= Solution
The <horizontally invariant magnetized shearing-sheet equations> are linear in the horizontal fields, so perturbations about the equilibrium satisfy the same equations. The fixed surface <boundary conditions> require $\delta B_x=\delta B_y=0$ at $z=\pm H$. Choose a <normal mode> with
$$
(\delta B_x,\delta B_y)=(b_x,b_y)e^{st}\sin[k(z+H)],\qquad
(\delta v_x,\delta v_y)=(u,v)e^{st}\cos[k(z+H)],
$$
where the <vertical wavenumber> is $k=m\pi/(2H)$, $m=1,2,\ldots$. With the <Alfvén frequency>
$$
\omega_a^2=\frac{k^2B_z^2}{\mu_0\rho}=k^2v_{Az}^2,
$$
the four amplitude equations become
$$
su-2\Omega v=\frac{kB_z}{\mu_0\rho}b_x,\qquad
sv+(2-q)\Omega u=\frac{kB_z}{\mu_0\rho}b_y,
$$
$$
sb_x=-kB_zu,\qquad sb_y+q\Omega b_x=-kB_zv.
$$
Eliminating the velocities leaves
$$
(s^2+\omega_a^2-2q\Omega^2)b_x-2\Omega s b_y=0,\qquad
2\Omega s b_x+(s^2+\omega_a^2)b_y=0.
$$
A nonzero amplitude requires the <determinant> of this system to vanish, yielding the <ideal magnetorotational dispersion relation>
$$
\boxed{(s^2+\omega_a^2)(s^2+\omega_a^2-2q\Omega^2)+4\Omega^2s^2=0}.
$$
Equivalently, with $\kappa_r^2=2(2-q)\Omega^2$,
$$
s^4+(2\omega_a^2+\kappa_r^2)s^2+\omega_a^2(\omega_a^2-2q\Omega^2)=0.
$$
As a <quadratic equation> for $s^2$, its <discriminant> is $\kappa_r^4+16\Omega^2\omega_a^2>0$. If $0<\omega_a^2<2q\Omega^2$, its constant term is negative, so one root $s^2$ is positive and there is an exponentially growing mode. If $\omega_a^2>2q\Omega^2$, both the constant term and the coefficient of $s^2$ are positive, giving two negative roots and only oscillatory modes. Equality is marginal.
The lowest allowed <vertical wavenumber>, $k=\pi/(2H)$, is the last to be stabilized as $|B_z|$ increases. Therefore the <finite-thickness magnetorotational instability criterion> is
$$
\boxed{0<\frac{\pi^2B_z^2}{8q\mu_0\rho H^2\Omega^2}<1}.
$$
There is also a vertically uniform velocity mode with zero magnetic perturbation; for the usual orbitally stable regime $q<2$, it is just stable <epicyclic motion>. For $q>2$, that uniform mode is already hydrodynamically unstable, independently of the magnetic criterion. At $q=2$ it is marginal. The criterion above concerns the magnetic modes.
Back to article page
