Spiral-driven accretion in protoplanetary discs: II. Self-similar solutions

Context. Accretion discs are ubiquitous in the Universe, and it is crucial to understand how angular momentum and mass are radially transported in these objects. Aims. Here, we study the role played by non-linear spiral patterns within hydrodynamical and non-self-gravitating accretion discs assuming that external disturbances such as infall onto the disc may trigger them. Methods. To do so, we computed self-similar solutions that describe discs in which a spiral wave propagates. These solutions present shocks and critical sonic points that were analyzed. Results. We calculated the wave structure for all allowed temperatures and for several spiral shocks. In particular, we inferred the angle of the spiral pattern, the stress it exerts on the disc, and the associated flux of mass and angular momentum as a function of temperature. We quantified the rate of angular momentum transport by means of the dimensionless α parameter. For the thickest disc we considered (corresponding to h/r values of about one-third), we found values of α as high as 0.1 that scaled with the temperature T such that α ∝ T^3/2 ∝ (h/r)³. The spiral angle scales with the temperature as arctan(r/h). Conclusions. These solutions suggests that perturbations occurring at disc outer boundaries, such as perturbations due to infall motions, can propagate deep inside the disc and therefore should not be ignored, even when considering small radii.