2007MNRAS.381.1287L

In this paper we explore numerically the evolution of a warped accretion disc. While previous analyses have concentrated on the case where the disc is thick enough that the warp propagates as a wave, we focus here on the opposite regime of a thin disc, where the warp evolves diffusively. By comparing the numerical results to a simple diffusion model, we are able to determine the diffusion coefficient of the warp, α₂, as a function of the relevant disc parameters, such as its thickness and especially its viscosity. We find that while in general the disc behaviour is well reproduced by the diffusion model and for relatively large viscosities the warp diffusion is well described by the linear theory (in particular confirming that the warp diffusion coefficient is inversely proportional to viscosity), significant non-linear effects are present as the viscosity becomes smaller, but still dominates over wave-propagation effects. In particular, we find that the inverse dependence of the diffusion coefficient on viscosity breaks down at low viscosities, so that α₂ never becomes larger than a saturation value α_max of the order of unity. This can have major consequences in the evolution of systems where a warped disc is present. In particular, it affects the location of the warp radius in the Bardeen-Petterson effect and therefore the spin-up (or spin-down) of supermassive black holes in the nuclei of galaxies. Additionally, we also find that while the rate of warp diffusion does not depend significantly on the detailed viscosity formulation, the rate of internal precession generated by the warp is strongly affected by it. Such effects should be considered with care when modelling the evolution of warped discs. This emphasizes the need to test the above results using different numerical schemes, and with higher resolution, in order to investigate the degree to which numerical simulations are able to provide accurate modelling of the complex fluid dynamics of warped discs.