  Magnetic interactions between a protostar and its accretion disc tend to
induce warping in the disc and produce secular changes in the stellar spin
direction, so that the spin axis may not always be perpendicular to the disc.
This may help explain the recently observed spin-orbit misalignment in a number
of exoplanetary systems. We study the dynamics of warped protoplanetary discs
under the combined effects of magnetic warping/precession torques and internal
stresses in the disc, including viscous damping of warps and propagation of
bending waves. We show that when the outer disc axis is misaligned with the
stellar spin axis, the disc evolves towards a warped steady-state on a
timescale that depends on the disc viscosity or the bending wave propagation
speed, but in all cases is much shorter than the timescale for the spin
evolution (of order of a million years). Moreover, for the most likely physical
parameters characterizing magnetic protostars, circumstellar discs and their
interactions, the steady-state disc has a rather small warp, such that the
whole disc lies approximately in a single plane determined by the outer disc
boundary conditions, although more extreme parameters may give rise to larger
disc warps. In agreement with our recent analysis (Lai et al. 2010) based on
flat discs, we find that the back-reaction magnetic torques of the slightly
warped disc on the star can either align the stellar spin axis with the disc
axis or push it towards misalignment, depending on the parameters of the
star-disc system. This implies that newly formed planetary systems may have a
range of inclination angles between the stellar spin axis and the symmetry axis
of the planetary orbits.
