velocity is simply $\mathbf{V}_p$ and the pattern is carried by the particle fluid alone. This is because, in this limit, the particles’ influence on the gas is so minor that $\mathbf{V}_g$ is effectively zero. On the other hand, when there is virtually no gas, $f_g\to 0$, the advection approaches zero, because $\mathbf{V}_g\to 0$ (as well as $\mathbf{V}_p$) in this limit.

Instability term
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We have already mentioned that instability was related to pressure maxima (or technically, pressure *perturbation* maxima, as there is a background gradient, which however has zero Laplacian ). This is reminiscent of the concentration mechanism of test particles in long-lived, two-dimensional vortices [e.g. @BargeSommeria1995; @KlahrBodenheimer2006], which in a Keplerian flow are anticyclonic.

In the streaming instability, this concentration is an active process, in the sense that feedback participates in the preservation of the high-pressure, particle-trapping zones. How does the instability grow? Schematically, as we have said, pressure maxima attract dust particles, as can be seen from the terminal velocity approximation given by equation . Then, dust drags the gas toward these same maxima (see the gas Euler equation , where the terminal velocity approximation may be injected). This creates convergent flows that *strengthen* these pressure maxima. These 