Photon bubbles in accretion discs

We show that radiation-dominated accretion discs are likely to suffer from a 'photon bubble' instability similar to that described by Arons in the context of accretion on to neutron star polar caps. The instability requires a magnetic field for its existence. In an asymptotic regime appropriate to accretion discs, we find that the overstable modes obey the remarkably simple dispersion relation
ο²=−i gkF ( B , k ).
Here g is the vertical gravitational acceleration, B is the magnetic field, and F is a geometric factor of order unity that depends on the relative orientation of the magnetic field and the wavevector. In the non-linear outcome it seems likely that the instability will enhance vertical energy transport and thereby change the structure of the innermost parts of relativistic accretion discs.     

Fractal features in accretion discs

Fractal concepts have been introduced in the accretion disc as a new feature. Due to the fractal nature of the flow, its continuity condition undergoes modifications. The conserved stationary fractal flow admits only saddle points and centre-type points in its phase portrait. Completely analytical solutions of the equilibrium point conditions indicate that the fractal properties enable the flow to behave like an effective continuum of lesser density, and facilitate the generation of transonicity. However, strongly fractal flows inhibit multitransonicity from developing. The mass accretion rate exhibits a fractal scaling behaviour, and the entire fractal accretion disc is stable under linearized dynamic perturbations.     

Outbursts of irradiated accretion discs

I solve analytically the viscous evolution of an irradiated accretion disc, as seen during outbursts of soft X-ray transients. The solutions predict steep power-law X-ray decays L _X ∼ (1 + t/t_visc)⁻⁴, changing to L _X ∼ (1 − t/t'_visc)⁴ at late times, where t _visc, t '_visc are viscous time-scales. These forms closely resemble the approximate exponential and linear decays inferred by King and Ritter in these two regimes. The decays are much steeper than for unirradiated discs because the viscosity is a function of the central accretion rate rather than of local conditions in the disc.     

Non-linear bending waves in Keplerian accretion discs

The non-linear dynamics of a warped accretion disc is investigated in the important case of a thin Keplerian disc with negligible viscosity and self-gravity. A one-dimensional evolutionary equation is formally derived that describes the primary non-linear and dispersive effects on propagating bending waves other than parametric instabilities. It has the form of a derivative non-linear Schrödinger (DNLS) equation with coefficients that are obtained explicitly for a particular model of a disc. The properties of this equation are analysed in some detail and illustrative numerical solutions are presented. The non-linear and dispersive effects both depend on the compressibility of the gas through its adiabatic index Γ. In the physically realistic case Γ < 3, non-linearity does not lead to the steepening of bending waves but instead enhances their linear dispersion. In the opposite case Γ > 3, non-linearity leads to wave steepening and solitary waves are supported. The effects of a small effective viscosity, which may suppress parametric instabilities, are also considered. This analysis may provide a useful point of comparison between theory and numerical simulations of warped accretion discs.     

Magnetic shear-flow instability in thin accretion discs

The possibility that the magnetic shear-flow instability (also known as the 'Balbus–Hawley' instability) might give rise to turbulence in a thin accretion disc is investigated through numerical simulations. The study is linear and the fluid disc is supposed to be incompressible and differentially rotating with a simple velocity profile with Ω∝ R ^{− q} . The simplicity of the model is counterbalanced by the fact that the study is fully global in all three spatial directions with boundaries on each side; finite diffusivities are also allowed. The investigation is also carried out for several values of the azimuthal wavenumber of the perturbations in order to analyse whether non-axisymmetric modes might be preferred, which may produce, in a non-linear extension of the study, a self-sustained magnetic field.
  We find the final pattern steady, with similar kinetic and magnetic energies and the angular momentum always transported outwards. Despite the differential rotation, there are only small differences for the eigenvalues for various non-axisymmetric eigensolutions. Axisymmetric instabilities are by no means preferred; in fact for Prandtl numbers between 0.1 and 1, the azimuthal wavenumbers m =0,1,2(10¹⁶ g s^-1). All three quantities appear to be equally readily excited. The equatorial symmetry is quadrupolar for the magnetic field and dipolar for the flow field system. The maximal magnetic field strength required to cause the instability is almost independent of the magnetic Prandtl number. With typical white dwarf values, a magnetic amplitude of 10⁵ G is estimated.     

