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 Incoming links: A Powerful Local Shear Instability in Weakly Magnetized Disks. I. Linear Analysis by Steven A. Balbus and John F. Hawley

John F. Hawley helped explain angular-momentum transport in accretion disks through the magnetorotational instability. With Steven A. Balbus, he showed in 1991 that even a weak magnetic field can destabilize differentially rotating disks whose angular velocity decreases outward.^[ref]

In the spring model of magnetorotational instability, magnetic tension couples displaced fluid elements and transfers angular momentum. This can amplify their separation despite stability under the corresponding unmagnetized rotational test.

Magnetorotational instability Created 2026-09-24 Updated 2026-11-05

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^{s t + ik z}

. Let

v_{A} = ∣ B_{z} ∣/ μ_{0} ρ

and

κ

κ^{2} = 4 Ω^{2} + \frac{d Ω ^{2}}{d l o g R} .

Eliminating the horizontal velocity and magnetic perturbations gives

s^{4} + (κ^{2} + 2 k^{2} v_{A}^{2}) s^{2} + k^{2} v_{A}^{2} (k^{2} v_{A}^{2} + \frac{d Ω ^{2}}{d l o g R}) = 0.

For a hydrodynamically stable background with

κ^{2} > 0

, a growing root exists exactly when

0 < k^{2} v_{A}^{2} < - \frac{d Ω ^{2}}{d l o g R} .

Indeed, 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

κ^{4} + 16 Ω^{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 Ω^{2}

. Maximizing the positive root gives

k^{2} v_{A}^{2} = 15 Ω^{2} /16

s_{m a x} = 3Ω/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.

Steven A. Balbus, working with John F. Hawley, established the astrophysical importance of the magnetorotational instability in weakly magnetized accretion disks. Their local linear stability analysis identified an instability associated with outward-decreasing angular velocity, even where the unmagnetized flow would pass the Rayleigh centrifugal stability criterion.^[ref]

The result offered a route from differential rotation to fluid fluctuations and outward angular momentum transport, allowing gas to accrete without attributing the transport to molecular viscosity alone.