Advective accretion disks and related problems including magnetic fields

Accretion disk theory was first developed as a theory with the local heat balance, where the whole energy produced by a viscous heating was emitted to the sides of the disk. One of the most important new invention of this theory was the phenomenological treatment of the turbulent viscosity , known the " alpha " prescription, where the (rφ) component of the stress tensor was approximated by (αp) with a unknown constant α. This prescription played the role in the accretion disk theory as well important as the mixing-length theory of convection for stellar evolution. Sources of turbulence in the accretion disk are discussed, including hydrodynamical turbulence, convection and magnetic field role. In parallel to the optically thick geometrically thin accretion disk models, a new branch of the optically thin accretion disk models was discovered, with a larger thickness for the same total luminosity. The choice between these solutions should be done of the base of a stability analysis. The ideas underlying the necessity to include advection into the accretion disk theory are presented and first models with advection are reviewed. The present status of the solution for a low-luminous optically thin accretion disk model with advection is discussed and the limits for an advection dominated accretion flows (ADAF) imposed by the presence of magnetic field are analysed. Related problems of mass ejection from accretion disks and jet formation are discussed.