We now develop a framework with which to interpret the streaming instability.

Particle concentration in pressure maxima
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We start with the equation governing the evolution of the dust mass fraction. Combining equations (\[continuity-dust\]) and (\[continuity-gas\]) (we use here the full equations rather than those of the reduced system), we obtain: $$\mathcal{D}_p\mathrm{ln}\frac{\rho_p}{\rho}+\frac{\nabla\cdot\left(\rho_g(\mathbf{V}_p-\mathbf{V}_g)\right)}{\rho}=0.
\label{Jacquet continuity}$$ It is clear from this equation that dust is advected by $\mathbf{V}_p$ but is also subject to a mass flux associated with the relative drift (the second term on the left-hand-side of equation (\[Jacquet continuity\])). We might expect then to observe dust clumps correlated with $\Delta\mathbf{V}$ (as noted by YG05 in the caption of their figure 6). In the terminal velocity approximation, equation (\[Jacquet continuity\]) becomes: $$\begin{aligned}
\mathcal{D}_p\mathrm{ln}\frac{\rho_p}{\rho}&=&-\frac{1}{\rho}\nabla\cdot\left(\rho_gt_{\mathrm{stop}}\frac{\nabla
P}{\rho}\right)\nonumber\\
&=&\frac{\rho_g}{\rho}\left(\Delta\mathbf{V}\cdot\nabla
\mathrm{ln}\rho-t_{\mathrm{stop}}\frac{\nabla^2 P}{\rho}\right),
\label{Jacquet-theorem-TVA}\end{aligned}$$ where the latter equality assumes the constancy of $\rho_g t_{\mathrm{stop}}$. There are two terms on the right-hand-side: one due to density variation, and one to pressure variation. We shall show below that the latter is responsible for growth, not the former (as was also ruled out by YJ07). Equation may be rewritten in the useful form: $$\left(\frac{\partial}{\partial
t}+\left(f_p\mathbf{V}_g+f_g\mathbf{V}_p\right)\cdot\nabla\right)\mathrm{ln}\frac{\rho_p}{\rho}=-\frac{\rho_gt_{\rm
stop}}{\rho^2}\nabla^2P
\label{new Jacquet theorem TVA}$$ in which we have used incompressibility. Note that we have 