= Solution
Assume a hydrogen-rich gas with approximately solar elemental ratios and local <thermochemical equilibrium> when identifying the limiting compositions. At $T\lesssim500\,\mathrm K$, a representative set of dominant molecules is \b[$\mathrm H_2,\mathrm H_2O,\mathrm{CH}_4,\mathrm{NH}_3$]. In the deeper $T\gtrsim1300\,\mathrm K$ regime, it is \b[$\mathrm H_2,\mathrm H_2O,\mathrm{CO},\mathrm N_2$]. <Helium> is an abundant atom, not a molecule. The carbon and nitrogen switches, <carbon monoxide–methane quenching> and <nitrogen–ammonia quenching> when frozen by mixing, are represented by
$$
\mathrm{CO}+3\mathrm H_2\rightleftharpoons\mathrm{CH}_4+\mathrm H_2O,\qquad
\mathrm N_2+3\mathrm H_2\rightleftharpoons2\mathrm{NH}_3.
$$
Low temperature and high pressure favor the right-hand sides, while hotter gas favors <carbon monoxide> and $\mathrm N_2$. Exact boundaries depend on pressure and composition; $\mathrm{CO}_2$ can become important at high metallicity. If mixing is strong, the cool upper atmosphere need not retain its local-equilibrium four-species ordering.
To preserve the hot-region reactant $A_1$ aloft, require transport to beat its conversion at the quench region. With a mixing length $\ell$ and vertical <eddy diffusivity> $K_{zz}$,
$$
\tau_{\rm mix}\sim\frac{\ell^2}{K_{zz}}\lesssim10^5\,\mathrm s,\qquad
\boxed{K_{zz}\gtrsim\ell^2/(10^5\,\mathrm s).}
$$
For the simplest use of the supplied <atmospheric scale height> reference, assume the same temperature and <mean molecular weight> as the reference atmosphere and take $\ell=H$. Since $H\propto1/g$, $H\simeq30/0.4=75\,\mathrm{km}$, giving \b[$K_{zz}\gtrsim5.6\times10^4\,\mathrm{m^2\,s^{-1}}=5.6\times10^8\,\mathrm{cm^2\,s^{-1}}$].
The hotter quench layer requires a temperature correction if the reference is ordinary Jupiter; the temperature of that scale-height reference was not specified. An explicit estimate using $T_q=1300\,\mathrm K$, dimensionless <mean molecular weight> $\mu=2.3$, and $g\simeq0.4\times25=10\,\mathrm{m\,s^{-2}}$ gives
$$
H=\frac{k_BT_q}{\mu m_Hg}\simeq4.7\times10^5\,\mathrm m,
\qquad K_{zz}\gtrsim2.2\times10^6\,\mathrm{m^2\,s^{-1}}
=2.2\times10^{10}\,\mathrm{cm^2\,s^{-1}}.
$$
Thus \b[a scale-height-based estimate using the hot layer is of order $10^{10}\,\mathrm{cm^2\,s^{-1}}$]. Both estimates state their assumptions: the reference scaling alone does not include the temperature ratio. Choosing $\ell=0.1H$ lowers the threshold by a factor of $100$, and a full quench calculation needs the reaction timescale along the profile. The <dependence of a quench diffusivity on mixing length> shows why the stated reaction timescale supplies a mixing constraint, not a unique measured coefficient.
Back to article page
