Hydrodynamic simulations of propagating warps and bending waves in accretion discs

We present the results of a study of propagating warp or bending waves in accretion discs. Three-dimensional hydrodynamic simulations were performed using smoothed particle hydrodynamics (SPH), and the results are compared with calculations based on the linear theory of warped discs. We examine the response of a gaseous disc to an initially imposed warping disturbance under a variety of physical conditions. We consider primarily the physical regime in which the dimensionless viscosity parameter alpha < H/r, where H/r is the disc aspect ratio, so that bending waves are expected to propagate. We also performed calculations for disc models in which alpha > H/r, where the warps are expected to evolve diffusively. Small-amplitude (linear) perturbations are studied in both Keplerian and slightly non-Keplerian discs, and we find that the results of the SPH calculations can be reasonably well fitted by those of the linear theory. The main results of these calculations are: (i) the warp in Keplerian discs when alpha < H/r propagates with little dispersion, and damps at a rate expected from estimates of the code viscosity; (ii) warps evolve diffusively when alpha > H/r; (iii) the slightly non-Keplerian discs lead to a substantially more dispersive behaviour of the warps, which damp at a similar rate to the Keplerian case, when alpha < H/r. Initially imposed higher amplitude, non-linear warping disturbances were studied in Keplerian discs. The results indicate that non-linear warps can lead to the formation of shocks, and that the evolution of the warp becomes less wave-like and more diffusive in character. This work is relevant to the study of the warped accretion discs that may occur around Kerr black holes or in misaligned binary systems, and is mainly concerned with discs in which alpha < H/r. The results indicate that SPH can model the hydrodynamics of warped discs, even when using rather modest numbers of particles.