  The strength and structure of the large-scale magnetic field in
protoplanetary discs are still unknown, although they could have important
consequences for the dynamics and evolution of the disc. Using a mean-field
approach in which we model the effects of turbulence through enhanced diffusion
coefficients, we study the time-evolution of the large-scale poloidal magnetic
field in a global model of a thin accretion disc, with particular attention to
protoplanetary discs. With the transport coefficients usually assumed, the
magnetic field strength does not significantly increase radially inwards,
leading to a relatively weak magnetic field in the inner part of the disc. We
show that with more realistic transport coefficients that take into account the
vertical structure of the disc and the back-reaction of the magnetic field on
the flow as obtained by Guilet & Ogilvie (2012), the magnetic field can
significantly increase radially inwards. The magnetic-field profile adjusts to
reach an equilibrium value of the plasma $\beta$ parameter (the ratio of
midplane thermal pressure to magnetic pressure) in the inner part of the disc.
This value of $\beta$ depends strongly on the aspect ratio of the disc and on
the turbulent magnetic Prandtl number, and lies in the range $10^4-10^7$ for
protoplanetary discs. Such a magnetic field is expected to affect significantly
the dynamics of protoplanetary discs by increasing the strength of MHD
turbulence and launching an outflow. We discuss the implications of our results
for the evolution of protoplanetary discs and for the formation of powerful
jets as observed in T-Tauri star systems.
