= Ideal magnetorotational dispersion relation
For a vertical <magnetic field> in a homogeneous <shearing sheet>, a horizontal <normal mode> with growth rate $s$ and vertical <Alfvén frequency> $\omega_a=kv_{Az}$ satisfies
$$
(s^2+\omega_a^2)(s^2+\omega_a^2-2q\Omega^2)+4\Omega^2s^2=0.
$$
The two roots for $s^2$ are real. For $q>0$, a growing magnetic mode exists precisely when $0<\omega_a^2<2q\Omega^2$. <Magnetic tension> is strong enough to couple radially displaced elements, yet weak enough to allow their separation to increase. https://www.damtp.cam.ac.uk/user/gio10/dad14.pdf[Ogilvie's lecture on magnetorotational instability] derives this relation and its finite-thickness boundary conditions.
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