  We present the results of global 3-D MHD simulations of stratified and
turbulent protoplanetary disc models. The aim of this work is to develop thin
disc models capable of sustaining turbulence for long run times, which can be
used for on-going studies of planet formation in turbulent discs. The results
are obtained using two codes written in spherical coordinates: GLOBAL and
NIRVANA. Both are time--explicit and use finite differences along with the
Constrained Transport algorithm to evolve the equations of MHD. In the presence
of a weak toroidal magnetic field, a thin protoplanetary disc in hydrostatic
equilibrium is destabilised by the magnetorotational instability (MRI). When
the resolution is large enough (25 vertical grid cells per scale height), the
entire disc settles into a turbulent quasi steady-state after about 300 orbits.
Angular momentum is transported outward such that the standard alpha parameter
is roughly 4-6*10^{-3}. We find that the initial toroidal flux is expelled from
the disc midplane and that the disc behaves essentially as a quasi-zero net
flux disc for the remainder of the simulation. As in previous studies, the disc
develops a dual structure composed of an MRI--driven turbulent core around its
midplane, and a magnetised corona stable to the MRI near its surface. By
varying disc parameters and boundary conditions, we show that these basic
properties of the models are robust. The high resolution disc models we present
in this paper achieve a quasi--steady state and sustain turbulence for hundreds
of orbits. As such, they are ideally suited to the study of outstanding
problems in planet formation such as disc--planet interactions and dust
dynamics.
