Hydrodynamic instability in warped astrophysical discs

Warped astrophysical discs are usually treated as laminar viscous flows, which have anomalous properties when the disc is nearly Keplerian and the viscosity is small: fast horizontal shearing motions and large torques are generated, which cause the warp to evolve rapidly, in some cases at a rate that is inversely proportional to the viscosity. However, these flows are often subject to a linear hydrodynamic instability, which may produce small-scale turbulence and modify the large-scale dynamics of the disc. We use a warped shearing sheet to compute the oscillatory laminar flows in a warped disc and to analyse their linear stability by the Floquet method. We find widespread hydrodynamic instability deriving from the parametric resonance of inertial waves. Even very small, unobservable warps in nearly Keplerian discs of low viscosity can be expected to generate hydrodynamic turbulence, or at least wave activity, by this mechanism.Comment: 17 pages, 7 figures, revised version, to be published in MNRA

The linear stability of dilute particulate rings

Irregular structure in planetary rings is often attributed to the intrinsic instabilities of a homogeneous state undergoing Keplerian shear. Previously these have been analysed with simple hydrodynamic models. We instead employ a kinetic theory, in which we solve the linearised moment equations derived in Shu and Stewart 1985 for a dilute ring. This facilitates an examination of velocity anisotropy and non-Newtonian stress, and their effects on the viscous and viscous/gravitational instabilities thought to occur in Saturn's rings. Because we adopt a dilute gas model, the applicability of our results to the actual dense rings of Saturn are significantly curtailled. Nevertheless this study is a necessary preliminary before an attack on the difficult problem of dense ring dynamics. We find the Shu and Stewart formalism admits analytic stability criteria for the viscous overstability, viscous instability, and thermal instability. These criteria are compared with those of a hydrodynamic model incorporating the effective viscosity and cooling function computed from the kinetic steady state. We find the two agree in the `hydrodynamic limit' (i.e. many collisions per orbit) but disagree when collisions are less frequent, when we expect the viscous stress to be increasingly non-Newtonian and the velocity distribution increasingly anisotropic. In particular, hydrodynamics predicts viscous overstability for a larger portion of parameter space. We also numerically solve the linearised equations of the more accurate Goldreich and Tremaine 1978 kinetic model and discover its linear stability to be qualitatively the same as that of Shu and Stewart's. Thus the simple collision operator adopted in the latter would appear to be an adequate approximation for dilute rings, at least in the linear regime