and $k_xg_e/\kappa^2\ll 1$ (which imply the conditions of ), one is allowed to modify the quartic term as $-if_pt_{\mathrm{stop}}\left(k/k_z\right)^2\omega^4$ since $if_p\left(k_x/k_z\right)^2t_{\mathrm{stop}}\omega^4 \ll \omega^3$ and the quadratic one as $(if_p\kappa^2+k_xg_e(f_g-f_p))t_{\mathrm{stop}}\omega^2$ since $k_xg_ef_p\ll if_p\kappa^2$. Under these conditions, the quartic may be factored as: $$\begin{aligned}
\left(\omega^2-\left(\kappa\frac{k_z}{k}\right)^2\right)\left(-if_pt_{\rm stop}\left(\frac{k}{k_z}\right)^2\omega^2+\omega+k_xg_et_{\mathrm{stop}}(f_g-f_p)\right)\nonumber\end{aligned}$$ such that we retrieve the quadratic dispersion relation of the reduced system.

Hence, it appears that the mathematical essence of the instability can be isolated if we make the following set of assumptions: (a) the terminal velocity approximation, in which stopping time is short and so the relative velocity is determined from the steady balance (\[TVA\]); (b) geostrophic balance holds in the remaining momentum equations, which account for the horizontal centre of mass velocity (the centre of mass executes modified epicycles in the orbital plane); and (c) that radial wavelengths of perturbed quantities are long (of order $L$ or longer). In so doing, we have reduced the order of the system from 6 to 2.

The ingredients for instability, apparently, are: a vertical velocity generated by a vertical pressure gradient (equation \[\[quadratic system Euler z\]\]); accumulation of particles by the associated vertical flux of background particles (equation \[\[quadratic system COM continuity\]\]); and finally gas incompressibility (equation 