Magnetorotational instability

Remarks

The relation is a quadratic equation in $\omega^2$ with real roots, since its discriminant is $\kappa^4 + 16 \Omega^2 k^2 v_A^2 \ge 0$. Its two roots have sum $2 k^2 v_A^2 + \kappa^2 \ge 0$ and product $k^2 v_A^2 (k^2 v_A^2 + d\Omega^2/d\ln R)$, so one root is negative exactly when this product is negative. Physically, the field acts like a spring between two fluid elements at different radii. The inner element is pulled back, loses angular momentum and sinks further in, while the outer element gains angular momentum and moves out.

For a Keplerian disk, $\Omega \propto R^{-3/2}$, so $d\ln\Omega/d\ln R = -3/2$ and the largest growth rate is $\frac{3}{4}\Omega$. The instability therefore acts in every weakly magnetized accretion disk and is believed to drive the turbulence that transports angular momentum.

Without the field the Rayleigh criterion applies: the flow is stable to axisymmetric perturbations when $\kappa^2 > 0$, and a Keplerian disk is stable. The limit $B \to 0$ is singular, because the band of unstable $k$ only moves to larger wavenumbers as $v_A \to 0$. In a real fluid, resistivity and viscosity damp these short waves, so the field must not be too weak.

Going deeper

For a student

The result is a local linear stability analysis in the spirit of the Rayleigh criterion for rotating flows. Compared with Rayleigh, the stability condition changes from "specific angular momentum increases outward" to "angular velocity increases outward". Its growth rate is of the order of the orbital frequency, so it is fast on astrophysical scales. It gives a physical basis to the turbulent viscosity in the Shakura–Sunyaev alpha disk model, and numerical simulations of its nonlinear stage are used to estimate $\alpha$.