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 _ { th } ) , mimicking a flip of field polarity , and AD is substantially reduced .
Applying to PPDs , we find that when small \sim 0.1 ( 0.01 ) \mu m grains are sufficiently abundant [ mass ratio \sim 0.01 ( 10 ^ { -4 } ) ] , \eta _ { H } can change sign up to \sim 2 - 3 scale heights above midplane at modest field strength ( plasma \beta \sim 100 ) 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 _ { th } .
Together with previous studies , we conclude by discussing the grain-abundance-dependent phenomenology of PPD gas dynamics .