  Rings and gaps are commonly observed in the dust continuum emission of young
stellar disks. Previous studies have shown that substructures naturally develop
in the weakly ionized gas of magnetized, non-ideal MHD disks. The gas rings are
expected to trap large mm/cm-sized grains through pressure gradient-induced
radial dust-gas drift. Using 2D (axisymmetric) MHD simulations that include
ambipolar diffusion and dust grains of three representative sizes (1~mm,
3.3~mm, and 1~cm), we show that the grains indeed tend to drift radially
relative to the gas towards the centers of the gas rings, at speeds much higher
than in a smooth disk because of steeper pressure gradients. However, their
spatial distribution is primarily controlled by meridional gas motions, which
are typically much faster than the dust-gas drift. In particular, the grains
that have settled near the midplane are carried rapidly inwards by a fast
accretion stream to the inner edges of the gas rings, where they are lifted up
by the gas flows diverted away from the midplane by a strong poloidal magnetic
field. The flow pattern in our simulation provides an attractive explanation
for the meridional flows recently inferred in HD 163296 and other disks,
including both "collapsing" regions where the gas near the disk surface
converges towards the midplane and a disk wind. Our study highlights the
prevalence of the potentially observable meridional flows associated with the
gas substructure formation in non-ideal MHD disks and their crucial role in
generating rings and gaps in dust.
