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.     

Super-reflection in fluid discs: corotation amplifier, corotation resonance, Rossby waves and overstable modes

In differentially rotating discs with no self-gravity, density waves cannot propagate around the corotation, where the wave pattern rotation speed equals the fluid rotation rate. Waves incident upon the corotation barrier may be super-reflected (commonly referred to as corotation amplifier), but the reflection can be strongly affected by wave absorptions at the corotation resonance/singularity. The sign of the absorption is related to the Rossby wave zone very near the corotation radius. We derive the explicit expressions for the complex reflection and transmission coefficients, taking into account wave absorption at the corotation resonance. We show that for generic discs, this absorption plays a much more important role than wave transmission across the corotation barrier. Depending on the sign of the gradient of the vortensity of the disc,  ζ=κ²/(2ΩΣ)  (where Ω is the rotation rate, κ is the epicyclic frequency and Σ is the surface density), the corotation resonance can either enhance or diminish the super-reflectivity, and this can be understood in terms of the location of the Rossby wave zone relative to the corotation radius. Our results provide the explicit conditions (in terms of disc thickness, rotation profile and vortensity gradient) for which super-reflection can be achieved. Global overstable disc modes may be possible for discs with super-reflection at the corotation barrier.     

Warping modes in discs around accreting neutron stars

The origin and stability of a thin sheet of plasma in the magnetosphere of an accreting neutron star are investigated. First, the radial extension of such a magnetospheric disc is explored. Then a mechanism for magnetospheric accretion is proposed, reconsidering the bending wave explored by Agapitou, Papaloizou & Terquem, that was found to be stable in ideal magnetohydrodynamics. We show that this warping becomes unstable and can reach high amplitudes, in a variant of Pringle's radiation-driven model for the warping of active galactic nucleus accretion discs. Finally, we discuss how this mechanism might give a clue to explain the observed X-ray kilohertz quasi-periodic oscillation of neutron star binaries.     

Unstable modes of Keplerian discs

We derive the softened analogue of the Laplace–Lagrange secular theory of planetary motion, and use it to show that a small fraction of counter-rotating stars is all it takes for a hot Keplerian disc to grow unstable lopsided 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.     

Self-regulated gravitational accretion in protostellar discs

We present a numerical model for the evolution of a protostellar disc that has formed self-consistently from the collapse of a molecular cloud core. The global evolution of the disc is followed for several million years after its formation. The capture of a wide range of spatial and temporal scales is made possible by use of the thin-disc approximation. We focus on the role of gravitational torques in transporting mass inward and angular momentum outward during different evolutionary phases of a protostellar disc with disc-to-star mass ratio of order 0.1. In the early phase, when the infall of matter from the surrounding envelope is substantial, mass is transported inward by the gravitational torques from spiral arms that are a manifestation of the envelope-induced gravitational instability in the disc. In the late phase, when the gas reservoir of the envelope is depleted, the distinct spiral structure is replaced by ongoing irregular non-axisymmetric density perturbations. The amplitude of these density perturbations decreases with time, though this process is moderated by swing amplification aided by the existence of the disc's sharp outer edge. Our global modelling of the protostellar disc reveals that there is typically a residual non-zero gravitational torque from these density perturbations, i.e. their effects do not exactly cancel out in each region. In particular, the net gravitational torque in the inner disc tends to be negative during first several million years of the evolution, while the outer disc has a net positive gravitational torque. Our global model of a self-consistently formed disc shows that it is also self-regulated in the late phase, so that it is near the Toomre stability limit, with a near-uniform Toomre parameter Q ≈ 1.5–2.0. Since the disc also has near-Keplerian rotation, and comparatively weak temperature variation, it maintains a near-power-law surface density profile proportional to r ^−3/2.     

Precessing warped accretion discs in X-ray binaries

We study the radiation-driven warping of accretion discs in the context of X-ray binaries. The latest evolutionary equations are adopted, which extend the classical alpha theory to time-dependent thin discs with non-linear warps. We also develop accurate, analytical expressions for the tidal torque and the radiation torque, including self-shadowing.
We investigate the possible non-linear dynamics of the system within the framework of bifurcation theory. First, we re-examine the stability of an initially flat disc to the Pringle instability. Then we compute directly the branches of non-linear solutions representing steadily precessing discs. Finally, we determine the stability of the non-linear solutions. Each problem involves only ordinary differential equations, allowing a rapid, accurate and well-resolved solution.
We find that radiation-driven warping is probably not a common occurrence in low-mass X-ray binaries. We also find that stable, steadily precessing discs exist for a narrow range of parameters close to the stability limit. This could explain why so few systems show clear, repeatable 'superorbital' variations. The best examples of such systems, Her X-1, SS 433 and LMC X-4, all lie close to the stability limit for a reasonable choice of parameters. Systems far from the stability limit, including Cyg X-2, Cen X-3 and SMC X-1, probably experience quasi-periodic or chaotic variability as first noticed recently by Wijers and Pringle. We show that radiation-driven warping provides a coherent and persuasive framework but that it does not provide a generic explanation for the long-term variabilities in all X-ray binaries.     

The paradox of the scale-free discs

Scale-free discs have no preferred length or time-scale. The question has been raised whether such discs have a continuum of unstable linear modes or perhaps no unstable modes at all. We resolve this paradox by analysing the particular case of a gaseous, isentropic disc with a completely flat rotation curve (the Mestel disc) exactly . The heart of the matter is this: what are the correct boundary conditions to impose at the origin or central cusp? We argue that the linear stability problem is ill-posed and that similar ambiguities may afflict general disc models with power-law central cusps. From any finite radius, waves reach the origin after finite time but with logarithmically divergent phase. Instabilities exist, but their pattern speeds depend upon an undetermined phase with which waves are reflected from the origin. For any definite choice of this phase, there is an infinite but discrete set of growing modes. The ratio of growth rate to pattern speed is independent of the central phase. This ratio is derived in closed form for non-self-gravitating normal modes and is shown to agree with approximate results obtained from the shearing sheet in the short-wavelength limit. This provides the first exact, analytically solved stability analysis for a differentially rotating disc. For self-gravitating normal modes, the ratio of growth rate to pattern is found numerically by solving recurrence relations in Mellin-transform space.     

Luminous hot accretion discs

We find a new two-temperature hot branch of equilibrium solutions for stationary accretion discs around black holes. In units of Eddington accretion rate defined as 10 L _Edd c ², the accretion rates to which these solutions correspond are within the range ̇ ₁≲ ̇ ≲1, where ̇ ₁ is the critical rate of advection-dominated accretion flow (ADAF). In these solutions, the energy loss rate of the ions by Coulomb energy transfer between the ions and electrons is larger than the viscously heating rate and it is the advective heating together with the viscous dissipation that balances the Coulomb cooling of ions. When ̇ ₁≲ ̇ ≲ ̇ ₂, where ̇ ₂∼5 ̇ ₁<1, the accretion flow remains hot throughout the disc. When ̇ ₂≲ ̇ ≲1, Coulomb interaction will cool the inner region of the disc within a certain radius ( r _tr∼several tens of Schwarzschild radii or larger depending on the accretion rate and the outer boundary condition) and the disc will collapse on to the equatorial plane and form an optically thick cold annulus. Compared with ADAF, these hot solutions are much more luminous because of the high accretion rate and efficiency; therefore, we call them luminous hot accretion discs.     

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 non-linear 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.