arXiv subjects

Gordon I. Ogilvie

Publications and source records attributed to Gordon I. Ogilvie.

82 records · Page 5Linked to original sources

arXiv2006

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.

astro-ph↗

arXiv2006

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.

astro-ph↗

arXiv2005

Wave attractors and the asymptotic dissipation rate of tidal disturbances

Linear waves in bounded inviscid fluids do not generally form normal modes with regular eigenfunctions. Examples are provided by inertial waves in a rotating fluid contained in a spherical annulus, and internal gravity waves in a stratified fluid contained in a tank with a non-rectangular cross-section. For wave frequencies in the ranges of interest, the inviscid linearized equations are spatially hyperbolic and their characteristic rays are typically focused on to wave attractors. When these systems experience periodic forcing, for example of tidal origin, the response of the fluid can become localized in the neighbourhood of a wave attractor. In this paper I define a prototypical problem of this form and construct analytically the long-term response to a periodic body force in the asymptotic limit of small viscosity. The vorticity of the fluid is localized in a detached shear layer close to the wave attractor in such a way that the total rate of dissipation of energy is asymptotically independent of the viscosity. I further demonstrate that the same asymptotic dissipation rate is obtained if a non-viscous damping force is substituted for the Navier-Stokes viscosity. I discuss the application of these results to the problem of tidal forcing in giant planets and stars, where the excitation and dissipation of inertial waves may make a dominant, or at least important, contribution to the orbital and spin evolution.

astro-ph↗

arXiv2005

A model for the nonlinear dynamics of turbulent shear flows

We investigate the nonlinear dynamics of turbulent shear flows, with and without rotation, in the context of a simple but physically motivated closure of the equation governing the evolution of the Reynolds stress tensor. We show that the model naturally accounts for some familiar phenomena in parallel shear flows, such as the subcritical transition to turbulence at a finite Reynolds number and the occurrence of a universal velocity profile close to a wall at large Reynolds number. For rotating shear flows we find that, depending on the Rayleigh discriminant of the system, the model predicts either linear instability or nonlinear instability or complete stability as the Reynolds number is increased to large values. We investigate the properties of Couette-Taylor flows for varying inner and outer cylinder rotation rates and identify the region of linear instability (similar to Taylor's), as well as regions of finite-amplitude instability qualitatively compatible with recent experiments. We also discuss quantitative predictions of the model in comparison with a range of experimental torque measurements. Finally, we consider the relevance of this work to the question of the hydrodynamic stability of astrophysical accretion discs.

astro-ph↗

arXiv2002

On the relation between viscoelastic and magnetohydrodynamic flows and their instabilities

We demonstrate a close analogy between a viscoelastic medium and an electrically conducting fluid containing a magnetic field. Specifically, the dynamics of the Oldroyd-B fluid in the limit of large Deborah number corresponds to that of a magnetohydrodynamic (MHD) fluid in the limit of large magnetic Reynolds number. As a definite example of this analogy, we compare the stability properties of differentially rotating viscoelastic and MHD flows. We show that there is an instability of the Oldroyd-B fluid that is physically distinct from both the inertial and elastic instabilities described previously in the literature, but is directly equivalent to the magnetorotational instability in MHD. It occurs even when the specific angular momentum increases outwards, provided that the angular velocity decreases outwards; it derives from the kinetic energy of the shear flow and does not depend on the curvature of the streamlines. However, we argue that the elastic instability of viscoelastic Couette flow has no direct equivalent in MHD.

astro-ph↗

arXiv2002

Interactions of High and Low-Mass Planets with Protoplanetary Disks

We present three-dimensional numerical simulations of the interaction of a circular-orbit planet with a protoplanetary disk. We calculate the flow pattern, the accretion rate, and torques on the planet. We consider planet masses ranging from 1 Earth mass to 1 Jupiter mass.

