  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
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 initial state in our simulations,
are presented in great detail. The angular momentum of the discs is assumed to
increase outwards with the radial distance according to a power law. The main
simplifying assumptions of our approach are not to include magnetic fields and
self-gravity in the discs. Furthermore, the dynamics of the spacetime is
accounted for by computing the transfer of mass and angular momentum from the
disc to the black hole through the event horizon : the evolution of the central
black hole is assumed to follow a sequence of Kerr black holes of increasing
mass and spin. 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 timescale. The
comparison with the results of paper I shows that the effect of the angular
momentum transfer from the torus to the black hole is to make constant angular
momentum discs less unstable, increasing the timescale of the instability.
However, we find that non-constant angular momentum discs are dramatically
stabilized for very small values of the angular momentum slope. Our
time-dependent simulations confirm, thus, the predictions of stationary studies
concerning the stabilizing effect of non-constant angular momentum
distributions.
