  This paper is devoted to self-consistent modeling of the magnetically
supported accretion disk with optically thick warm corona based on first
principles. In our model, we consider the gas heating by magneto-rotational
instability (MRI) dynamo. Our goal is to show that the proper calculation of
the gas heating by magnetic dynamo can build up the warm, optically thick
corona above the accretion disk around black hole of stellar mass. Using
vertical model of the disk supported and heated by the magnetic field together
with radiative transfer in hydrostatic and radiative equilibrium we developed
relaxation numerical scheme which allows us to compute the transition form the
disk to corona in a self consistent way. We demonstrate here that the warm (up
to 5 keV), optically thick (up to 10 Thompson optical depths), Compton cooled
corona can form due to the magnetic heating. Such warm corona is stronger for
higher accretion rate and larger magnetic field strength. The radial extent of
the warm corona is limited by the occurrence of the local thermal instability,
which purely depends on radiative processes. The obtained coronal parameters
are in agreement with those constrained from X-ray observations. The warm
magnetically supported corona is tends to appear in the inner disk regions. It
may be responsible for Soft X-ray excess seen in accreting sources. For lower
accretion rates and weaker magnetic field parameters, thermal instability
prevents warm corona to exist, giving rise to eventual clumpiness or ionized
outflow.
