= Solution
\b[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.https://maths.ucd.ie/~plynch/eniac/CFvN-1950.pdf[Abstract and introductory development: filtered geostrophic description; four 24-hour forecasts at 500 mb; forecast-error discussion]{ref}
Near the north pole put $\delta=\pi/2-\theta$ and $r\simeq a\delta$. Then
$$
f=2\Omega\sin\theta=2\Omega\cos\delta\simeq2\Omega-\Omega\delta^2
=\boxed{2\Omega-\frac{\gamma r^2}2},\qquad \gamma=2\Omega/a^2.
$$
Using 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 $u_r\partial_rQ+(u_\phi/r)\partial_\phi Q$. Substituting the specified <streamfunction> gives
$$
\boxed{\partial_tQ+J(\psi,Q)=0,\qquad
J(\psi,Q)=\frac1r(\psi_rQ_\phi-\psi_\phi Q_r)}.
$$
The factors of $1/r$ are required by cylindrical geometry.
\b[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 $J(\psi,f)=\gamma\psi_\phi$. Hence
$$
-\alpha^2\psi_t+\gamma\psi_\phi=0.
$$
For $\psi=F(r)\cos(\omega t-n\phi)$, this yields
$$
\boxed{\omega=-\frac{n\gamma}{\alpha^2}}.
$$
The wave is retrograde relative to the balanced mean flow, since $\omega/n<0$ in this convention.
Multiplying the radial equation by $r^2$ gives
$$
r^2F''+rF'+[(\alpha^2-L_d^{-2})r^2-n^2]F=0.
$$
For $K^2=\alpha^2-L_d^{-2}>0$, setting $y=Kr$ gives the standard <Bessel equation> with derivatives with respect to $y$. Thus
$$
\boxed{F(r)=C_1J_n(Kr)+C_2Y_n(Kr),\qquad K=\sqrt{\alpha^2-L_d^{-2}}}.
$$
Regularity 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.
\b[What the hexagon fixes.] With the north-polar approximation and the supplied rotation period,
$$
\Omega=\frac{2\pi}{37980\ \mathrm s}=1.654\times10^{-4}\ \mathrm{s^{-1}},\quad
r_0\simeq a\frac{14\pi}{180}=1.417\times10^7\ \mathrm m,\quad
\gamma=9.84\times10^{-20}\ \mathrm{m^{-2}\,s^{-1}}.
$$
The intrinsic phase speed at that radius is $c_\phi=r_0\omega/n=-\gamma r_0/\alpha^2$. Therefore the measured $-4\ \mathrm{m\,s^{-1}}$ fixes
$$
\boxed{\alpha^2=\frac{\gamma r_0}{4}=3.48\times10^{-13}\ \mathrm{m^{-2}},\qquad
L_d=(\alpha^2-K^2)^{-1/2}}.
$$
The <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 $F''+F'/r$ is small relative to $n^2F/r_0^2$. Then $K^2\simeq n^2/r_0^2$ and
$$
\boxed{L_d\simeq\left(\alpha^2-\frac{36}{r_0^2}\right)^{-1/2}\simeq2.4\times10^6\ \mathrm m}.
$$
This is a conditional estimate, not a uniquely determined observation. The general real-$K$ bound is $L_d\ge\alpha^{-1}\simeq1.7\times10^6$ m. Different radial structures give different values: for example $K=3.0\times10^{-7}$ and $5.0\times10^{-7}\ \mathrm{m^{-1}}$ yield about $2.0\times10^6$ and $3.2\times10^6$ m, respectively, at the same measured phase speed. Their amplitudes remain free. The given $5^\circ$ meander corresponds to a radial half-amplitude about $2.5\times10^6$ m and helps characterize the pattern's size, but alone removes none of this eigenvalue ambiguity.
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