Viscous overstability in Saturn's B ring - 1. Direct simulations and measurement of transport coefficients

Local simulations with up to 60,000 self-gravitating dissipatively collidin g particles indicate that dense unperturbed ring systems with optical depth tau > 1 can exhibit spontaneous viscous oscillatory instability (overstabi lity), with parameter values appropriate for Saturn's B ring. These axisymm etric oscillations, with scale similar to 100 m and frequency close to the orbital period, generally coexist with inclined Julian-Toomre type wakes fo rming in gravitating disks. The onset of overstability depends on the inter nal density of particles, their elasticity, and the size distribution. The same type of oscillatory behavior is also obtained in an approximation wher e the particle-particle gravity is replaced by an enhanced frequency of ver tical oscillations, Omega (z) / Omega > 1. This has the advantage that thes e systems can be more easily studied analytically, as in the absence of wak es the system has a spatially uniform ground state. For Omega (z) / Omega = 3.6 overstability again starts at tau similar to 1. Also, nongravitating s ystems, Omega (z) / Omega = 1, show overstability, but this requires tau si milar to 4. To facilitate a quantitative hydrodynamical study of overstabil ity we have measured the transport coefficients (kinematic shear viscosity upsilon, kinematic bulk viscosity zeta, and kinematic heat conductivity kap pa) in simulations with Omega (z) / Omega 3.6, 2.0, and 1.0. Both local and nonlocal (collisional) contributions to the momentum and energy flux are t aken into account, the latter being dominant in dense systems with large im pact frequency. In this limit we find zeta/upsilon approximate to 2, kappa/ upsilon approximate to 4. The dependence of pressure, viscosity, and dissip ation on density and kinetic temperature changes is also estimated. Prelimi nary comparisons indicate that the condition for overstability is beta > be ta (cr) similar to 1, where beta : = partial derivative log(upsilon) / part ial derivative log(tau). This limit is clearly larger than the beta (cr) si milar to 0 suggested by the linear stability analysis in Schmit and Tscharn uter (1995), where the system was assumed to stay isothermal even when pert urbed. However, it agrees with the nonisothermal analysis in Spahn et al. ( 2000). This increased stability is in part due to the inclusion of temperat ure oscillations in the analysis, and in part:due to bulk viscosity exceedi ng shear viscosity. A detailed comparison between simulations and hydrodyna mical analysis is presented in a separate paper (Schmidt et al. 2001). (C) 2001 Academic Press.