 Radially stratified magnetorotational dispersion relation

For an axisymmetric vertical mode of a shearing sheet with a vertical magnetic field, radial buoyancy frequency $N_r$, and $A=k^2{v}_A^2>0$, the nonstationary amplitudes obey ${\sigma }^4+(2A+N_r^2+{\Omega }^2){\sigma }^2+A(A+N_r^2-3{\Omega }^2)=0$. The two roots for ${\sigma }^2$ are real. A growing mode exists exactly when $A+N_r^2<3{\Omega }^2$. If the advected buoyancy displacement variable is retained as an independent amplitude, the full characteristic polynomial also has a stationary factor $\sigma$.