in turn can attract more dust particles, and the process thus runs away. This is sketched in Figure \[SI\].

Indeed, inspection of equation shows that growth is correlated with $f_pP'/\rho_p'$. $f_p$ is a measure of the feedback of dust on the gas, while $P'/\rho_p'$ measures how well dust density and pressure maxima correlate with each other, or in other words, the efficiency of the dust-driven pressure loading. Growth is positive if both pressure and density perturbations are not out of phase by more than $90\,^{\circ}$. For example, YJ07’s eigenvector “linA” has a pressure phase relative to particle density of $43\,^{\circ}$ and a complex frequency $\omega=(-0.3480127+0.4190204i)\Omega$ while that same phase for “linB” is $-95\,^{\circ}$ with a complex frequency $\omega=(0.4998786+0.0145764i)\Omega$.[^4] For the secular mode in the reduced model, $P'/\rho_p'=h/\delta=(\omega/k_z)^2$ (see equation ) is real and positive to leading order.

While the above identifies the positive feedback loop leading to instability, we have yet to account for two other physical ingredients that are necessary condition to it. Indeed, drag of the dust on the gas is actually insufficient alone to offset the pressure force that tends to accelerate gas *away* from pressure maxima. In the terminal velocity approximation, we have: $$\frac{\rho_p}{\rho_g}\frac{\mathbf{V}_p-\mathbf{V}_g}{t_{\mathrm{stop}}}-\frac{\nabla P}{\rho_g}=-\frac{\nabla P}{\rho},
