rest replaces the material derivative by ${\partial }_t$, so the quasi-geostrophic potential-vorticity equation becomesInsert the separation of variables $\psi =\varphi (x,y,t)P(z)$ and imposeThe horizontal amplitude then satisfiesThe Linear Rossby-wave equation therefore governs each vertical normal mode, whose Rossby deformation radius is $c/f_0$.

Potential-vorticity dynamics. The quasi-geostrophic potential-vorticity equation expresses material conservation of a balanced potential vorticity by a horizontally nondivergent geostrophic flow. Its terms represent relative vorticity, planetary vorticity and the layer-thickness/free-surface contribution. The Rossby deformation radius $L_d$ measures the scale at which that thickness contribution competes with the horizontal-vorticity term. The quasi-geostrophic approximation filters fast acoustic waves and gravity waves, making the slower balanced evolution accessible to early digital forecasts. The 1950 barotropic-vorticity forecasting study demonstrated practical twenty-four-hour forecasts from observed mid-tropospheric fields, while identifying discretization and omitted baroclinicity as error sources.^[ref]

Near the north pole put $\delta =\pi /2-\theta$ and $r\simeq a\delta$. ThenUsing projected rather than arc distance gives the same leading polar expansion. It is a north-polar expression; the sign structure is changed consistently at the opposite pole.

In cylindrical coordinates, horizontal advection is $u_r{\partial }_{r}Q+(u_{\varphi }/r){\partial }_{\varphi }Q$. Substituting the specified streamfunction givesThe factors of $1/r$ are required by cylindrical geometry.

An exact traveling eigenfunction. This is a polar gamma-plane Rossby wave. The radial equation implies ${\nabla }_h^2\psi -\psi /L_d^2=-{\alpha }^2\psi$, so $Q=f-{\alpha }^2\psi$. Since $J(\psi ,\psi )=0$, its nonlinear self-advection vanishes exactly, regardless of amplitude. Also $f_r=-\gamma r$, giving $J(\psi ,f)=\gamma {\psi }_{\varphi }$. HenceFor $\psi =F(r)\cos (\omega t-n\varphi )$, this yieldsThe wave is retrograde relative to the balanced mean flow, since $\omega /n<0$ in this convention.

Multiplying the radial equation by $r^2$ givesFor $K^2={\alpha }^2-L_d^{-2}>0$, setting $y=Kr$ gives the standard Bessel equation with derivatives with respect to $y$. ThusRegularity at the pole excludes $Y_n$ for the usual nonnegative integers. If $K^2<0$, the corresponding general solutions are modified Bessel functions; at $K=0$ they are $r^{\pm n}$, with a constant and $\ln r$ for $n=0$. The printed Bessel hint mixes its independent-variable labels; using $y=Kr$ and derivatives in $y$ resolves that notation.

What the hexagon fixes. With the north-polar approximation and the supplied rotation period,The intrinsic phase speed at that radius is $c_{\varphi }=r_0\omega /n=-\gamma r_0/{\alpha }^2$. Therefore the measured $-4\mathrm{m}{\mathrm{s}}^{-1}$ fixesThe polar-wave deformation-radius identifiability issue is that a unique $L_d$ additionally requires a radial wavenumber or boundary condition. The quoted peak-to-peak meander is an amplitude, not a radial wavelength: multiplying the exact eigenfunction by a constant changes its amplitude without changing $K,\alpha$ or $\omega$. Thus it cannot be silently converted into $K=2\pi /\Delta r$.