The runaway instability of thick discs around black holes - II. Non-constant angular momentum discs

We present results from a comprehensive number of relativistic, time-dependent, axisymmetric simulations of the runaway instability of non-constant angular momentum thick discs around black holes. This second paper in the series extends earlier results where only constant angular momentum discs were considered. All relevant aspects of the theory of stationary thick discs around rotating black holes, necessary to build the equilibrium initial data used in our simulations, are presented in great detail. The angular momentum of the evolved discs is assumed to increase outwards with the radial distance according to a power law, l = Kr-alpha, where K > 0 corresponds to prograde discs (with respect to the black hole rotation) and K < 0 to retro-grade discs. The main simplifying assumptions of our approach are not to include magnetic fields or self-gravity in the discs (test-fluid approximation). Furthermore, the dynamics of the space-time is accounted for by computing the transfer of mass and angular momentum from the disc to the black hole through the event horizon. In this approximation the evolution of the central black hole, which initially is non-rotating, is assumed to follow a sequence of Kerr black holes of increasing mass and spin. All discs we build slightly overflow the potential barrier at the cusp, departing from equilibrium, so that accretion is possible. In agreement with previous results based on stationary models we find that by allowing the mass and the spin of the black hole to grow, constant angular momentum discs rapidly become unstable on a dynamical time-scale (a few orbital periods). The comparison with the results of our first paper shows that the effect of the angular momentum transfer from the tori to the black hole is to make constant angular momentum discs less unstable, increasing the time-scale for the runaway instability to grow. However, we find that non-constant angular momentum discs are dramatically stabilized for very small values of the angular momentum slope alpha, much smaller than the Keplerian value alpha = 1/2. Our fully relativistic and time-dependent simulations thus confirm the predictions of stationary studies concerning the stabilizing effect of non-constant angular momentum distributions. For the various disc-to-hole mass ratios considered, we systematically find that the critical values of a below which the runaway instability can exist are slightly smaller than those reported previously in the literature based on stationary studies.