This is an independent, purely 2D model — not a flattened version of the 3D scene. Instead of only tracking a scalar displacement, it solves the actual local shearing-box linear system for the magnetorotational instability (Balbus & Hawley 1991, 1998): radial velocity ux, azimuthal velocity uy, and the two perturbed field components Bx, By (in Alfvén-speed units), coupled by Coriolis force, magnetic tension and Keplerian shear stretching.
σu_x = 2Ωu_y + k·v_A·B_x
σu_y = -(2-q)Ωu_x + k·v_A·B_y q = 3/2 (Keplerian shear)
σB_x = -k·v_A·u_x
σB_y = -k·v_A·u_y - qΩB_x
σ² = -(κ²+2k²v_A²)/2 + √[(κ²/2)² + 4Ω²k²v_A²] κ = Ω for Keplerian
This closed-form σ(k·v_A/Ω) is the exact quartic dispersion relation (not the parabolic stand-in some simpler visualizations use) — it reproduces the rigorous maximum growth rate of exactly 0.75Ω at k·v_A/Ω = √15/4 ≈ 0.968, and the exact stability cutoff at k·v_A/Ω = √3 ≈ 1.732 where magnetic tension fully overwhelms the shear.
- Left panel — a radial–vertical (x–z) cross-section: field lines threading the disk kink as ξ(z,t) = A(t)·sin(kz); tracer dots are colored by their real azimuthal velocity uy (computed from the eigenvector above, not just the sign of the displacement) — red leads the shear, blue lags it.
- Right panel — a top-down view of the annulus: rings at different radii rotate at their own real Keplerian Ω(r), demonstrating the differential rotation that powers the instability, with a small radial nudge from the same growing mode.
- Orbital radius, field strength (vA) and wavenumber (k) control the same physical quantities as in the 3D shearing-box scene, so results are directly comparable — but every number here (growth rate, polarization ratios, Maxwell stress) comes from solving the full linear system, verified against the known analytic results before this page was written.
Real-world relevance: the sign of −BxBy (shown live) is what makes MRI turbulence special — it is guaranteed positive for the growing mode, meaning the instability transports angular momentum outward by construction, not by chance correlation, which is why it — not molecular viscosity — is what actually lets matter accrete onto black holes on astrophysically short timescales.