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.     

Wave excitation in three-dimensional discs by external potential

We study the excitation of density and bending waves and the associated angular momentum transfer in gaseous discs with finite thickness by a rotating external potential. The disc is assumed to be isothermal in the vertical direction and has no self-gravity. The disc perturbations are decomposed into different modes, each characterized by the azimuthal index m and the vertical index n , which specifies the nodal number of the density perturbation along the disc normal direction. The   n = 0  modes correspond to the two-dimensional density waves previously studied by Goldreich & Tremaine and others. In a three-dimensional disc, waves can be excited at both Lindblad resonances (LRs; for modes with   n = 0, 1, 2, …  ) and vertical resonances (VRs; for the   n ≥ 1  modes only). The torque on the disc is positive for waves excited at outer Lindblad/vertical resonances and negative at inner Lindblad/vertical resonances. While the   n = 0  modes are evanescent around corotation, the   n ≥ 1  modes can propagate into the corotation region where they are damped and deposit their angular momenta. We have derived analytical expressions for the amplitudes of different wave modes excited at LRs and/or VRs and the resulting torques on the disc. It is found that for   n ≥ 1  , angular momentum transfer through VRs is much more efficient than LRs. This implies that in some situations (e.g. a circumstellar disc perturbed by a planet in an inclined orbit), VRs may be an important channel of angular momentum transfer between the disc and the external potential. We have also derived new formulae for the angular momentum deposition at corotation and studied wave excitations at disc boundaries.     

On corotation torques, horseshoe drag and the possibility of sustained stalled or outward protoplanetary migration

We study the torque on low-mass protoplanets on fixed circular orbits, embedded in a protoplanetary disc in the isothermal limit. We consider a wide range of surface density distributions including cases where the surface density increases smoothly outwards. We perform both linear disc response calculations and non-linear numerical simulations. We consider a large range of viscosities, including the inviscid limit, as well as a range of protoplanet mass ratios, with special emphasis on the co-orbital region and the corotation torque acting between disc and protoplanet.
For low-mass protoplanets and large viscosity, the corotation torque behaves as expected from linear theory. However, when the viscosity becomes small enough to enable horseshoe turns to occur, the linear corotation torque exists only temporarily after insertion of a planet into the disc, being replaced by the horseshoe drag first discussed by Ward. This happens after a time that is equal to the horseshoe libration period reduced by a factor amounting to about twice the disc aspect ratio. This torque scales with the radial gradient of specific vorticity, as does the linear torque, but we find it to be many times larger. If the viscosity is large enough for viscous diffusion across the co-orbital region to occur within a libration period, we find that the horseshoe drag may be sustained. If not, the corotation torque saturates leaving only the linear Lindblad torques. As the magnitude of the non-linear co-orbital torque (horseshoe drag) is always found to be larger than the linear torque, we find that the sign of the total torque may change even for mildly positive surface density gradients. In combination with a kinematic viscosity large enough to keep the torque from saturating, strong sustained deviations from linear theory and outward or stalled migration may occur in such cases.     

Forced oscillations in a hydrodynamical accretion disk and QPOs

This is the second of a series of papers aimed to look for an explanation on the generation of high frequency quasi-periodic oscillations (QPOs) in accretion disks around neutron star, black hole, and white dwarf binaries. The model is inspired by the general idea of a resonance mechanism in the accretion disk oscillations as was already pointed out by Abramowicz and Klu’zniak (2001). In a first paper (P'etri, 2005a, paper I), we showed that a rotating misaligned magnetic field of a neutron star gives rise to some resonances close to the inner edge of the accretion disk. In this second paper, we suggest that this process does also exist for an asymmetry in the gravitational potential of the compact object. We prove that the same physics applies, at least in the linear stage of the response to the disturbance in the system. This kind of asymmetry is well suited for neutron stars or white dwarfs possessing an inhomogeneous interior allowing for a deviation from a perfectly spherically symmetric gravitational field. After a discussion on the magnitude of this deformation applied to neutron stars, we show by a linear analysis that the disk initially in a cylindrically symmetric stationary state is subject to {three kinds of resonances: a corotation resonance, a Lindblad resonance due to a driven force and a parametric resonance}. In a second part, we focus on the linear response of a thin accretion disk in the 2D limit. {Waves are launched at the aforementioned resonance positions and propagate in some permitted regions inside the disk, according to the dispersion relation obtained by a WKB analysis}. In a last part, these results are confirmed and extended via non linear hydrodynamical numerical simulations performed with a pseudo-spectral code solving Euler's equations in a 2D cylindrical coordinate frame. {We found that for a weak potential perturbation, the Lindblad resonance is the only effective mechanism producing a significant density fluctuation}. In a last step, we replaced the Newtonian potential by the so called logarithmically modified pseudo-Newtonian potential in order to take into account some general-relativistic effects like the innermost stable circular orbit (ISCO). The latter potential is better suited to describe the close vicinity of a neutron star or a black hole. However, from a qualitative point of view, the resonance conditions remain the same. The highest kHz QPOs are then interpreted as the orbital frequency of the disk at locations where the response to the resonances are maximal. It is also found that strong gravity is not required to excite the resonances.     

