A weak magnetic field threading a differentially rotating plasma is violently unstable: the magnetorotational instability (MRI), discovered by Balbus & Hawley (1991), is the leading mechanism by which real accretion disks around black holes transport angular momentum outward and accrete inward.
This is a local "shearing box" model — a small patch of disk co-rotating with the flow at angular frequency Ω(r). A vertical field line displaced radially by ξ is stretched by Keplerian shear (dΩ/dlnr = −3/2 Ω); magnetic tension then pulls back on the lagging fluid element, converting orbital shear energy into growing radial and azimuthal motion.
Ω(r) = √(GM/r³) Keplerian angular frequency
q = k·v_A / Ω dimensionless wavenumber
γ(q) = (3/4)·Ω·q·(2−q) growth rate, 0 < q < 2
γ_max = (3/4)Ω at q ≈ 1 rigorous Balbus–Hawley result
This γ(q) curve is a simplified stand-in for the full quartic MRI dispersion relation — it is built to reproduce the two facts that are rigorously established for a Keplerian disk: the maximum growth rate of exactly (3/4)Ω, and the stability cutoff once k·v_A exceeds ~2Ω (tension overwhelms shear at short wavelength).
- Orbital radius — sets the real Keplerian Ω(r) for a 10 M☉ black hole via Ω=√(GM/r³); closer to the hole, everything grows faster in absolute time.
- Field strength (vA) and wavenumber (k) — together set q = k·v_A/Ω. Tune them so q ≈ 1 to hit the fastest-growing mode; push q past 2 and the mode is stable (field lines are too stiff for the shear to bend).
- Field lines and tracer particles show the same displacement ξ(z,t) = A(t)·sin(kz), amplitude growing as A(t) = A₀·e^(γt) until it saturates — real disks saturate this growth nonlinearly into MHD turbulence, which this linear model does not simulate.
Real-world relevance: MRI-driven turbulence is why accretion disks accrete at all on astrophysically short timescales — without it, viscous transport alone would take far longer than the age of the universe.