Effects of magnetohydrodynamic accretion flows on global structure of a Kerr black-hole magnetosphere

In relation to black-hole models for central engines in active galactic nuclei, we discuss some effects of accreting matter on the global structure of the Kerr black-hole magnetosphere. Based on the steady axisymmetric magnetohydrodynamic (MHD) equations in Kerr geometry, we analytically study the general-relativistic Grad-Shafranov equation and the poloidal wind equation. The field line geometry is clearly given in the region where the plasma can be magnetically supported. Near the black hole, the magnetic support breaks down and the plasma begins to flow toward the event horizon. We consider the magnetohydrodynamic conditions for the plasma inflows which should pass through the Alfv\'en critical point. Then it is found that the accumulation of magnetic field lines threading the horizon is suppressed for the rapidly rotating holes. This result means that the global shape of the field lines and the MHD energy extraction from the rotating black hole can drastically change as the spin down of the black hole proceeds. We present an evolutionary model for the activity and the structure of the black-hole magnetosphere.