take the dust for example, and use the terminal velocity approximation (or higher-order corrections) as a proxy to eliminate the gas velocity (we cannot treat it as *imposed*—this is indeed the point of the streaming instability), the drag acceleration becomes $-\nabla P/\rho$ which makes no reference to the dust velocity. Same holds for the gas. Finally, as we have noted earlier, the variation of the relative drift with density is not responsible for growth in this approximation and hence does not conform to the notion of a collective drag underlying the instability.

Conclusion
==========

Through successive approximations, we have obtained a reduced system giving rise to the streaming instability. It would appear that the essence of the instability can be effectively established given:

-   the terminal velocity approximation

-   that a form of geostrophic balance holds for the planar centre of mass velocity

-   that radial wavelengths are sufficiently long.

The leading order (in the stopping time) expansion of the growth rate is the same as the fourth-order system with the terminal velocity approximation.

We interpret the streaming instability as arising from dust pileup in pressure perturbation maxima, with the dust then dragging the gas and hence strengthening the maxima. 