confused with $\rho_p$) and $a$ the grain radius. Regardless of the relevant drag law, one defines a dimensionless stopping time measuring the coupling of dust to dynamical disturbances: $$\tau_s\equiv\Omega t_{\mathrm{stop}},$$ By virtue of Newton’s third law, an all-important feedback term of the dust on the gas appears in the Euler equation for the gas (\[Euler-gas\]). The system of equations is closed by assuming gas incompressibility (the Boussinesq approximation), as in YG05.

It is convenient to express equations (\[continuity-dust\])-(\[Euler-gas\]) in terms of centre-of-mass velocity $\mathbf{V}\equiv (\rho_p\mathbf{V}_p+\rho_g\mathbf{V}_g)/\rho$ (with $\rho\equiv\rho_p+\rho_g$ the total density) and the relative dust-to-gas drift $\Delta\mathbf{V}\equiv\mathbf{V}_p-\mathbf{V}_g$ (YG05): $$\frac{\partial \rho}{\partial t}+\nabla\cdot(\rho\mathbf{V})=0
\label{continuity-COM}$$ $$\nabla\cdot(\mathbf{V}-\frac{\rho_p}{\rho}\Delta\mathbf{V})=0
\label{incompressibility}$$ $$\frac{\partial \mathbf{V}}{\partial
t}+\mathbf{V}\cdot\nabla\mathbf{V}+\mathbf{F}(\Delta\mathbf{V}^2)=-\Omega^2\mathbf{R}-\frac{\nabla
P}{\rho}
\label{Euler-COM}$$ $$\frac{\partial\Delta\mathbf{V}}{\partial
t}+\mathbf{V}\cdot\nabla(\Delta\mathbf{V})+(\Delta\mathbf{V}\cdot\nabla)\mathbf{V}+\mathbf{G}(\Delta\mathbf{V}^2)=-\frac{\rho}{\rho_g}\frac{\Delta\mathbf{V}}{t_{\mathrm{stop}}}+\frac{\nabla
P}{\rho_g},
\label{Euler-DeltaV}$$ with: $$\mathbf{F}(\Delta\mathbf{V}^2)\equiv\frac{1}{\rho}\nabla\cdot\left(\frac{\rho_g\rho_p}{\rho}\Delta\mathbf{V}\Delta\mathbf{V}\right)$$ $$\mathbf{G}(\Delta\mathbf{V}^2)\equiv\frac{\rho_g}{\rho}\Delta\mathbf{V}\cdot\nabla\left(\frac{\rho_g}{\rho}\Delta\mathbf{V}\right)-\frac{\rho_p}{\rho}\Delta\mathbf{V}\cdot\nabla\left(\frac{\rho_p}{\rho}\Delta\mathbf{V}\right).$$ We shall now work in the so-called shearing-sheet approximation: We neglect all curvature terms and disc-scale gradients since the lengthscales of interest are much smaller than the heliocentric distance, or even the pressure scale height. We also adopt Cartesian coordinates in a frame corotating with the Keplerian flow at a fixed heliocentric distance $R_0$, at angular velocity $\Omega$. We denote by $x$, $y$ and $z$ coordinates in the radial, azimuthal, and vertical (that is, perpendicular to the midplane) directions, respectively, and $\mathbf{e}_x$, $\mathbf{e}_y$ and $\mathbf{e}_z$ 