= Radially stratified magnetorotational dispersion relation
{title2=$\sigma^4+(2A+N_r^2+\Omega^2)\sigma^2+A(A+N_r^2-3\Omega^2)=0$}
For an <axisymmetric vertical mode of a shearing sheet> with a vertical <magnetic field>, <radial buoyancy frequency> $N_r$, and $A=k^2v_A^2>0$, 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 $\sigma$.
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