Instability of warped discs

Accretion discs are generally warped. If a warp in a disc is too large, the disc can `break' apart into two or more distinct planes, with only tenuous connections between them. Further if an initially planar disc is subject to a strong differential precession, then it can be torn apart into discrete annuli that precess effectively independently. In previous investigations, torque-balance formulae have been used to predict where and when the disc breaks into distinct parts. In this work, focusing on discs with Keplerian rotation and where the shearing motions driving the radial communication of the warp are damped locally by turbulence (the `diffusive' regime), we investigate the stability of warped discs to determine the precise criterion for an isolated warped disc to break. We find and solve the dispersion relation, which in general yields three roots. We provide a comprehensive analysis of this viscous-warp instability and the emergent growth rates and their dependence on disc parameters. The physics of the instability can be understood as a combination of (1) a term which would generally encapsulate the classical Lightman-Eardley instability in planar discs (given by $\partial(\nu\Sigma)/\partial\Sigma < 0$) but is here modified by the warp to include $\partial(\nu_1|\psi|)/\partial|\psi| < 0$ and (2) a similar condition acting on the diffusion of the warp amplitude given in simplified form by $\partial(\nu_2|\psi|)/\partial|\psi| < 0$. We discuss our findings in the context of discs with an imposed precession, and comment on the implications for different astrophysical systems.