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

TL;DR

This study introduces a new simulation scheme to accurately model magnetically driven accretion in protoplanetary disks, establishing scalable relations between accretion rates and disk physical parameters.

Contribution

The paper develops the super-box-scale diffusion scheme for shearing-box simulations, enabling systematic analysis of wind-driven accretion and deriving predictive power-law relations.

Findings

The SBD scheme maintains magnetic field symmetry over 500 orbits.

Accretion rate scales with plasma beta, Elsasser number, and active layer thickness.

Derived relations predict accretion rates within a factor of 2-3 across parameter space.

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.