Viscous overstability in Saturn's B-Ring - II. Hydrodynamic theory and comparison to simulations

We investigate the viscous oscillatory instability (overstability) of an un perturbed dense planetary ring, an instability that might play a role in th e formation of radial structure in Saturn's B-ring. We generalize existing hydrodynamic models by including the heat flow equation in the analysis and compare our results to the development of overstable modes in local partic le simulations. With the heat flow, in addition to the balance equations fo r mass and momentum, we take into account the balance law for the energy of the random motion; i.e., we allow for a thermal mode in a stability analys is of the stationary Keplerian flow. We also incorporate the effects of non local transport of momentum and energy on the stability of the ring. In a c ompanion paper (Salo, H., J. Schmidt, and F. Spahn 2001. Icarus, doi:10.100 6/icar.2001.6680) we describe the determination of the local and nonlocal p arts of the viscosity, the heat conductivity, the pressure, as well as the collisional cooling, together with their dependences on temperature and den sity, in local event-driven simulations of a planetary ring. The ring's sel f-gravity is taken into account in these simulations by an enhancement of t he frequency of vertical oscillations Omega (z) > Omega. We use these value s as parameters in our hydrodynamic model for the comparison to overstabili ty in simulated rings of meter-sized inelastic particles of large optical d epth with Omega (z)/Omega = 3.6. We find that the inclusion of the energy-b alance equation has a stabilizing influence on the overstable modes, shifti ng the stability boundary to higher optical depths, and moderating the grow th rates of the instability, as compared to a purely isothermal treatment. The non-isothermal model predicts correctly the growth rates and oscillatio n frequencies of overstable modes in the simulations, as well as the phase shifts and relative amplitudes of the perturbations in density and radial a nd tangential velocity. (C) 2001 Academic Press.