  We study the 2-D, time-dependent hydrodynamics of radiation-driven winds from
luminous accretion disks threaded by a strong, large-scale, ordered magnetic
field. The radiation force is due to spectral lines and is calculated using a
generalized multidimensional formulation of the Sobolev approximation. The
effects of the magnetic field are approximated by adding a force that emulates
a magnetocentrifugal force. Our approach allows us to calculate disk winds when
the magnetic field controls the flow geometry, forces the flow to corotate with
the disk, or both. In particular, we calculate models where the lines of the
poloidal component of the field are straight and inclined to the disk at a
fixed angle. Our numerical calculations show that flows which corotate with the
disk have a larger mass loss rate than their counterparts which conserve
specific angular momentum. The difference in the mass loss rate between these
two types of winds can be several orders of magnitude for low disk luminosities
but vanishes for high disk luminosities. Winds which corotate with the disk
have much higher velocities than angular momentum conserving winds. Fixing the
wind geometry stabilizes winds which are unsteady when the geometry is derived
self-consistently. The inclination angle between the poloidal velocity and the
normal to the disk midplane is important. Non-zero inclination angles allow the
magnetocentrifugal force to increase the mass loss rate for low luminosities,
and increase the wind velocity for all luminosities. Our calculations also show
that the radiation force can launch winds from magnetized disks. The line force
can be essential in producing MHD winds from disks where the thermal energy is
too low to launch winds or where the field lines make an angle of < 30^o with
respect to the normal to the disk.
