the corresponding unit vectors.

Using this model, @Nakagawaetal1986 computed the equilibrium solution to equations -, $$\mathbf{V}=(-\frac{g_e}{2\Omega}-\frac{3}{2}\Omega x) \mathbf{e_y}
\label{NSH-V}$$ $$\Delta\mathbf{V}=-\frac{g_et_{stop}}{1+(f_g\tau_s)^2}\mathbf{e_x}+\frac{f_gg_e\Omega
t_{stop}^2}{2(1+(f_g\tau_s)^2)}\mathbf{e_y},
\label{NSH-DeltaV}$$ where $f_{g,p}\equiv\rho_{g,p}/\rho$ the mass fraction of gas and particles (of uniform densities in this equilibrium flow), and $$g_e\equiv-\frac{1}{\rho}\frac{dP}{dR}$$ is the pressure-induced acceleration on the gas+dust fluid. It is the free energy stored in this gradient that the streaming instability can access (YG05). This gradient, consequently, introduces a lengthscale which helps determine the characteristic scale of the instability. We denote this scale by $L$ and quantify it through: $$\label{LLL}
L= \frac{g_e}{\Omega^2} \sim \left(\frac{H}{R}\right)H,$$ where $H$ is the gas disc semi-thickness. It follows that $L$ is a small fraction of $H$.

We now decompose the variables in equilibrium value and perturbation as follows: $$\mathbf{V}=\mathbf{V}_0+\mathbf{v}$$ $$\Delta\mathbf{V}=\Delta\mathbf{V}_0+\Delta\mathbf{v}$$ $$\rho=\rho_0(1+\delta)$$ $$P=P_0(x)+\rho_0h,$$ where the zero subscripts refer to the unperturbed background. Coordinates of the velocity perturbations are given by: $\mathbf{v}=u\mathbf{e}_x+v\mathbf{e}_y+w\mathbf{e}_z$, $\Delta\mathbf{v}=\Delta u\mathbf{e}_x+\Delta v\mathbf{e}_y+\Delta
w\mathbf{e}_z$. We then decompose the perturbation into axisymmetric Fourier modes $\propto \exp{[i(k_xx+k_zz-\omega t)]}$. Here, $k_x$ and $k_z$ are real wavenumbers and $\omega\equiv\omega_{\Re}+is$ is a complex frequency, with $\omega_{\Re}$ a wave frequency and $s$ a growth rate (if $s<0$, the perturbation is damped).

YG05 studied the resulting sixth-order system numerically and found unconditional 