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 & 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.

References in corpus (8)

• The World of the Complex Ginzburg-Landau Equation
• Cassini Imaging of Saturn's Rings II: A Wavelet Technique for Analysis of Density Waves and Other Radial Structure in the Rings
• Spatial Structure and Coherent Motion in Dense Planetary Rings Induced by Self-Gravitational Instability
• The B-ring's surface mass density from hidden density waves: Less than meets the eye?
• The viscous overstability, nonlinear wavetrains, and finescale structure in dense planetary rings
• Hydrodynamical simulations of viscous overstability in Saturn's rings
• Exploring overstabilities in Saturn's A ring using two stellar occultations
• A procedure to analyze nonlinear density waves in Saturn's rings using several occultation profiles

Cited by in corpus (3)

• Density Waves and the Viscous Overstability in Saturn's Rings
• Sharp gap edges in dense planetary rings: An axisymmetric diffusion model
• Viscous Overstability in Saturn's Rings: Influence of Collective Self-gravity