The Generalized Nonlinear Ohm's Law: How a Strong Electric Field Influences Non-ideal MHD Effects in Dusty Protoplanetary Disks

Abstract

The magnetohydrodynamics (MHD) of protoplanetary disks are strongly subject to the non-ideal MHD effects arising from the low ionization fraction of the disk gas. A strong electric field induced by gas motions can heat ionized gas particles and can thereby affect the ionization balance in the disks. Our previous studies revealed that in dusty protoplanetary disks, the Ohmic conductivity decreases with increasing electric field strength until the electrical breakdown of the disk gas occurs. In this study, we extend our previous work to more general cases where both electric and magnetic fields affect the motion of plasma particles, allowing us to study the impacts of plasma heating on all non-ideal MHD effects: Ohmic, Hall, and ambipolar diffusion. We find that the upper limit on the electric current we previously derived applies even in the presence of magnetic fields. Although the Hall and ambipolar resistivities can either increase or decrease with electric field strength depending on the abundance of charged dust grains, the Ohmic resistivity always increases with electric field strength. An order-of-magnitude estimate suggests that a large-scale electric current generated by gas motions in the inner part of protoplanetary disks could exceed the upper limit. This implies that MHD motions of the inner disk, such as the motion driven by the Hall-shear instability, could either get suppressed or trigger electrical breakdown (lightning discharge). This may have important implications for gas accretion and chondrule formation in the inner part of protoplanetary disks.