Radially stratified magnetorotational dispersion relation (${\sigma }^4+(2A+N_r^2+{\Omega }^2){\sigma }^2+A(A+N_r^2-3{\Omega }^2)=0$)

For an axisymmetric vertical mode of a shearing sheet with a vertical magnetic field, radial buoyancy frequency $N_r$, and $A=k^2{v}_A^2>0$, 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 $\sigma$.

For $-5{\Omega }^2<N_r^2<3{\Omega }^2$ and a continuum of vertical wavenumbers, the growing branch of the radially stratified magnetorotational dispersion relation is maximized at $k^2{v}_A^2={\Omega }^2-(N_r^2+{\Omega }^2{)}^2/(16{\Omega }^2)$, with ${\sigma }_{\max }=(3{\Omega }^2-N_r^2)/(4\Omega )$. For $N_r^2\le -5{\Omega }^2$ the optimum moves to the $k\to 0$ boundary and the limiting rate is $\sqrt{-N_r^2-{\Omega }^2}$. For $N_r^2\ge 3{\Omega }^2$ there is no exponential instability. Boundaries that discretize the allowed wavenumbers can alter the actual maximum.