Wave interference as the origin of the cyclic magnetorotational dynamo in accretion disks: insights from weakly nonlinear theory and local shearing box simulations

Long-period cyclic reversals of the large-scale magnetic field are a prominent feature of the dynamo associated with the magnetorotational instability (MRI) in accretion disks, but their physical origin remains unclear. We develop a quasilinear theory (QLT) of the MRI dynamo where the electromotive force (emf) is computed from the linear eigenfunctions under the WKB approximation. The emf depends on the mean field B more generally than standard mean-field closures allow. In the unstratified case, the leading order contribution to the large-scale dynamo is the shear-current effect: the emf depends on the current J as = times J, with a tensor (B,t) that oscillates with time t and whose off-diagonal components generate the mean field. The oscillations arise from beats between the two branches of eigenfrequencies. Since the beat frequency varies only weakly with wavenumber, the beats remain coherent and drive the long-period butterfly cycle seen in local shearing box simulations. We predict a dominant cycle period about 30 (1+a 2) 1/2,t orb, with a the vertical-to-radial aspect ratio and t orb the orbital period, and an amplitude scaling about a squared before saturation at a 5. Both trends agree with zero-net-flux unstratified shearing box simulations with Athena++. A carrier-envelope analysis of the simulation spectra shows that the same interference mechanism extends beyond strict QLT, through higher-order linear combinations of the eigenfrequencies, with observed cycles arising from pairwise beats within this spectral network. These results identify coherent interference between nearly degenerate eigenfrequencies as a key mechanism behind large-scale cyclic dynamos, with implications for magnetic variability in protoplanetary disks, X-ray binaries, and AGNs.