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 150-220m, revealed by the Cassini UVIS and RSS instruments, and the irregular variations in brightness on 1-10km, 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.
Deep Dive into 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 150-220m, revealed by the Cassini UVIS and RSS instruments, and the irregular variations in brightness on 1-10km, 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 a
Saturn's A and B-rings sport an abundance of irregular radial structure which, though aesthetically pleasing, presents something of a puzzle to the theoretician. The instruments aboard the Cassini space probe show that these patterns manifest on a vast range of length-scales and can take quite different forms. For instance, there exist quasi-periodic microstructure on scales of 0.1 km (Colwell et al. 2007;Thomson et al. 2007), discontinuous and irregular striations on the 1-10 km intermediate scale, and much broader 100 km undulations (Porco et al. 2005). In addition to the difficulties involved in tackling these three orders of magnitude, there are the formidable modelling questions posed by a cold disk of densely-packed, inelastic, and infrequently colliding particles (Stewart et al. 1984, Araki and Tremaine 1986, Salo 1991, Hämeen-Antilla and Salo 1993, Schmidt et al. 2001, Latter and Ogilvie 2008). This paper will focus on only a subset of these phenomena, the smaller-scale variations, and will not linger especially on the modelling issues. Specifically, it investigates how the quasi-periodic microstructure relates to the viscous overstability, on one hand, and to structure formation on the intermediate scales, on the other. The variations on the much longer 100km scale are not examined, and we suspect that they have their origin in a different mechanism entirely, perhaps ballistic transport (Durisen 1995).
Our starting point is the viscous overstability, which is now regarded as a key player in the short scale radial dynamics of Saturn’s rings (Schmit and Tscharnuter 1995, Schmidt et al. 2001, Spahn and Schmidt 2006, Latter and Ogilvie 2008). The viscous overstability is an axisymmetric oscillatory instability that afflicts the homogeneous state of Keplerian shear. Growing modes rely on the alliance of the fluid disk’s inertial-acoustic oscillations with the disk’s stress oscillations: variations in the stress extract energy from the Keplerian shear and inject it into the inertial-acoustic wave; but the increased motion this induces magnifies the stress oscillation itself which can extract even more energy, and so the process runs away. The feedback loop requires (a) the stress to efficiently remove energy from the Keplerian flow and (b) the two oscillations to communicate effectively, in particular for them to be in phase. The first condition is tied to the stress’s sensitivity to surface density. The second condition is often violated in dilute rings and turbulent disks, where the stress can lag behind the epicycles and the overstability fails to work (Ogilvie 2001, Latter andOgilvie 2006a). The instability’s linear theory has been well established by a variety of theoretical approaches: hydrodynamics (Schmit andTscharnuter 1995, Schmidt et al. 2001), N-body simulations (Salo 2001, Salo et al. 2001), and kinetics (Latter andOgilvie 2006a, 2008). However, its nonlinear theory has received surprisingly little attention. Two hydrodynamical studies exist, a large-scale nonlinear simulation of a ring annulus in 1D (Schmit and Tscharnuter 1999) and a weakly nonlinear analysis (Schmidt and Salo 2003). The simulations show that the nonlinear evolution of an overstable disk is characterised by significant disorder. In contrast, the latter study suggests that an overstable ring may exhibit simple coherent structures which take the form of travelling waves. Because Schmit and Tscharnuter impose reflecting boundary conditions, and hence break translational symmetry, such coherent structures may have been difficult to observe in their simulations. Certainly, more simulations need to be undertaken and the influence of the boundary conditions better understood. Concurrently, a fully nonlinear theory extending the work of Schmidt and Salo is required to fully establish the existence of the nonlinear solutions. The former project we present elsewhere, the latter we present here.
First, we demonstrate that steady nonlinear travelling wavetrains are exact solutions to the governing nonlinear equations of a viscously overstable disk, and second, that these solutions are the loci of a rich secondary set of dynamics which generate irregular variations in the waves’ amplitude and wavenumber. We propose that viscously overstable regions in the A and B-rings support a bed of travelling wavetrain solutions and these structures correspond to the quasi-periodic variations registered by Cassini’s UVIS and RSS instruments (the radial ‘microstructure’). Modulations and defects in the amplitudes and wavelengths of these wavetrains may yield different optical properties which can be traced by the Cassini cameras. Consequently, we hypothesise that these larger-scale variations are associated with the intermediate 1-10 km structures observed (Porco et al. 2005). Our hypotheses are investigated with a one-dimensional, isothermal fluid model endowed with a Navier-Stokes stress. Though simplistic, it should predict qualitatively correc
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