The structure and evolution of protoplanetary disks are primarily governed by disk angular momentum transport via magnetic ?eld. Previous magnetohydrodynamical simulations of protoplanetary disks have shown that the strength of the net vertical magnetic field (the large-scale magnetic field threading the disk) determines the efficiency of disk accretion. However, how the distribution of the net vertical magnetic fields in the disks evolves is highly uncertain both observationally and theoretically.
The goal of this study is to study the coupled evolution of the mass and net vertical magnetic field in protoplanetary disks. We construct a one-dimensional disk model that treats the radial advection and diffusion of the gas surface density and net vertical magnetic field simultaneously. We consider accretion by angular momentum transport driven by magnetic winds. Based on the results of previous non-ideal magnetohydrodynamical simulations, the mass accretion rate is given as a function of magnetic field strength and distance from the central star.
We find that when magnetic diffusion outside the disk is inefficient and net flux inside the disk is conserved, the mass accretion rate increases with time and disk dispersal can occur on a timescale comparable to observations. We also find that the late stage of the disk evolution is self-similar, with the surface density and net field strength approaching a power law of orbital radius and time. We analytically derive a self-similar solution that describes the late-stage accretion. Our results suggest that magnetically driven accretion can explain the observed lifetimes of protoplanetary disks if the disks lose vertical magnetic flux slowly. Still, the stellar accretion rate predicted from our model is more than an order of magnitude smaller than the observed stellar accretion rate. This indicates that mechanisms other than magnetic disk winds are also needed to account for disk accretion around young stars.