Beyond the $\alpha$ model: scaling the wind-driven accretion rate in protoplanetary disks using systematic non-ideal magnetohydrodynamical simulations

Abstract

Magnetically driven mass accretion in protoplanetary disks plays a crucial role in understanding disk evolution and planet formation. However, the $\alpha$ prescription lacks a direct connection to physical processes, and no systematic scaling law yet exists for the accretion rate as a function of disk quantities. While local shearing-box simulations offer a powerful approach to analyzing accretion structure at low computational cost, they suffer from a problem: the toroidal magnetic field generated by Keplerian shear accumulates within the computational domain, disrupting a geometry consistent with global wind-driven accretion. In this study, we introduce the super-box-scale diffusion (SBD) scheme into non-ideal MHD shearing-box simulations. The SBD scheme continuously damps the horizontally averaged horizontal magnetic field components, thereby mitigating this problem and maintaining the field-line symmetry required for global wind-driven accretion for more than 500 orbital periods. Comparison with self-similar solutions supports the SBD method, with the vertical structure and plasma-beta dependence of the accretion rate agreeing to within 23--28\%. We then conduct a parameter survey of 46 cases using a magnetic diffusivity table constructed from ionization equilibrium calculations, covering disk radius, surface density, magnetic field strength, and dust-to-gas ratio. We find that the surface field-line pitch and mass accretion rate follow power-law scalings with the midplane plasma beta, an effective ambipolar Elsasser number, and the normalized thickness of the magnetically active layer. These relations reproduce the numerical results to within a factor of 2--3 across the explored parameter space and, in most cases, to within a factor of 2. They provide a framework for predicting the mass accretion rate from local disk physical quantities without invoking an $\alpha$ parameter.