The global structure of a self-excited magnetic field arising from the magn
etic shear instability has been simulated in spherical geometry by a 3D ful
ly non-linear approach. In order to model the structure of an accretion dis
c we prescribe a rotation profile of the Brandt type which is Keplerian in
the outer regions but yields rigid rotation at the inner core. We performed
a whole series of runs at different dynamo numbers with an increasing numb
er of modes in spectral space, thereby checking the influence of the numeri
cal resolution in our simulations. Starting from arbitrary small perturbati
ons, the magnetic and kinetic energies grow by several orders of magnitude
as soon as a certain azimuthal resolution of at least m=15 was used at a dy
namo number of order C-Omega=10(5). Several phases of the transition to tur
bulence are realized and interpretations are given for the respective effec
ts occurring at each stage. The resulting magnetic field is highly non-axis
ymmetric and possesses a pronounced inhomogeneous vortex structure of twist
ed flux tubes. The flow is almost axisymmetric but shows a Kolmogorov-like
behaviour for small scales. The outer surface of the shell is penetrated by
magnetic field lines in spot-like regions, which are located mainly in the
equatorial plane. For very high dynamo numbers we find a cyclic behaviour
of the averaged magnetic field amplitude. The problem of angular momentum t
ransport is discussed in terms of the Shakura-Sunyaev viscosity alpha, whic
h depends exponentially on the radial distance and adopts values in the ran
ge 10(-3)-10(-5).