 Magnetorotational instability

The magnetorotational instability allows a weak magnetic field to destabilize some differentially rotating flows that are stable to axisymmetric hydrodynamic displacements. A useful picture connects two fluid elements by magnetic tension: the inner element loses angular momentum and moves inward, while the outer element gains angular momentum and moves outward. Decreasing angular velocity amplifies their separation. This makes the instability relevant to angular momentum transport in an accretion disk.

For a local, unstratified, incompressible ideal-MHD model with vertical magnetic field $B_z$, take an axisymmetric perturbation proportional to $e^{st+ikz}$. Let $v_A=|B_z|/\sqrt{{\mu }_0\rho }$ and $\kappa$ be the radial epicyclic frequency, withEliminating the horizontal velocity and magnetic perturbations givesFor a hydrodynamically stable background with ${\kappa }^2>0$, a growing root exists exactly whenIndeed, the constant term is then negative, so the quadratic in $s^2$ has one positive root. If the constant term is nonnegative, both roots are nonpositive: their sum is negative and their discriminant is ${\kappa }^4+16{\Omega }^2{k}^2{v}_A^2$. The vertical-field local limit follows from the original linear analysis.^[ref]

In a Keplerian disk, the unstable interval is $0<k^2{v}_A^2<3{\Omega }^2$. Maximizing the positive root gives $k^2{v}_A^2=15{\Omega }^2/16$ and growth rate $s_{\max }=3\Omega /4$. Strong tension stabilizes shorter wavelengths. In a finite disk the unstable wavelengths must also fit the vertical domain; magnetic diffusivity, viscosity and nonideal magnetic coupling can modify or suppress them. Thus a negative angular-velocity gradient alone is not a universal sufficient condition for every magnetized flow. Linear growth also does not by itself specify the eventual turbulent stress.

Magnetorotational instability (MRI) is a fundamental instability that occurs in differentially rotating astrophysical fluids in the presence of a magnetic field. It plays a critical role in the dynamics of accretion disks, such as those found around black holes, neutron stars, and young stellar objects, as well as in the formation of stars and planets.