The ballistic transport instability in Saturn's rings - II. Non-linear wave dynamics

The ejecta discharged by impacting meteorites redistribute a planetary ring's mass and angular momentum. This 'ballistic transport' of ring properties instigates a linear instability that could generate the 100-1000-km undulations observed in Saturn's inner B-ring and in its C-ring. We present semi-analytic results demonstrating how the instability sustains steadily travelling non-linear wavetrains. At low optical depths, the instability produces approximately sinusoidal waves of low amplitude, which we identify with those observed between radii 77 000 and 86 000 km in the C-ring. On the other hand, optical depths of 1 or more exhibit hysteresis, whereby the ring falls into multiple stable states: the homogeneous background equilibrium or large-amplitude wave states. Possibly the 'flat zones' and 'wave zones' between radii 93 000 and 98 000 km in the B-ring correspond to the stable homogeneous and wave states, respectively. In addition, we test the linear stability of the wavetrains and show that only a small subset are stable. Stable solutions all possess wavelengths greater than the length-scale of fastest linear growth. We supplement our calculations with a weakly non-linear analysis that suggests that the C-ring reproduces some of the dynamics of the complex Ginzburg-Landau equation. In the third paper in the series, these results will be tested and extended with numerical simulations.