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We investigate the influence of collective self-gravity forces on thenonlinear, large-scale evolution of the viscous overstability in Saturn'srings. We numerically solve the axisymmetric hydrodynamic equations in theisothermal and non-isothermal approximation, including radial self-gravity andemploying transport coefficients derived by Salo et al. We assume opticaldepths of 1.5-2 to model Saturn's dense rings. Furthermore, local N-bodysimulations, incorporating vertical and radial collective self-gravity areperformed. Vertical self-gravity is mimicked through an increased frequency ofvertical oscillations, while radial self-gravity is approximated by solving thePoisson equation for an axisymmetric thin disk with a Fourier method. Directparticle-particle forces are omitted, which prevents small-scale gravitationalinstabilities (self-gravity wakes) from forming, an approximation that allowsus to study long radial scales and to compare directly the hydrodynamic modeland the N-body simulations. Our isothermal and non-isothermal hydrodynamicmodel results with vanishing self-gravity compare very well with results ofLatter & Ogilvie and Rein & Latter, respectively. In contrast, for rings withradial self-gravity we find that the wavelengths of saturated overstable wavessettle close to the frequency minimum of the nonlinear dispersion relation,i.e. close to a state of vanishing group velocities of the waves. Goodagreement is found between non-isothermal hydrodynamics and N-body simulationsfor moderate and strong radial self-gravity, while the largest deviations occurfor weak self-gravity. The resulting saturation wavelengths of viscousoverstability for moderate and strong self-gravity (100-300m) agree reasonablywell with the length scales of axisymmetric periodic micro-structure inSaturn's inner A-ring and the B-ring, as found by Cassini.

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