An effective model for magnetic field amplification by the magnetorotational and parasitic instabilities

The magnetorotational instability (MRI) is considered a leading mechanism for driving angular momentum transport in differentially rotating astrophysical flows, including accretion disks and protoneutron stars. This process is mediated by the exponential amplification of the magnetic field whose final amplitude is envisioned to be limited by secondary (parasitic) instabilities. In this paper, we investigated the saturation of the MRI via parasitic modes relaxing previous approximations. We carried out the first systematic analysis of the evolution of parasitic modes as they feed off the exponentially growing MRI while being advected by the background shear flow. We provide the most accurate calculation of the amplification factor to which the MRI can grow before the fastest parasitic modes reach a comparable amplitude. We find that this amplification factor is remarkably robust, depending only logarithmically on the initial amplitude of the parasitic modes, in reasonable agreement with numerical simulations. Based on these insights, and guided by numerical simulations, we provide a simple analytical expression for the amplification of magnetic fields responsible for MRI-driven angular momentum transport. Our effective model for magnetic field amplification may enable going beyond the standard prescription for viscous transport currently employed in numerical simulations when the MRI cannot be explicitly resolved.