  Jetted astrophysical phenomena with black hole (BH) engines, including binary
mergers, jetted tidal disruption events, and X-ray binaries, require a
large-scale vertical magnetic field for efficient jet formation. However, a
dynamo mechanism that could generate these crucial large-scale magnetic fields
has not been identified and characterized. We have employed 3D global general
relativistic magnetohydrodynamical (MHD) simulations of accretion disks to
quantify, for the first time, a dynamo mechanism that generates large-scale
magnetic fields. This dynamo mechanism primarily arises from the nonlinear
evolution of the magnetorotational instability (MRI). In this mechanism, large
non-axisymmetric MRI-amplified shearing wave modes, mediated by the
axisymmetric azimuthal magnetic field, generate and sustain the large-scale
vertical magnetic field through their nonlinear interactions. We identify the
advection of magnetic loops as a crucial feature, transporting the large-scale
vertical magnetic field from the outer regions to the inner regions of the
accretion disk. This leads to a larger characteristic size of the, now
advected, magnetic field when compared to the local disk height. We
characterize the complete dynamo mechanism with two timescales: one for the
local magnetic field generation, $t_{\rm g}$, and one for the large-scale scale
advection, $t_{\rm adv}$. Whereas the dynamo we describe is nonlinear, we
explore the potential of linear mean field models to replicate its core
features. Our findings indicate that traditional $\alpha$-dynamo models, often
computed in stratified shearing box simulations, are inadequate and that the
effective large-scale dynamics is better described by the shear current effects
or stochastic $\alpha$-dynamos.
