Hydrodynamical Simulations of Narrow Planetary Rings. I. Scaling

Abstract

We investigate the global dynamics of the uranian rings using a modified 2-D smoothed particle hydrodynamic code combined with a 2-D tree code used to compute the particle-to-particle gravitational interactions. This code includes epicyclic fluid motion, nonaxisymmetric flow, local and nonlocal shear viscosity, self-consistent scale height evolution, ring–satellite gravitational interaction and coevolution, and ring self-gravity. To follow the scale height of each SPH particle we solve the vertical momentum equation for the flow using a Runge-Kutta scheme with a second order polynomial fit to the vertical behavior of the fluid pressure (N. Borderies et al., 1985, Icarus63, 406–420). The behavior of the fluid viscosity is obtained from I. Mosqueira (1996, Ph.D. thesis, Cornell University, Ithaca, NY), who found good agreement between a low optical depth extension of the high optical depth nonlocal viscosity model of Borderies et al. (1985) with the results of a local patch-code ring simulation. Our present viscosity model includes a further correction which accounts for the epicyclic limit to the mean free path (P. Goldreich and S. Tremaine, 1978, Icarus34, 227–239). This treatment covers both the high and the low ring density regimes so long as the fluid treatment remains valid. All energy source and sink terms in the presence of a nonzero fluid velocity divergence are self-consistently considered, including terms not considered in prior studies of rings. Furthermore, a new method has been tested and used to remove the noise contribution to the viscous momentum flux which occurs in traditional treatments of the viscosity within the SPH framework. In the present context this correction is needed to assure a physical behavior for the ring. Even within a 2-D framework the uranian rings are so narrow compared to their semimajor axes that radial resolution requires too many particles given our present computer resources. To address this issue we have developed a physical scaling that reduces the semimajor axis of the ring while preserving its width and, we believe, retains the relevant global satellite–ring dynamics. In the case of ring resolution on the order of kilometers, with a value of the scaling parameter that reduces the ring's semimajor axis by a factor of 20, our scaling allows for savings between a factor of 500 in the case of synodic time scales to a factor of 10,000 for viscous time scales. We expect this scaling to figure significantly in numerical studies of the global dynamics of planetary rings in the future. In the present study we use this scaling to follow the evolution of an epsilon-like ring in the presence of an m=2 mode (but no satellites) with and without self-gravity. In these runs, chosen as an initial demonstration of the usefulness and applicability of our scaling, the ring can be seen to evolve in the presence of shear.