On the width and shape of the corotation region for low-mass planets

We study the coorbital flow for embedded, low-mass planets. We provide a simple semi-analytic model for the corotation region, which is subsequently compared to high-resolution numerical simulations. The model is used to derive an expression for the half-width of the horseshoe region, x _s, which in the limit of zero softening is given by   x _s/ r _p= 1.68( q / h )^1/2  , where q is the planet to central star mass ratio, h is the disc aspect ratio and   r _p  is the orbital radius. This is in very good agreement with the same quantity measured from simulations. This result is used to show that horseshoe drag is about an order of magnitude larger than the linear corotation torque in the zero-softening limit. Thus, the horseshoe drag, the sign of which depends on the gradient of specific vorticity, is important for estimates of the total torque acting on the planet. We further show that phenomena, such as the Lindblad wakes, with a radial separation from corotation of approximately a pressure scaleheight H can affect x _s, even though for low-mass planets   x _s≪ H   . The effect is to distort streamlines and reduce x _s through the action of a back pressure. This effect is reduced for smaller gravitational softening parameters and planets of higher mass, for which x _s becomes comparable to H .     

The radial disc structure around a magnetic neutron star: analytic and semi-analytic solutions

The radial structure of a thin accretion disc is calculated in the presence of a central dipole magnetic field aligned with the rotation axis. The problem is treated using a modified expression for the turbulent magnetic diffusion, which allows the angular momentum equation to be integrated analytically. The governing algebraic equations are solved iteratively between 1 and 10⁴ stellar radii. An analytic approximation is provided that is valid near the disruption radius at about 100 stellar radii. At that point, which is approximately 60 per cent of the Alfvén radius and typically about 30 per cent of the corotation radius, the disc becomes viscously unstable. This instability results from the fact that both radiation pressure and opacity caused by electron scattering become important. This in turn is a consequence of the magnetic field which leads to an enhanced temperature in the inner parts. This is because the magnetic field gives rise to a strongly enhanced vertically integrated viscosity, so that the viscous torque can balance the magnetic torque.     

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.     

The spin period of EX Hydrae

We show that the spin period of the white dwarf in the magnetic cataclysmic variable (CV) EX Hydrae represents an equilibrium state in which the corotation radius is comparable with the distance from the white dwarf to the inner Lagrange point. We also show that a continuum of spin equilibria exists at which P _spin is significantly longer than ∼0.1 P _orb. Most systems occupying these equilibrium states should have orbital periods below the CV period gap, as observed.     

Temporal behaviour of global perturbations in compressible axisymmetric flows with free boundaries

The dynamics of small global perturbations in the form of a linear combination of a finite number of non‐axisymmetric eigenmodes is studied in the two‐dimensional approximation. The background flow is assumed to be an axisymmetric perfect fluid with adiabatic index γ = 5/3 rotating with a power law angular velocity distribution Γ ∝ r^–q , 1.5 < q < 2.0, confined by free boundaries in the radial direction. The substantial transient growth of acoustic energy of optimized perturbations is discovered. An optimal energy growth G is calculated numerically for a variety of parameters. Its value depends essentially on the perturbation azimuthal wavenumber m and increases for higher values of m. The closer the rotation profile to the Keplerian law, the larger growth factors can be obtained but over a longer time. The highest acoustic energy increase found numerically is of order ∼10² over ∼6 typical Keplerian periods. Slow neutral eigenmodes with corotation radius beyond the outer boundary mostly contribute to the transient growth. The revealed linear temporal behaviour of perturbations may play an important role in angular momentum transfer in toroidal flows near compact relativistic objects (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Implications of mean field accretion disc theory for vorticity and magnetic field growth

