= Solution
Work locally at $R_0$ with $\Omega=\Omega(R_0)$, and define
$$
D=\left.\frac{d\Omega^2}{d\ln R}\right|_{R_0},\qquad a=k^2v_A^2,\qquad v_A=\frac{B_0}{\sqrt{\mu_0\rho}},\qquad\kappa^2=4\Omega^2+D.
$$
Here $\kappa$ is the <radial epicyclic frequency>. Combining the radial imbalance, the <Coriolis force> and the magnetic spring gives the <spring model of magnetorotational instability>:
$$
\ddot\xi_R-2\Omega\dot\xi_\phi=-(D+a)\xi_R,\qquad
\ddot\xi_\phi+2\Omega\dot\xi_R=-a\xi_\phi.
$$
For a <normal mode> proportional to $e^{-i\omega t}$, the coefficient <matrix> is
$$
\begin{pmatrix}a+D-\omega^2&2i\Omega\omega\\-2i\Omega\omega&a-\omega^2\end{pmatrix}.
$$
Setting its <determinant> to zero yields the <ideal magnetorotational dispersion relation>,
$$
\boxed{\omega^4-\omega^2(2a+4\Omega^2+D)+a(a+D)=0.}
$$
Its two values of $\omega^2$ are
$$
\omega_\pm^2=a+\frac{\kappa^2}{2}\pm\frac12\sqrt{(\kappa^2)^2+16a\Omega^2},
$$
so they are real. For $a>0$, their product is negative exactly when $a+D<0$, giving an exponentially growing <normal mode>. Conversely, $a+D>0$ also makes their sum positive and therefore gives two positive squared <frequencies>. \b[A fixed nonzero magnetic mode is exponentially unstable for $k^2v_A^2+D<0$, oscillatory for $k^2v_A^2+D>0$, and marginal at equality.] Stability against every allowed nonzero <wavenumber> in the unbounded ideal model requires $D\ge0$; finite-domain restrictions can remove the unstable band.
For $\Phi=-GM/R$, the equilibrium relation gives $\Omega^2=GM/R^3$, so $D=-3\Omega^2$ and $\kappa^2=\Omega^2$. Hence the <Keplerian disk> has the <magnetorotational instability> band
$$
\boxed{0<k^2v_A^2<3\Omega^2.}
$$
Without the <magnetic field>, the polynomial becomes $\omega^2(\omega^2-\kappa^2)=0$. The <Rayleigh centrifugal stability criterion> instead requires $\kappa^2>0$, equivalently $d(R^4\Omega^2)/dR>0$. A <Keplerian disk> satisfies that hydrodynamic criterion. Magnetic coupling therefore destabilizes a disk that has a restoring hydrodynamic <epicyclic oscillation>; the field couples angular-momentum exchange between elements, rather than acting only as an isolated restoring spring.
\Image[/past-exam-of-the-advanced-fluid-dynamics-course-of-the-university-of-oxford/2024-keplerian-magnetorotational-growth.png]
{title=Unstable band and fastest growth of the ideal vertical-field Keplerian mode}
The plotted maximum is the <maximum growth rate of ideal magnetorotational instability>, $3\Omega/4$ at $k^2v_A^2=15\Omega^2/16$.
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