A Weakly Nonlinear Model for the Damping of Resonantly Forced Density Waves in Dense Planetary Rings

Abstract

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.

Viscous Overstability in Saturn’s Rings: Influence of Collective Self-gravity

“…The nonlinear frequency reduction due to self-gravity is analogous to the nonlinear wavenumber reduction found for resonant spiral density waves in a dense ring, for which pressure forces play a minor role ; Lehmann et al (2016)). In the isothermal model (panel d), with the ideal gas equation of state, the presence of any substantial self-gravity force results in a nonlinear reduction of the oscillation frequencies.…”

Density waves and the viscous overstability in Saturn’s rings

“…That being said, hydrodynamic models have been found useful for at least a qualitative description of various structures in planetary rings, mainly those of Saturn. Prominent examples are the propagation of satellite induced spiral density waves (Goldreich & Tremaine 1978b,c, 1979Shu 1984;Shu et al 985a;Shu et al 985b;Borderies et al 1985Borderies et al , 1986Schmidt et al 2016;Lehmann et al 2016Lehmann et al , 2019, the formation of moonlet induced gaps and 'propeller' structures (Goldreich & Tremaine 1980;Showalter et al 1986;Borderies et al 1989;Spahn et al 1992;Hahn et al 2009;Hoffmann et al 2015;Spahn et al 2018;Grätz et al 2019;Seiß et al 2019;Seiler et al 2019) or the small-scale axisymmetric viscous overstability (Schmit & Tscharnuter 1995Spahn et al 2000;Schmidt et al 2001;Salo et al 2001;Schmidt & Salo 2003;Latter & Ogilvie 2009, 2010Lehmann et al 2017Lehmann et al , 2019. It is the latter to which this study is devoted.…”

Planetary Ring Dynamics -- The Streamline Formalism -- 2. Theory of Narrow Rings and Sharp Edges

“…Orders of magnitude for B a , B 1 and B 2 can be obtained from the comparison of these relations with specific models. This model gives good semi-quantitative agreement with observed properties, in particular in the analysis of density wave damping, although a detailed quantitative comparison probably calls for more sophisticated models (Borderies et al, 1986;Shu et al, 1985a;Lehmann et al, 2016). Another related issue is to investigate to which extent it is possible to connect this type of modelling with phenomenological physical constraints on ring rheology.…”