  The planet-forming region of protoplanetary disks is cold, dense, and
therefore weakly ionized. For this reason, magnetohydrodynamic (MHD) turbulence
is thought to be mostly absent, and another mechanism has to be found to
explain gas accretion. It has been proposed that magnetized winds, launched
from the ionized disk surface, could drive accretion in the presence of a
large-scale magnetic field. The efficiency and the impact of these surface
winds on the disk structure is still highly uncertain. We present the first
global simulations of a weakly ionized disk that exhibits large-scale
magnetized winds. We also study the impact of self-organization, which was
previously demonstrated only in non-stratified models. We perform numerical
simulations of stratified disks with the PLUTO code. We compute the ionization
fraction dynamically, and account for all three non-ideal MHD effects: ohmic
and ambipolar diffusions, and the Hall drift. Simplified heating and cooling
due to non-thermal radiation is also taken into account in the disk atmosphere.
We find that disks can be accreting or not, depending on the configuration of
the large-scale magnetic field. Magnetothermal winds, driven both by magnetic
acceleration and heating of the atmosphere, are obtained in the accreting case.
In some cases, these winds are asymmetric, ejecting predominantly on one side
of the disk. The wind mass loss rate depends primarily on the average ratio of
magnetic to thermal pressure in the disk midplane. The non-accreting case is
characterized by a meridional circulation, with accretion layers at the disk
surface and decretion in the midplane. Finally, we observe self-organization,
resulting in axisymmetric rings of density and associated pressure "bumps". The
underlying mechanism and its impact on observable structures are discussed.
