We investigate the influence of collective self-gravity forces on the nonlinear, large-scale evolution of the viscous overstability in Saturn's rings. We numerically solve the axisymmetric nonlinear hydrodynamic equations in the isothermal and non-isothermal approximation, including radial self-gravity and employing transport coefficients derived by Salo et al. (2001). We assume optical depths τ = 1.5 − 2 to model Saturn's dense rings. Furthermore, local N-body simulations, incorporating vertical and radial collective self-gravity are performed. Vertical self-gravity is mimicked through an increased frequency of vertical oscillations, while radial self-gravity is approximated by solving the Poisson equation for an axisymmetric thin disk with a Fourier method. Direct particle-particle forces are omitted, which prevents small-scale gravitational instabilities (self-gravity wakes) from forming, an approximation that allows us to study long radial scales and to compare directly the hydrodynamic model and the N-body simulations. Our isothermal and non-isothermal hydrodynamic model results with vanishing self-gravity compare very well with results of Latter and Ogilvie (2010) and Rein and Latter (2013), respectively. In contrast, for rings with radial self-gravity we find that the wavelengths of saturated overstable waves settle close to the frequency minimum of the nonlinear dispersion relation, i.e. close to a state of vanishing group velocities of the waves. Good agreement is found between non-isothermal hydrodynamics and N-body simulations for moderate and strong radial self-gravity, while the largest deviations occur for weak self-gravity. The resulting saturation wavelengths of viscous overstability for moderate and strong self-gravity (λ ∼ 100 − 300m) agree reasonably well with the length scales of axisymmetric periodic micro-structure in Saturn's inner A-ring and the B-ring, as found by Cassini.

This paper addresses resonantly forced spiral density waves in a dense planetary ring which is close to the threshold for viscous overstability. We solve 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 it is found that, depending on the relative magnitude of both wave types, the presence of viscous overstability can lead to a damping of an unstable density wave and vice versa. The damping of the short-scale viscous overstability by a density wave is 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 are performed and support the results of our large-scale hydrodynamical integrations.

We revisit the equation for viscous damping of density waves derived from linearized theory and show that the damping is not only determined by the magnitudes of shear and bulk viscosity. Modifications arise from the dependence of the viscosity on the ring’s surface mass density. This was noted more than 30 years ago by Goldreich & Tremaine (1978b). Still, to date the consequences have not been explored. In the literature these terms have been neglected throughout when fitting the rings’ viscosity from observations of wave damping. Therefore, one must suspect that these viscosities, as well as the dispersion velocities inferred from them, suffer from systematic bias, which might be small or significant, depending on the local conditions in the ring. We show that the modified damping formula, to linear order, is related to the stability threshold for viscous overstability and argue that the appearance of density waves may be altered by this instability.

In this paper we address the stability of resonantly forced density waves in dense planetary rings. Already by Goldreich and Tremaine (1978) it has been argued that density waves might be unstable, depending on the relationship between the ring's viscosity and the surface mass density. In the recent paper Schmidt et al. (2016) we have pointed out that when -within a fluid description of the ring dynamics -the criterion for viscous overstability is satisfied, forced spiral density waves become unstable as well. In this case, linear theory fails to describe the damping, but nonlinearity of the underlying equations guarantees a finite amplitude and eventually a damping of the wave. We apply the multiple scale formalism to derive a weakly nonlinear damping relation from a hydrodynamical model. This relation describes the resonant excitation and nonlinear viscous damping of spiral density waves in a vertically integrated fluid disk with density dependent transport coefficients. The model consistently predicts density waves to be (linearly) unstable in a ring region where the conditions for viscous overstability are met. Sufficiently far away from the Lindblad resonance, the surface mass density perturbation is predicted to saturate to a constant value due to nonlinear viscous damping. The wave's damping lengths of the model depend on certain input parameters, such as the distance to the threshold for viscous overstability in parameter space and the ground state surface mass density.

Protoplanetary discs (PPDs) can host a number of instabilities that may partake directly or indirectly in the process of planetesimal formation. These include the Vertical Shear Instability (VSI), Convective Overstability (COS), Streaming Instability (SI), and Dust Settling Instability (DSI), to name a few. Notably, the VSI and COS have mostly been studied in purely gaseous discs, while the SI and DSI have only been analyzed in isothermal discs. How these instabilities operate under more general conditions is therefore unclear. To this end, we devise a local model of a PPD describing a non-isothermal gas interacting with a single species of dust via drag forces. Using this, we find that dust drag sets minimum length scales below which the VSI and COS are suppressed. Similarly, we find that the SI can be suppressed on sufficiently small scales by the gas’ radial buoyancy if it cools on roughly a dynamical timescale. We show that the DSI can be effectively stabilized by vertical buoyancy, except at special radial and vertical length scales. We also find novel instabilities unique to a dusty, non-isothermal gas. These result in a dusty analog of the COS that operates in slowly cooled discs, and a dusty version of the VSI that is strongly enhanced by dust settling. We briefly discuss the possible implications of our results on planetesimal formation.