astro-ph↗

arXiv2001

Launching of jets and the vertical structure of accretion disks

The launching of magnetohydrodynamic outflows from accretion disks is considered. We formulate a model for the local vertical structure of a thin disk threaded by a poloidal magnetic field of dipolar symmetry. The model consists of an optically thick disk matched to an isothermal atmosphere. The disk is supposed to be turbulent and possesses an effective viscosity and an effective magnetic diffusivity. In the atmosphere, if the magnetic field lines are inclined sufficiently to the vertical, a magnetocentrifugal outflow is driven and passes through a slow magnetosonic point close to the surface. We determine how the rate of mass loss varies with the strength and inclination of the magnetic field. In particular, we find that for disks in which the mean poloidal field is sufficiently strong to stabilize the disk against the magnetorotational instability, the mass loss rate decreases extremely rapidly with increasing field strength, and is maximal at an inclination angle of 40-50 degrees. For turbulent disks with weaker mean fields, the mass loss rate increases monotonically with increasing strength and inclination of the field, but the solution branch terminates before achieving excessive mass loss rates. Our results suggest that efficient jet launching occurs for a limited range of field strengths, and a limited range of inclination angles in excess of 30 degrees. In addition, we determine the direction and rate of radial migration of the poloidal magnetic flux, and discuss whether configurations suitable for jet launching can be maintained against dissipation.

astro-ph↗

arXiv2000

Linear and non-linear theory of a parametric instability of hydrodynamic warps in Keplerian discs

We consider the stability of warping modes in Keplerian discs. We find them to be parametrically unstable using two lines of attack, one based on three-mode couplings and the other on Floquet theory. We confirm the existence of the instability, and investigate its nonlinear development in three dimensions, via numerical experiment. The most rapidly growing non-axisymmetric disturbances are the most nearly axisymmetric (low m) ones. Finally, we offer a simple, somewhat speculative model for the interaction of the parametric instability with the warp. We apply this model to the masing disc in NGC 4258 and show that, provided the warp is not forced too strongly, parametric instability can fix the amplitude of the warp.

astro-ph↗

arXiv2000

The response of a turbulent accretion disc to an imposed epicyclic shearing motion

We excite an epicyclic motion, whose amplitude depends on the vertical position, $z$, in a simulation of a turbulent accretion disc. An epicyclic motion of this kind may be caused by a warping of the disc. By studying how the epicyclic motion decays we can obtain information about the interaction between the warp and the disc turbulence. A high amplitude epicyclic motion decays first by exciting inertial waves through a parametric instability, but its subsequent exponential damping may be reproduced by a turbulent viscosity. We estimate the effective viscosity parameter, $α_{\rm v}$, pertaining to such a vertical shear. We also gain new information on the properties of the disc turbulence in general, and measure the usual viscosity parameter, $α_{\rm h}$, pertaining to a horizontal (Keplerian) shear. We find that, as is often assumed in theoretical studies, $α_{\rm v}$ is approximately equal to $α_{\rm h}$ and both are much less than unity, for the field strengths achieved in our local box calculations of turbulence. In view of the smallness ($\sim 0.01$) of $α_{\rm v}$ and $α_{\rm h}$ we conclude that for $β= p_{\rm gas}/p_{\rm mag} \sim 10$ the timescale for diffusion or damping of a warp is much shorter than the usual viscous timescale. Finally, we review the astrophysical implications.

astro-ph↗

arXiv1997

On the difficulty of launching an outflow from an accretion disk

We solve for the local vertical structure of a thin accretion disk threaded by a poloidal magnetic field. The angular velocity deviates from the Keplerian value as a result of the radial Lorentz force, but is constant on magnetic surfaces. Angular momentum transport and energy dissipation in the disk are parametrized by an alpha-prescription, and a Kramers opacity law is assumed to hold. We also determine the stability of the equilibria with respect to the magnetorotational (or Balbus-Hawley) instability. If the magnetic field is sufficiently strong, stable equilibria can be found in which the angle of inclination, i, of the magnetic field to the vertical at the surface of the disk has any value in the range [0,90 degrees). By analyzing the dynamics of a transonic outflow in the corona of the disk, we show that a certain potential difference must be overcome even when i > 30 degrees. We determine this potential difference as a function of i for increasing values of the vertical magnetic field strength. For magnetorotationally stable equilibria, the potential difference increases faster than the fourth power of the magnetic field strength, quickly exceeding a value corresponding to the central temperature of the disk, and is minimized with respect to i at i approximately equal to 38 degrees. We show that this property is relatively insensitive to the form of the opacity law. Our results suggest that an additional source of energy, such as coronal heating, may be required for the launching of an outflow from a magnetized disk.

astro-ph↗