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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 $K=2\pi /\Delta r$.

A transparent order-of-magnitude closure is a slowly varying radial envelope near $r_0$, so $F^{\prime \prime }+F^{\prime }/r$ is small relative to $n^{2}F/r_0^2$. Then $K^2\simeq n^2/r_0^2$ andThis is a conditional estimate, not a uniquely determined observation. The general real-$K$ bound is $L_d\ge {\alpha }^{-1}\simeq 1.7\times 10^6$ m. Different radial structures give different values: for example $K=3.0\times 10^{-7}$ and $5.0\times 10^{-7}{\mathrm{m}}^{-1}$ yield about $2.0\times 10^6$ and $3.2\times 10^6$ m, respectively, at the same measured phase speed. Their amplitudes remain free. The given $5^{\circ }$ meander corresponds to a radial half-amplitude about $2.5\times 10^6$ m and helps characterize the pattern's size, but alone removes none of this eigenvalue ambiguity.