Active galactic nuclei are interesting astrophysical objects which contain a super-massive black hole at the central region as an engine of their activity. Plasma accretion onto the black hole is the fundamental process for producing the high and variable x-ray luminosity from the central region, and the magnetic field is expected to be a crucial component in the accretion models to explain various observed features. This motivates us to study general-relativistic effects in a black-hole magnetosphere, in particular, in relation to the important role of the spinning motion. The stationary and axisymmetric magnetospheric structure based on the general-relativistic magnetohydrodynamic treatment under the assumption of infinite conductivity is briefly reviewed. We then turn to analyse the time evolution of axisymmetric magnetic fields in the vicinity of the black hole, taking into account the contribution of a finite magnetic diffusivity . Our approach is to solve the general-relativistic Maxwell equations supplemented by the generalized Ohm's law for a given velocity field of accreting and rotating plasma. Numerical examples of time-dependent solutions which show self-excitation of dipolar and quadrupolar modes of the poloidal magnetic field were recently given by Khanna and Camenzind. However, the validity of the dynamo action remains controversial. Here, we develop the analytical perturbation method under the approximation of small diffusivity , where is the black-hole radius, and our analysis is limited to the narrow boundary region very close to the black-hole surface. The condition for growing multipolar seed fields is expressed by rotation parameters. We discuss our simple analytical results in comparison with the numerical calculations.
Astrophysical dynamos and magnetic helicity conservation