On the Grain-Modified Magnetic Diffusivities in Protoplanetary Disks

TL;DR

This paper analytically and numerically investigates how small grains in protoplanetary disks influence magnetic diffusivities, revealing sign changes in Hall effect and reduced ambipolar diffusion that impact disk dynamics and stability.

Contribution

It provides new analytical expressions for magnetic diffusivities considering grain effects and demonstrates their impact on disk magnetic behavior through simulations.

Findings

Hall diffusivity can change sign at a threshold magnetic field.

Small grains significantly reduce ambipolar diffusion in outer disk regions.

Magnetic field amplification saturates at low levels with abundant small grains.

Abstract

Weakly ionized protoplanetary disks (PPDs) are subject to non-ideal-magnetohydrodynamic (MHD) effects including Ohmic resistivity, the Hall effect and ambipolar diffusion (AD), and the resulting magnetic diffusivities ($\eta_O, \eta_H$ and $\eta_A$) largely control the disk gas dynamics. The presence of grains not only strongly reduces disk ionization fraction, but also modify the scalings of $\eta_H$ and $\eta_A$ with magnetic field strength. We derive analytically asymptotic expressions of $\eta_H$ and $\eta_A$ in both strong and weak field limits and show that towards strong field, $\eta_H$ can change sign (at a threshold field strength $B_{\rm th}$), mimicking a flip of field polarity, and AD is substantially reduced. Applying to PPDs, we find that when small $\sim0.1$ ($0.01$)$\mu$m grains are sufficiently abundant [mass ratio $\sim0.01$ ($10^{-4}$)], $\eta_H$ can change sign up to $\sim2-3$ scale heights above midplane at modest field strength (plasma $\beta\sim100$) over a wide range of disk radii. Reduction of AD is also substantial towards the AD dominated outer disk and may activate the magneto-rotational instability. We further perform local non-ideal MHD simulations of the inner disk (within 10 AU) and show that with sufficiently abundant small grains, magnetic field amplification due to the Hall-shear instability saturates at very low level near the threshold field strength $B_{\rm th}$. Together with previous studies, we conclude by discussing the grain-abundance-dependent phenomenology of PPD gas dynamics.