  Many accretion discs are thought to be warped. Recent hydrodynamical
simulations show that (i) discs can break into distinct planes when the
amplitude of an imposed warp is sufficiently high and the viscosity
sufficiently low, and that (ii) discs can tear up into discrete rings when an
initially planar disc is subject to a forced precession. Previously, we
investigated the local stability of isolated, Keplerian, warped discs in order
to understand the physics causing an accretion disc to break into distinct
planes, finding that anti-diffusion of the warp amplitude is the underlying
cause. Here, we explore the behaviour of this instability in disc regions where
the rotation profile deviates from Keplerian. We find that at small warp
amplitudes non-Keplerian rotation can stabilize the disc by increasing the
critical warp amplitude for instability, while at large warp amplitudes
non-Keplerian rotation can lead to an increased growth rate for discs that are
unstable. Tidal effects on discs in binary systems are typically weak enough
such that the disc remains close to Keplerian rotation. However, the inner
regions of discs around black holes are strongly affected, with the smallest
radius at which the disc can break into discrete planes being a function of the
black hole spin. We suggest that interpreting observed frequencies in the power
spectra of light curves from accreting compact objects as nodal and apsidal
precession of discrete orbits requires an instability that can break the disc
into discrete rings such as the one explored here.
