General features of the linear crystalline morphology of accretion disks

Abstract: In this paper, we analyze the so-called Master Equation of the linear backreaction of a plasma disk in the central object magnetic field, when small scale ripples are considered. This study allows to single out two relevant physical properties of the linear disk backreaction: (i) the appearance of a vertical growth of the magnetic flux perturbations; (ii) the emergence of sequence of magnetic field O-points, crucial for the triggering of local plasma instabilities. We first analyze a general Fourier approach to the solution of the addressed linear partial differential problem. This technique allows to show how the vertical gradient of the backreaction is, in general, inverted with respect to the background one. Instead, the fundamental harmonic solution constitutes a specific exception for which the background and the perturbed profiles are both decaying. Then, we study the linear partial differential system from the point of view of a general variable separation method. The obtained profile describes the crystalline behavior of the disk. Using a simple rescaling, the governing equation is reduced to the second-order differential Whittaker equation. The zeros of the radial magnetic field are found by using the solution written in terms Kummer functions. The possible implications of the obtained morphology of the disk magnetic profile are then discussed in view of the jet formation. GraphicAbstract: [Figure not available: see fulltext.]