On the formation of the fine-scale structure in Saturn's B ring

In a previous paper (U. Schmit und W. M. Tscharnuter 1995, Iacarus 115, 304-319; hereafter referred to as Paper I) we developed a linear stability analysis of Saturn's B ring on the basis of the hydrodynamic approximation. The ring is thus treated as a very thin layer of a differentially rotating viscous fluid with cylindrical symmetry. In this paper we follow the evolution into the nonlinear regime by solving the basic hydrodynamic equations numerically. We demonstrate that, for the viscous instability proposed by D. N. C. Lin and P Bodenheimer (1981, Astrophys. J. 248, L83-L86), J. Lukkari (1981, Nature 292, 433-435), and W. Ward (1981, Geophys. Res. Lett. 8, 641-643) to explain the fine-scale structure of Saturn's B ring, which was discovered by the Voyager mission, even in the nonlinear frame the amplitudes grow indefinitely. This is also true for the secular instability in the sense of D. Lynden-Bell and J. E. Pringle (1974, Mon. Not. R. Astron. Sec. 168, 603-637), which is driven by the ring's self-gravitation. The only mechanism that evolves out of an unstable initial state into an oscillating (in space and time), quasi-stable final state is the viscous overstability we discussed in Paper I. This is due to the fact that the increase of the dynamic shear viscosity coefficient with increasing surface density, e.g., as calculated by J. Wisdom and S. Tremaine (1988, Astron. J. 95, 925-940) is sufficiently steep. Based on extensive calculations (up to 10(4) orbital periods) for a "hoop-like" section spanning 40 km in the radial direction within the B ring, we suggest that the ring's irregular structure is generated by nonlinear wave-wave interactions, although there exists a narrow range of preferred wavelengths between 200 and 250 m (about four times the Jeans length of the ring) when self-gravitation of the ring material is taken into account. By contrast, without self-gravitation, we have found systematic generation of modes with continuously increasing wavelengths, at least during 2 x 104 orbital periods. Thus, self-gravitation plays a key role in the formation of the fine-scale structure in Saturn's B ring, (C) 1999 Academic Press.