83 research outputs found
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
Viscous overstability and eccentricity evolution in three-dimensional gaseous discs
We investigate the growth or decay rate of the fundamental mode of even
symmetry in a viscous accretion disc. This mode occurs in eccentric discs and
is known to be potentially overstable. We determine the vertical structure of
the disc and its modes, treating radiative energy transport in the diffusion
approximation. In the limit of very long radial wavelength, an analytical
criterion for viscous overstability is obtained, which involves the effective
shear and bulk viscosity, the adiabatic exponent and the opacity law of the
disc. This differs from the prediction of a two-dimensional model. On shorter
wavelengths (a few times the disc thickness), the criterion for overstability
is more difficult to satisfy because of the different vertical structure of the
mode. In a low-viscosity disc a third regime of intermediate wavelengths
appears, in which the overstability is suppressed as the horizontal velocity
perturbations develop significant vertical shear. We suggest that this effect
determines the damping rate of eccentricity in protoplanetary discs, for which
the long-wavelength analysis is inapplicable and overstability is unlikely to
occur on any scale. In thinner accretion discs and in decretion discs around Be
stars overstability may occur only on the longest wavelengths, leading to the
preferential excitation of global eccentric modes.Comment: 11 pages, 8 figure
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