Formation of spirals in early-stage protoplanetary discs

Class II protoplanetary discs feature numerous non-axisymmetric substructures such as spirals, and the underlying mechanisms for their formation are still highly debated. Coincidentally, early-stage massive discs are subject to gravitational instability, which causes them to collapse into denser substructures. However, like for most instabilities, real systems usually remain marginally stable, here with a Toomre parameter Q ≳ 1. We studied the growth of spiral structures in the disc triggered by the self-gravity of the gas. We specifically focused on discs that are considered stable, that is, with respect to the gravitational instability (with Q > 1), as these discs remain unstable to non-axisymmetric perturbations such as from spirals. After a linear stability analysis, we produced high-resolution 2D shearing sheet simulations with the GPU-accelerated code of self-gravitating discs. We probed different initial densities and thermodynamical models of Toomre-stable discs. The initial transient growth of the spiral wave matches the linear theory when the time dependence of the amplification is taken into account. The spirals are then rapidly non-linearly amplified with a growth rate of about ten orbital timescales. After this time, the large-scale spiral modes are amplified up to 1000 times more than predicted by linear theory. At later times, low-density discs reach a weak gravito-turbulent state with α≈ 10^ -3 , and discs with a higher density undergo runaway collapse of the spiral arms. All discs exhibit dominant large-scale spirals.