not yet linearised the system—this equation holds equally well in the nonlinear regime—and also that in this subsection we have made no hypothesis on the centre of mass dynamics (e.g. presence of Coriolis forces), since we have only used mass conservation equations and the terminal velocity approximation. Its linearised form, however, with the Fourier dependence $\exp{[i(k_xx+k_zz-\omega t)]}$, yields $$\omega=\mathbf{k}\cdot(f_p\mathbf{V}_g+f_g\mathbf{V}_p) +
if_pk^2t_{\mathrm{stop}}\frac{P'}{\rho_p'},
\label{growth-gradP}$$ with $P'\equiv\rho h$ and $\rho_p'\equiv\rho\delta$ the perturbation in pressure and (particle) density, respectively.

What we have in is a form of ‘advective-reaction’ equation. Dust is advected at a velocity intermediate between $\mathbf{V}_p$ and $\mathbf{V}_g$, to wit $$\label{wavy}
f_p\mathbf{V}_{g}+f_g\mathbf{V}_{p}=\mathbf{V}_p+(\mathbf{V}_g-\mathbf{V})$$ which corresponds to the $U_{\rm sum}$ of YG05, as in the frame used, $\mathbf{k}\cdot\mathbf{V}=0$. On the right hand side we have a source term proportional to the pressure Laplacian. This term will tend to draw dust *towards* pressure maxima in the $x-z$ plane and *away* from pressure minima. Clearly, instability is intimately connected with this term, and the unstable mode works by concentrating dust at pressure maxima. We discuss both processes in turn.

Advection term
--------------

The left-hand side of equation gives rise to the streaming instability wave character: a density pattern in $x$ and $z$ will be advected at 