  We investigate accretion onto a central star, with the size, rotation rate,
and magnetic dipole of a young stellar object, to study the flow pattern
(velocity and density) of the fluid within and outside of the disc. We perform
resistive MHD simulations of thin $\alpha$-discs, varying the parameters such
as the stellar rotation rate and magnetic field, and (anomalous) coefficients
of viscosity and resistivity in the disc. To provide a benchmark for the
results and to compare with known analytic results, we also perform purely
hydrodynamic simulations (HD) for the same problem. Although obtained for a
different situation with differing inner boundary condition, the disc structure
in the HD simulations closely follows the analytic solution of Klu\'zniak and
Kita (2000) -- in particular a region of "midplane" backflow exists in the
right range of radii, depending on the viscosity parameter. In the MHD
solutions, whenever the magnetic Prandtl number does not exceed a certain
critical value, the midplane backflow exists throughout the accretion disc,
extending all the way down to the inner transition zone where the disc
transitions to a magnetic funnel flow. For values of the magnetic Prandtl
number close to the critical value the backflow and the inner disc undergo a
quasiperiodic radial oscillation, otherwise the backflow is steady, as is the
disc solution. From our results, supplemented by our reading of the literature,
we conclude that midplane backflow is a real feature of at least some accretion
discs, whether HD $\alpha$-discs or MHD discs, including ones driven by MRI
turbulence.
