Viscous Overstability in Saturn's B Ring: I. Direct Simulations and Measurement of Transport Coefficients

Abstract

Local simulations with up to 60,000 self-gravitating dissipatively colliding particles indicate that dense unperturbed ring systems with optical depth τ>1 can exhibit spontaneous viscous oscillatory instability (overstability), with parameter values appropriate for Saturn's B ring. These axisymmetric oscillations, with scale ∼100 m and frequency close to the orbital period, generally coexist with inclined Julian–Toomre type wakes forming in gravitating disks. The onset of overstability depends on the internal density of particles, their elasticity, and the size distribution. The same type of oscillatory behavior is also obtained in an approximation where the particle–particle gravity is replaced by an enhanced frequency of vertical oscillations, Ω z /Ω>1. This has the advantage that these systems can be more easily studied analytically, as in the absence of wakes the system has a spatially uniform ground state. For Ω z /Ω=3.6 overstability again starts at τ ∼1. Also, nongravitating systems, Ω z /Ω=1, show overstability, but this requires τ ∼4. To facilitate a quantitative hydrodynamical study of overstability we have measured the transport coefficients (kinematic shear viscosity ν, kinematic bulk viscosity ζ, and kinematic heat conductivity κ) in simulations with Ω z /Ω=3.6, 2.0, and 1.0. Both local and nonlocal (collisional) contributions to the momentum and energy flux are taken into account, the latter being dominant in dense systems with large impact frequency. In this limit we find ζ/ν≈2, κ/ν≈4. The dependence of pressure, viscosity, and dissipation on density and kinetic temperature changes is also estimated. Preliminary comparisons indicate that the condition for overstability is β>β cr ∼1, where β:=∂ log(ν)/∂ log(τ). This limit is clearly larger than the β cr ∼0 suggested by the linear stability analysis in Schmit and Tscharnuter (1995), where the system was assumed to stay isothermal even when perturbed. However, it agrees with the nonisothermal analysis in Spahn et al. (2000). This increased stability is in part due to the inclusion of temperature oscillations in the analysis, and in part due to bulk viscosity exceeding shear viscosity. A detailed comparison between simulations and hydrodynamical analysis is presented in a separate paper (Schmidt et al. 2001).