In addition to the scalar Shakura–Sunyaev α _ss turbulent viscosity transport term used in simple analytic accretion disc modelling, a pseudo-scalar transport term also arises. The essence of this term can be captured even in simple models for which vertical averaging is interpreted as integration over a half-thickness and each hemisphere is separately studied. The additional term highlights a complementarity between mean field magnetic dynamo theory and accretion disc theory treated as a mean field theory. Such pseudo-scalar terms have been studied, and can lead to large-scale magnetic field and vorticity growth. Here it is shown that vorticity can grow even in the simplest azimuthal and half-height integrated disc model, for which mean quantities depend only on radius. The simplest vorticity growth solutions seem to have scales and vortex survival times consistent with those required for facilitating planet formation. In addition, it is shown that, when the magnetic back-reaction is included to lowest order, the pseudo-scalar driving the magnetic field growth and that driving the vorticity growth will behave differently with respect to shearing and non-shearing flows: the former pseudo‐scalar can more easily reverse sign in the two cases.     

Interface modes and their instabilities in accretion disc boundary layers

We study global non-axisymmetric oscillation modes trapped near the inner boundary of an accretion disc. Observations indicate that some of the quasi-periodic oscillations (QPOs) observed in the luminosities of accreting compact objects (neutron stars, black holes and white dwarfs) are produced in the innermost regions of accretion discs or boundary layers. Two simple models are considered in this paper. The magnetosphere–disc model consists of a thin Keplerian disc in contact with a uniformly rotating magnetosphere with and low plasma density, while the star–disc model involves a Keplerian disc terminated at the stellar atmosphere with high density and small density scaleheight. We find that the interface modes at the magnetosphere–disc boundary are generally unstable due to Rayleigh–Taylor and/or Kelvin–Helmholtz instabilities. However, differential rotation of the disc tends to suppress Rayleigh–Taylor instability, and a sufficiently high disc sound speed (or temperature) is needed to overcome this suppression and to attain net mode growth. On the other hand, Kelvin–Helmholtz instability may be active at low disc sound speeds. We also find that the interface modes trapped at the boundary between a thin disc and an unmagnetized star do not suffer Rayleigh–Taylor or Kelvin–Helmholtz instability, but can become unstable due to wave leakage to large disc radii and, for sufficiently steep disc density distributions, due to wave absorption at the corotation resonance in the disc. The non-axisymmetric interface modes studied in this paper may be relevant to the high-frequency QPOs observed in some X-ray binaries and in cataclysmic variables.     

Studies of accretion flows around rotating black holes – II. Standing shocks in the pseudo-Kerr geometry

Standing, propagating or oscillating shock waves are common in accretion and winds around compact objects. We study the topology of all possible solutions using the pseudo-Kerr geometry. We present the parameter space spanned by the specific energy and angular momentum and compare it with that obtained from the full general relativity to show that the potential can work satisfactorily in fluid dynamics also, provided the polytropic index is suitably modified. We then divide the parameter space depending on the nature of the solution topology. We specifically study the nature of the standing Rankine–Hugoniot shocks. We also show that as the Kerr parameter is increased, the shock location generally moves closer to the black hole. In future, these solutions can be used as guidelines to test numerical simulations around compact objects.     

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.     

On an excitation mechanism for trapped inertial waves in discs around black holes

According to one model, high-frequency quasi-periodic oscillations (QPOs) can be identified with inertial waves, trapped in the inner regions of accretion discs around black holes due to relativistic effects. In order to be detected, their amplitudes need to reach large enough values via some excitation mechanism. We work out in detail a non-linear coupling mechanism suggested by Kato, in which a global warping or eccentricity of the disc has a fundamental role. These large-scale deformations combine with trapped modes to generate 'intermediate' waves of negative energy that are damped as they approach either their corotation resonance or the inner edge of the disc, resulting in amplification of the trapped waves. We determine the growth rates of the inertial modes, as well as their dependence on the spin of the black hole and the properties of the disc. Our results indicate that this coupling mechanism can provide an efficient excitation of trapped inertial waves, provided the global deformations reach the inner part of the disc with non-negligible amplitude.     

