  In this paper, we develop the analysis of a two-dimensional
magnetohydrodynamical configuration for an axially symmetric and rotating
plasma (embedded in a dipole like magnetic field), modeling the structure of a
thin accretion disk around a compact astrophysical object. Our study
investigates the global profile of the disk plasma, in order to fix the
conditions for the existence of a crystalline morphology and ring sequence, as
outlined by the local analysis pursued in [1, 2]. In the linear regime, when
the electromagnetic back-reaction of the plasma is small enough, we show the
existence of an oscillating radial behavior for the flux surface function which
very closely resembles the one outlined in the local model, apart from a radial
modulation of the amplitude. In the opposite limit, corresponding to a dominant
back-reaction in the magnetic structure over the field of central object, we
can recognize the existence of a ring-like decomposition of the disk, according
to the same modulation of the magnetic flux surface, and a smoother radial
decay of the disk density, with respect to the linear case. In this extreme
non-linear regime, the global model seems to predict a configuration very close
to that of the local analysis, but here the thermostatic pressure, crucial for
the equilibrium setting, is also radially modulated. Among the conditions
requested for the validity of such a global model, the confinement of the
radial coordinate within a given value sensitive to the disk temperature and to
the mass of the central objet, stands; however, this condition corresponds to
dealing with a thin disk configuration.
