  We study axisymmetric models of layered protoplanetary discs taking radiative
transfer effects into account, and allowing for a residual viscosity in the
dead zone. We also explore the effect of different viscosity prescriptions. In
addition to the ring instability reported in the first paper of the series we
find an oscillatory instability of the dead zone, accompanied by variations of
the accretion rate onto the central star. We provide a simplified analytical
description explaining the mechanism of the oscillations. Finally, we find that
the residual viscosity enables stationary accretion in large regions of layered
discs. Based on results obtained with the help of a simple 1-D hydrocode we
identify these regions, and discuss conditions in which layered discs can give
rise to FU~Orionis phenomena.
