the lengthscale $1/k$ of variation is longer than the “stopping length” $f_gg_et_{\mathrm{stop}}^2=f_g\tau_s^2L$ (see Appendix B for justification).

We will additionally neglect $\mathbf{F}$ and $\mathbf{G}$ because of the smallness of $\Delta\mathbf{V}$, an approximation YG05 find to be safe for $\omega\ll\Omega$.

The linearised system of equations is then: $$-i\omega u -2\Omega v +g_e\delta +ik_xh =0
\label{quartic system Euler x}$$ $$\frac{\kappa^2}{2\Omega}u -i\omega v=0
\label{quartic system Euler y}$$ $$-\omega w +k_zh =0
\label{quartic system Euler z}$$ $$ik_xu +ik_zw -i\omega\delta=0
\label{quartic system COM continuity}$$ $$ik_xu+ik_zw-ik_xg_e(f_p-f_g)t_{\mathrm{stop}}\delta +f_pk^2t_{\mathrm{stop}}h
=0
\label{quartic system incompressibility}$$ where zero subscripts have been dropped, $k^2=k_x^2+k_z^2$, and we have introduced the epicyclic frequency $\kappa$, given by $\kappa^2\equiv R^{-3}d(R^4\Omega^2)/dR$. In a Keplerian disc, $\kappa=\Omega$, but one could also imagine studying streaming instabilities in any rotating fluid, possibly even by direct experimentation. Equations - are derived from perturbation of the $x$, $y$ and $z$ components of equation (\[Euler-COM\]). Equations and stem from perturbing the continuity equations (\[continuity-COM\]) and (\[incompressibility\]) respectively. (In equation \[\[quartic system incompressibility\]\] we also make use of equation \[\[TVA\]\].) Notice that the important term proportional to ($f_p-f_g$) in equation derives from a sort of buoyancy force that arises in the two fluid systems, with an effective density of $\rho^2/\rho_p$. When $f_p=f_g$, this effective 