Chaotic zone boundary for low free eccentricity particles near an eccentric planet

We consider particles with low free or proper eccentricity that are orbiting near planets on eccentric orbits. Through collisionless particle integration, we numerically find the location of the boundary of the chaotic zone in the planet's corotation region. We find that the distance in semimajor axis between the planet and boundary depends on the planet mass to the 2/7 power and is independent of the planet eccentricity, at least for planet eccentricities below 0.3. Our integrations reveal a similarity between the dynamics of particles at zero eccentricity near a planet in a circular orbit and with zero free eccentricity particles near an eccentric planet. The 2/7th law has been previously explained by estimating the semimajor at which the first-order mean motion resonances are large enough to overlap. Orbital dynamics near an eccentric planet could differ due to first-order corotation resonances that have strength proportional to the planet's eccentricity. However, we find that the corotation resonance width at low free eccentricity is small; also the first-order resonance width at zero free eccentricity is the same as that for a zero-eccentricity particle near a planet in a circular orbit. This accounts for insensitivity of the chaotic zone width to planet eccentricity. Particles at zero free eccentricity near an eccentric planet have similar dynamics to those at zero eccentricity near a planet in a circular orbit.     

Non-axisymmetric instabilities in shocked adiabatic accretion flows

We investigate the linear stability of a shocked accretion flow on to a black hole in the adiabatic limit. Our linear analyses and numerical calculations show that, despite the post-shock deceleration, the shock is generally unstable to non-axisymmetric perturbations. The simulation results of Molteni, Tóth & Kuznetsov can be well explained by our linear eigenmodes. The mechanism of this instability is confirmed to be based on the cycle of acoustic waves between the corotation radius and the shock. We obtain an analytical formula to calculate the oscillation period from the physical parameters of the flow. We argue that the quasi-periodic oscillation should be a common phenomenon in accretion flows with angular momentum.     

On type-I migration near opacity transitions. A generalized Lindblad torque formula for planetary population synthesis

We give an expression for the Lindblad torque acting on a low-mass planet embedded in a protoplanetary disk that is valid even at locations where the surface density or temperature profile cannot be approximated by a power law, such as an opacity transition. At such locations, the Lindblad torque is known to suffer strong deviation from its standard value, with potentially important implications for type I migration, but the full treatment of the tidal interaction is cumbersome and not well suited to models of planetary population synthesis. The expression that we propose retains the simplicity of the standard Lindblad torque formula and gives results that accurately reproduce those of numerical simulations, even at locations where the disk temperature undergoes abrupt changes. Our study is conducted by means of customized numerical simulations in the low-mass regime, in locally isothermal disks, and compared to linear torque estimates obtained by summing fully analytic torque estimates at each Lindblad resonance. The functional dependence of our modified Lindblad torque expression is suggested by an estimate of the shift of the Lindblad resonances that mostly contribute to the torque, in a disk with sharp gradients of temperature or surface density, while the numerical coefficients of the new terms are adjusted to seek agreement with numerics. As side results, we find that the vortensity related corotation torque undergoes a boost at an opacity transition that can counteract migration, and we find evidence from numerical simulations that the linear corotation torque has a non-negligible dependency upon the temperature gradient, in a locally isothermal disk.     

A numerical solution of the dynamical instability of a thin accretion torus

We consider the non-axisymmetric, dynamical instability of a thin accretion torus with a non-zero shift of corotation radius. By numerical method we evaluated the wave number dependence of the linear rate of growth of instability and the co-rotation shift. The rate of growth is only slightly affected by the non-zero co-rotation shift, while the dispersion relation in the case of a shift is the same as that of the linear KdV equation. This shows that the “planet-like” solution found in numerical simulations of thin tori is very probably analogous to the soliton solution of the KdV equation.     

On spin-up/spin-down torque reversals in disc accreting pulsars

The recent BATSE observations of the spin-up and spin-down of accreting pulsars have shown that the standard formulation for the accretion torque as proposed by Ghosh &38; Lamb may need to be revised. The observations indicate alternate spin-up and spin-down phases driven by torques of similar magnitude and typically larger than the mean torque. The variations of the torque in systems such as Cen X-3 are difficult to explain in terms of changes of the mass accretion rate. The implication is that the torque does not depend on the accretion rate as in the GL model. In this paper we argue that the observed changes in the spin rate can result from stochastic transitions between two magnetospheric states. In particular, we show that intermediate magnetospheric systems are not admissible, because of a disc-induced magnetospheric instability which exists in a star–disc magnetic interaction system. This explains why torque reversal occurs in disc accreting pulsars with similar magnitudes.     

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.