density is stationary with respect to perturbations in $\rho_p$.

One can straightforwardly obtain a dispersion relation by setting the determinant of the above 5$\times$5 system to zero[^1]: $$\begin{aligned}
\label{22}
-if_pt_{\mathrm{stop}}\omega^4 + \omega^3 +
(if_p\kappa^2+k_xg_ef_g)t_{\mathrm{stop}}\omega^2-\left(\kappa\frac{k_z}{k}\right)^2\omega\nonumber\\
 + k_x\left(\kappa\frac{k_z}{k}\right)^2g_et_{\mathrm{stop}}(f_p-f_g)=0
\label{quartic}\end{aligned}$$ Series solutions in $\tau_s$ carried out by YG05 have shown that the roots fall in three branches, two epicycles giving rise to damping, and a secular mode, with: $$\omega_\Re=k_x(f_p-f_g)g_et_{\rm stop}+o(t_{\rm stop})
\label{omegaR}$$ $$s=if_pt_{\mathrm{stop}}^3\left((f_p-f_g)g_e\frac{kk_x}{k_z}\right)^2+o(t_{\mathrm{stop}}^3),
\label{growth-TVA}$$ to leading order in the stopping time[^2]. (The notation $o(X)$ indicates that the ratio $o(X)/X$ tends to zero as $X\rightarrow 0$). It can be seen that feedback is essential to the instability, as ignoring it would amount to setting $f_p=0$ everywhere. Also, the necessity of the background drift as measured by $g_e$ is evident. More rigorously, it may be seen that for $g_e=0$, the dispersion relation becomes, if we disregard the $\omega=0$ mode: $$f_p\kappa t_{\rm stop}X^3-X^2+f_p\kappa t_{\rm stop}X-\left(\frac{k_z}{k}\right)^2=0$$ with $X\equiv i\omega/\kappa$. One can show that, for sufficiently small $\kappa t_{\rm stop}$, this cubic has three real positive roots (in terms of $X$), corresponding to $\omega$ being a purely imaginary number of negative imaginary part .

The secular mode in a reduced system
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It is possible to simplify the 