= Solution
Take $x$ eastward, $y$ northward and $z$ upward. In a <rotating reference frame>, the horizontal <Cauchy momentum equation> is
$$
\frac{D\mathbf u_h}{Dt}+f\widehat{\mathbf z}\times\mathbf u_h=-\frac1\rho\nabla_h p+\mathbf F_h,
\qquad f=2\Omega\sin\phi.
$$
Here $\mathbf F_h$ represents other accelerations, such as <friction>, and $f$ is the <Coriolis parameter>. When the <Rossby number> $U/(|f|L)$ and the ratio $1/(|f|\tau)$ are small, and <friction> is weak, both advective and local <acceleration> can be neglected. The <Coriolis force> then balances the <pressure gradient force>: this is <geostrophic balance>, with $fv=p_x/\rho$ and $fu=-p_y/\rho$. The <geostrophic wind> runs along <atmospheric isobars>, with low <pressure> on its left in the northern hemisphere.
A rapid change of <pressure> need not bring the corresponding <geostrophic wind> into existence at once. An initially weak wind therefore leaves an inward <pressure gradient force> around the new low. For a smoothed anomaly of horizontal scale $L=5\times10^5\ \mathrm m$, <geostrophic balance> would require
$$
U_g\sim\frac{|p\prime|}{\rho |f|L}
\simeq33\ \mathrm{m\,s^{-1}}
\left(\frac{1.2\ \mathrm{kg\,m^{-3}}}{\rho}\right)
\left(\frac{10^{-4}\ \mathrm{s^{-1}}}{|f|}\right).
$$
For the illustrative midlatitude value $|f|=10^{-4}\ \mathrm{s^{-1}}$, the implied <Rossby number> is already about $0.67$, so curved-flow <acceleration> matters. With a small tropical value $|f|=10^{-5}\ \mathrm{s^{-1}}$, the inferred speed is $333\ \mathrm{m\,s^{-1}}$ and the <Rossby number> about $67$: the assumed <geostrophic balance> is untenable. <Geostrophic adjustment> redistributes the anomaly and emits <internal gravity waves>, leaving a weaker balanced component.
For a target speed $U_*=10\ \mathrm{m\,s^{-1}}$ on the same scale, the required scale estimate is
$$
\boxed{|p\prime_{\rm adj}|\sim\rho |f|LU_*
=6\ \mathrm{hPa}\left(\frac{\rho}{1.2\ \mathrm{kg\,m^{-3}}}\right)
\left(\frac{|f|}{10^{-4}\ \mathrm{s^{-1}}}\right).}
$$
Thus $6\ \mathrm{hPa}$, or $0.30$ of the initial magnitude, is the midlatitude estimate; at the illustrative tropical $|f|=10^{-5}\ \mathrm{s^{-1}}$ it is $0.6\ \mathrm{hPa}$, or $0.030$ of the initial magnitude. \b[The paper gives no latitude, so there is no unique numerical reduction.] At the equator itself one must use equatorial dynamics rather than divide by $f=0$. Even the tropical $10\ \mathrm{m\,s^{-1}}$ estimate has <Rossby number> $2$, so it describes the scale permitted by the stated balance, rather than a controlled tropical <geostrophic balance>.
There are two qualifications to “unstable”. A literal top-hat <pressure> anomaly has a discontinuous edge, where no finite classical <geostrophic wind> can balance its distributional <pressure gradient>. The $500\ \mathrm{km}$ estimate assumes a smoothed edge; a thinner edge requires stronger winds. More generally a large low is not, by its amplitude alone, proved to have an exponentially growing instability. For a concrete counterexample to the claim that such an amplitude cannot be stationary, set $p\prime(s)=-P\exp(-s^2/a^2)$, with cylindrical radius $s$, constant $\rho$ and $f>0$. The cyclonic <gradient wind>
$$
v_\theta(s)=\frac{-fs+\sqrt{f^2s^2+4s p_s/\rho}}2
$$
is real and satisfies $v_\theta^2/s+fv_\theta=p_s/\rho$ for every $P>0$, with a regular limit at the centre. This is a stationary <axisymmetric flow>; no claim about its perturbation stability is needed. Here the physically justified conclusion is that a rapidly imposed anomaly without its balancing wind is \b[unbalanced].
Small tropical <Coriolis parameter> allows only weak horizontal <pressure gradients> to coexist with ordinary large-scale winds. <Hydrostatic balance> connects the vertical variation of these <pressure gradients> to horizontal <temperature gradients>; equivalently the approximate <thermal wind> relation gives $f u_z\simeq-(g/T)T_y$. For comparable vertical wind shear, small $|f|$ means smaller $|T_y|$. <Internal gravity waves> can spread heating-induced <buoyancy> anomalies efficiently before rotation arrests the adjustment. This motivates the <weak temperature gradient approximation> for the tropical free troposphere; it does not require the land surface or atmospheric boundary layer to have uniform <temperature>.
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