Abstract

This paper considers resonantly forced spiral density waves in a dense planetary ring that is close to the threshold for viscous overstability. We solved numerically the hydrodynamical equations for a dense thin disk in the vicinity of an inner Lindblad resonance with a perturbing satellite. Our numerical scheme is one-dimensional so that the spiral shape of a density wave is taken into account through a suitable approximation of the advective terms arising from the fluid orbital motion. This paper is a first attempt to model the co-existence of resonantly forced density waves and short-scale free overstable wavetrains as observed in Saturn’s rings, by conducting large-scale hydrodynamical integrations. These integrations reveal that the two wave types undergo complex interactions, not taken into account in existing models for the damping of density waves. In particular we found that, depending on the relative magnitude of both wave types, the presence of viscous overstability can lead to the damping of an unstable density wave and vice versa. The damping of the short-scale viscous overstability by a density wave was investigated further by employing a simplified model of an axisymmetric ring perturbed by a nearby Lindblad resonance. A linear hydrodynamic stability analysis as well as local N-body simulations of this model system were performed and support the results of our large-scale hydrodynamical integrations.