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On the Grain-Modified Magnetic Diffusivities in Protoplanetary Disks

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 ($η_O, η_H$ and $η_A$) largely control the disk gas dynamics. The presence of grains not only strongly reduces disk ionization fraction, but also modify the scalings of $η_H$ and $η_A$ with magnetic field strength. We derive analytically asymptotic expressions of $η_H$ and $η_A$ in both strong and weak field limits and show that towards strong field, $η_H$ 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 $\sim0.1$ ($0.01$)$μ$m grains are sufficiently abundant [mass ratio $\sim0.01$ ($10^{-4}$)], $η_H$ can change sign up to $\sim2-3$ scale heights above midplane at modest field strength (plasma $β\sim100$) 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.