The viscous overstability, nonlinear wavetrains, and finescale structure in dense planetary rings

This paper addresses the fine-scale axisymmetric structure exhibited in Saturn's A and B-rings. We aim to explain both the periodic microstructure on 150–220 m, revealed by the Cassini UVIS and RSS instruments, and the irregular variations in brightness on 1–10 km, reported by the Cassini ISS. We propose that the former structures correspond to the peaks and troughs of the nonlinear wavetrains that form naturally in a viscously overstable disk. The latter variations on longer scales may correspond to modulations and defects in the wavetrains' amplitudes and wavelength. We explore these ideas using a simple hydrodynamical model which captures the correct qualitative behaviour of a disk of inelastically colliding particles, while also permitting us to make progress with analytic and semi-analytic techniques. Specifically, we calculate a family of travelling nonlinear density waves and determine their stability properties. Detailed numerical simulations that confirm our basic results will appear in a following paper.