42 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
Local and global dynamics of warped astrophysical discs
Astrophysical discs are warped whenever a misalignment is present in the
system, or when a flat disc is made unstable by external forces. The evolution
of the shape and mass distribution of a warped disc is driven not only by
external influences but also by an internal torque, which transports angular
momentum through the disc. This torque depends on internal flows driven by the
oscillating pressure gradient associated with the warp, and on physical
processes operating on smaller scales, which may include instability and
turbulence. We introduce a local model for the detailed study of warped discs.
Starting from the shearing sheet of Goldreich & Lynden-Bell, we impose the
oscillating geometry of the orbital plane by means of a coordinate
transformation. This warped shearing sheet (or box) is suitable for analytical
and computational treatments of fluid dynamics, magnetohydrodynamics, etc., and
it can be used to compute the internal torque that drives the large-scale
evolution of the disc. The simplest hydrodynamic states in the local model are
horizontally uniform laminar flows that oscillate at the orbital frequency.
These correspond to the nonlinear solutions for warped discs found in previous
work by Ogilvie, and we present an alternative derivation and generalization of
that theory. In a companion paper we show that these laminar flows are often
linearly unstable, especially if the disc is nearly Keplerian and of low
viscosity. The local model can be used in future work to determine the
nonlinear outcome of the hydrodynamic instability of warped discs, and its
interaction with others such as the magnetorotational instability.Comment: 17 pages, 10 figures, revised version, to be published in MNRA
Thermodynamics of the dead-zone inner edge in protoplanetary disks
In protoplanetary disks, the inner boundary between the turbulent and laminar
regions could be a promising site for planet formation, thanks to the trapping
of solids at the boundary itself or in vortices generated by the Rossby wave
instability. At the interface, the disk thermodynamics and the turbulent
dynamics are entwined because of the importance of turbulent dissipation and
thermal ionization. Numerical models of the boundary, however, have neglected
the thermodynamics, and thus miss a part of the physics. The aim of this paper
is to numerically investigate the interplay between thermodynamics and dynamics
in the inner regions of protoplanetary disks by properly accounting for
turbulent heating and the dependence of the resistivity on the local
temperature. Using the Godunov code RAMSES, we performed a series of 3D global
numerical simulations of protoplanetary disks in the cylindrical limit,
including turbulent heating and a simple prescription for radiative cooling. We
find that waves excited by the turbulence significantly heat the dead zone, and
we subsequently provide a simple theoretical framework for estimating the wave
heating and consequent temperature profile. In addition, our simulations reveal
that the dead-zone inner edge can propagate outward into the dead zone, before
staling at a critical radius that can be estimated from a mean-field model. The
engine driving the propagation is in fact density wave heating close to the
interface. A pressure maximum appears at the interface in all simulations, and
we note the emergence of the Rossby wave instability in simulations with
extended azimuth. Our simulations illustrate the complex interplay between
thermodynamics and turbulent dynamics in the inner regions of protoplanetary
disks. They also reveal how important activity at the dead-zone interface can
be for the dead-zone thermodynamic structure.Comment: 16 pages, 16 figures. Accepted in Astronomy and Astrophysic
Quasi-periodic oscillations, trapped inertial waves and strong toroidal magnetic fields in relativistic accretion discs
The excitation of trapped inertial waves (r-modes) by warps and
eccentricities in the inner regions of a black hole accretion disc may explain
the high-frequency quasi-periodic oscillations (HFQPOs) observed in the
emission of Galactic X-ray binaries. However, it has been suggested that strong
vertical magnetic fields push the oscillations' trapping region toward the
innermost stable circular orbit (ISCO), where conditions could be unfavourable
for their excitation. This paper explores the effects of large-scale magnetic
fields that exhibit \textit{both} toroidal and vertical components, through
local and global linear analyses. We find that a strong toroidal magnetic field
can reduce the detrimental effects of a vertical field: in fact, the isolation
of the trapping region from the ISCO may be restored by toroidal magnetic
fields approaching thermal strengths. The toroidal field couples the r-modes to
the disc's magneto-acoustic response and inflates the effective pressure within
the oscillations. As a consequence, the restoring force associated with the
vertical magnetic field's tension is reduced. Given the analytical and
numerical evidence that accretion discs threaded by poloidal magnetic field
lines develop a strong toroidal component, our result provides further evidence
that the detrimental effects of magnetic fields on trapped inertial modes are
not as great as previously thought.Comment: 16 pages, 6 figures, MNRAS accepte