Magnetocentrifugally Driven Winds: Comparison of MHD Simulations with Theory

Stationary MHD outflows from a rotating accretion disk are investigated numerically by time-dependent axisymmetric simulations. The initial magnetic field is taken to be a split-monopole poloidal field configuration frozen into the disk. The disk is treated as a perfectly conducting, time-independent density boundary [ρ(r)] in Keplerian rotation. The outflow velocity from this surface is not specified but rather is determined self-consistently from the MHD equations. The temperature of the matter outflowing from the disk is small in the region where the magnetic field is inclined away from the symmetry axis (c≪v) but relatively high (c ≲ v) at very small radii in the disk, where the magnetic field is not inclined away from the axis. We have found a large class of stationary MHD winds. Within the simulation region, the outflow accelerates from thermal velocity ( ~c_s) to a much larger asymptotic poloidal flow velocity of the order of ½, where M is the mass of the central object and r_i is the inner radius of the disk. This asymptotic velocity is much larger than the local escape speed and is larger than fast magnetosonic speed by a factor of ~1.75. The acceleration distance for the outflow, over which the flow accelerates from ~0% to, say, 90% of the asymptotic speed, occurs at a flow distance of about 80r_i. The outflows are approximately spherical, with only small collimation within the simulation region. The collimation distance over which the flow becomes collimated (with divergence less than, say, 10⁰) is much larger than the size of our simulation region. Close to the disk the outflow is driven by the centrifugal force, while at all larger distances the flow is driven by the magnetic force, which is proportional to -▽(rB_ϕ)², where B_ϕ is the toroidal field. Our stationary numerical solutions allow us to (1) compare the results with MHD theory of stationary flows, (2) investigate the influence of different outer boundary conditions on the flows, and (3) investigate the influence of the shape of the simulation region on the flows. Different comparisons were made with the theory. The ideal MHD integrals of motion (constants on flux surfaces) were calculated along magnetic field lines and were shown to be constants with an accuracy of 5%-15%. Other characteristics of the numerical solutions were compared with the theory, including conditions at the Alfvén surface. Different outer boundary conditions on the toroidal component of the magnetic field were investigated. We conclude that the commonly used "free" boundary condition on the toroidal field leads to artificial magnetic forces on the outer boundaries, which can significantly influence to the calculated flows. New outer boundary conditions are proposed and investigated that do not give artificial forces. We show that simulated flows may depend on the shape of the simulation region. Namely, if the simulation region is elongated in the z-direction, then Mach cones on the outer cylindrical boundary may be partially directed inside the simulation region. Because of this, the boundary can have an artificial influence on the calculated flow. This effect is reduced if the computational region is approximately square or if it is spherical. Simulations of MHD outflows with an elongated computational region can lead to artificial collimation of the flow.