Magnetic field bending in accretion discs with dynamos

We consider the problem of poloidal magnetic field advection and bending of an initially vertical field owing to radial inflow in thin accretion discs. For a ratio of kinematic viscosity to magnetic diffusivity of order unity, significant bending of an externally applied vertical field cannot occur in a disc with no internal dynamo. However, we show that if poloidal field is generated by a dynamo operating near its critical state, then significant field bending may be possible. Our results are of particular relevance to wind launching from accretion discs.     

An alpha theory of time-dependent warped accretion discs

The non-linear fluid dynamics of a warped accretion disc was investigated in an earlier paper by developing a theory of fully non-linear bending waves in a thin, viscous disc. That analysis is extended here to take proper account of thermal and radiative effects by solving an energy equation that includes viscous dissipation and radiative transport. The problem is reduced to simple one-dimensional evolutionary equations for mass and angular momentum, expressed in physical units and suitable for direct application. This result constitutes a logical generalization of the alpha theory of Shakura & Sunyaev to the case of a time-dependent warped accretion disc. The local thermal–viscous stability of such a disc is also investigated.     

Turbulence and its parameterization in accretion discs

Accretion disc turbulence is investigated in the framework of the shearing box approximation. The turbulence is either driven by the magneto‐rotational instability or, in the non‐magnetic case, by an explicit and artificial forcing term in the momentum equation. Unlike the magnetic case, where most of the dissipation occurs in the disc corona, in the forced hydrodynamic case most of the dissipation occurs near the midplane. In the hydrodynamic case evidence is presented for the stochastic excitation of epicycles. When the vertical and radial epicyclic frequencies are different (modeling the properties around rotating black holes), the beat frequency between these two frequencies appear to show up as a peak in the temporal power spectrum in some cases. Finally, the full turbulent resistivity tensor is determined and it is found that, if the turbulence is driven by a forcing term, the signs of its off‐diagonal components are such that this effect would not be capable of dynamo action by the shear–current effect. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Evolution of initially localized perturbations in stratified ionized discs

A detailed solution of an initial value problem of a vertically localized initial perturbation in rotating magnetized vertically stratified disc is presented. The appropriate linearized magnetohydrodynamics equations are solved by employing the Wentzel–Kramers–Brillouin (WKB) approximation and the results are verified numerically. The eigenfrequencies as well as eigenfunctions are explicitly obtained. It is demonstrated that the initial perturbation remains confined within the disc. It is further shown that thin enough discs are stable but as their thickness grows increasing number of unstable modes participate in the solution of the initial value problem. However, it is demonstrated that due to the localization of the initial perturbation, the growth time of the instability is significantly longer than the calculated inverse growth rate of the individual unstable eigenfunctions.     

On the precession of accretion discs in X-ray binaries

In this Letter, recent results on the nodal precession of accretion discs in close binaries are applied to the discs in some X-ray binary systems. The ratio between the tidally forced precession period and the binary orbital period is given, as well as the condition required for the rigid precession of gaseous Keplerian discs. Hence the minimum precessional period that may be supported by a fluid Keplerian disc is determined. It is concluded that near-rigid body precession of tilted accretion discs can occur and generally reproduce observationally inferred precession periods, for reasonable system parameters. In particular, long periods in SS 433, Her X-1, LMC X-4 and SMC X-1 can be fitted by the tidal model. It is also found that the precession period that has been tentatively put forward for Cyg X-2 cannot be accommodated by a tidally precessing disc model for any realistic choice of system parameters.     

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.     

Inertial waves near corotation in three-dimensional hydrodynamical discs

This paper concerns the interaction between non-axisymmetric inertial waves and their corotation resonances in a hydrodynamical disc. Inertial waves are of interest because they can localize in resonant cavities circumscribed by Lindblad radii and, as a consequence, can exhibit discrete oscillation frequencies that may be observed. It is often hypothesized that these trapped eigenmodes are affiliated with the poorly understood quasi-periodic oscillation phenomenon. We demonstrate that a large class of non-axisymmetric three-dimensional (3D) inertial waves cannot manifest as trapped normal modes. This class includes any inertial wave whose resonant cavity contains a corotation singularity. Instead, these 'singular' modes constitute a continuous spectrum and, as an ensemble, are convected with the flow, giving rise to shearing waves. Finally, we present a simple demonstration of how the corotation singularity stabilizes 3D perturbations in a slender torus.