Non-linear dynamics of the corotation torque

The excitation of spiral waves by an external perturbation in a disc deposits angular momentum in the vicinity of the corotation resonance (the radius where the speed of a rotating pattern matches the local rotation rate). We use matched asymptotic expansions to derive a reduced model that captures non-linear dynamics of the resulting torque and fluid motions. The model is similar to that derived for forced Rossby wave critical layers in geophysical fluid dynamics. Using the model we explore the saturation of the corotation torque, which occurs when the background potential (specific) vorticity is redistributed by the disturbance. We also consider the effects of dissipation. If there is a radial transport of potential vorticity, the corotation torque does not saturate. The main application is to the creation, growth and migration of protoplanets within discs like the primordial solar nebula. The disturbance also nucleates vortices in the vicinity of corotation, which may spark further epochs of planet formation.     

Corotational damping of discoseismic c modes in black hole accretion discs

Discoseismic c modes in accretion discs have been invoked to explain low-frequency variabilities observed in black hole X-ray binaries. These modes are trapped in the innermost region of the disc and have frequencies much lower than the rotation frequency at the disc inner radius. We show that because the trapped waves can tunnel through the evanescent barrier to the corotational wave zone, the c modes are damped due to wave absorption at the corotation resonance. We calculate the corotational damping rates of various c modes using the Wentzel-Kramers-Brillouin (WKB) approximation. The damping rate varies widely depending on the mode frequency, the black hole spin parameter and the disc sound speed, and is generally much less than 10 per cent of the mode frequency. A sufficiently strong excitation mechanism is needed to overcome this corotational damping and make the mode observable.     

The excitation, propagation and dissipation of waves in accretion discs: the non-linear axisymmetric case

We analyse the non-linear propagation and dissipation of axisymmetric waves in accretion discs using the ZEUS-2D hydrodynamics code. The waves are numerically resolved in the vertical and radial directions. Both vertically isothermal and thermally stratified accretion discs are considered. The waves are generated by means of resonant forcing, and several forms of forcing are considered. Compressional motions are taken to be locally adiabatic  ( γ =5/3)  . Prior to non-linear dissipation, the numerical results are in excellent agreement with the linear theory of wave channelling in predicting the types of modes that are excited, the energy flux by carried by each mode, and the vertical wave energy distribution as a function of radius. In all cases, waves are excited that propagate on both sides of the resonance (inwards and outwards). For vertically isothermal discs, non-linear dissipation occurs primarily through shocks that result from the classical steepening of acoustic waves. For discs that are substantially thermally stratified, wave channelling is the primary mechanism for shock generation. Wave channelling boosts the Mach number of the wave by vertically confining the wave to a small cool region at the base of the disc atmosphere. In general, outwardly propagating waves with Mach numbers near resonance  ℳ_r≳0.01  undergo shocks within a distance of order the resonance radius.     

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.     

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.     

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.     

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 .     

Magnetic and vertical shear instabilities in accretion discs

The stability properties of magnetized discs rotating with angular velocity Ω = Ω( s ,  z ), dependent on both the radial and the vertical coordinates s and z , are considered. Such a rotation law is adequate for many astrophysical discs (e.g., galactic and protoplanetary discs, as well as accretion discs in binaries). In general, the angular velocity depends on height, even in thin accretion discs. A linear stability analysis is performed in the Boussinesq approximation, and the dispersion relation is obtained for short-wavelength perturbations. Any dependence of Ω on z can destabilize the flow. This concerns primarily small-scale perturbations for which the stabilizing effect of buoyancy is strongly suppressed due to the energy exchange with the surrounding plasma. For a weak magnetic field, instability of discs is mainly associated with vertical shear, whilst for an intermediate magnetic field the magnetic shear instability, first considered by Chandrasekhar and Velikhov, is more efficient. This instability is caused by the radial shear which is typically much stronger than the vertical shear. Therefore the growth time for the magnetic shear instability is much shorter than for the vertical shear instability. A relatively strong magnetic field can suppress both these instabilities. The vertical shear instability could be the source of turbulence in protoplanetary discs, where the conductivity is low.     

Stability of accretion discs threaded by a strong magnetic field

We study the stability of poloidal magnetic fields anchored in a thin accretion disc. The two-dimensional hydrodynamics in the disc plane is followed by a grid-based numerical simulation including the vertically integrated magnetic forces. The three-dimensional magnetic field outside the disc is calculated in a potential field approximation from the magnetic flux density distribution in the disc. For uniformly rotating discs we confirm numerically the existence of the interchange instability as predicted by Spruit, Stehle & Papaloizou . In agreement with predictions from the shearing sheet model, discs with Keplerian rotation are found to be stabilized by the shear, as long as the contribution of magnetic forces to support against gravity is small. When this support becomes significant, we find a global instability which transports angular momentum outwardly and allows mass to accrete inwardly. The instability takes the form of a m =1 rotating 'crescent', reminiscent of the purely hydrodynamic non-linear instability previously found in pressure-supported discs. A model where the initial surface mass density Σ( r ) and B _z ( r ) decrease with radius as power laws shows transient mass accretion during about six orbital periods, and settles into a state with surface density and field strength decreasing approximately exponentially with radius. We argue that this instability is likely to be the main angular momentum transport mechanism in discs with a poloidal magnetic field sufficiently strong to suppress magnetic turbulence. It may be especially relevant in jet-producing discs.     

The photoevaporation of discs around young stars in massive clusters

We present models in which the photoevaporation of discs around young stars by an external ultraviolet source (as computed by Adams et al.) is coupled with the internal viscous evolution of the discs. These models are applied to the case of the Orion Nebula Cluster (ONC), where the presence of a strong ultraviolet field from the central OB stars, together with a detailed census of circumstellar discs and photoevaporative flows, is well established. In particular we investigate the constraints that are placed on the initial disc properties in the ONC by the twin requirement that most stars possess a disc on a scale of a few astronomical unit (au), but that only a minority (<20 per cent) are resolved by Hubble Space Telescope ( HST ) at a scale of 50 au. We find that these requirements place very weak constraints on the initial radius distribution of circumstellar discs: the resulting size distribution readily forgets the initial radius distribution, owing to the strong positive dependence of the photoevaporation rate on disc radius. Instead, the scarcity of large discs reflects the relative scarcity of initially massive discs (with mass  >0.1 M_⊙  ). The ubiquity of discs on a small scale, on the other hand, mainly constrains the time-span over which the discs have been exposed to the ultraviolet field (<2 Myr). We argue that the discs that are resolved by HST represent a population of discs in which self-gravity was important at the time that the dominant central OB star switched on, but that, according to our models, self-gravity is unlikely to be important in these discs at the present time. We discuss the implications of our results for the so-called proplyd lifetime problem.     

The excitation of spiral density waves through turbulent fluctuations in accretion discs – I. WKBJ theory

We study and elucidate the mechanism of spiral density wave excitation in a differentially rotating flow with turbulence which could result from the magneto-rotational instability. We formulate a set of wave equations with sources that are only non-zero in the presence of turbulent fluctuations. We solve these in a shearing box domain, subject to the boundary conditions of periodicity in shearing coordinates, using a WKBJ method. It is found that, for a particular azimuthal wavelength, the wave excitation occurs through a sequence of regularly spaced swings during which the wave changes from leading to trailing form. This is a generic process that is expected to occur in shearing discs with turbulence. Trailing waves of equal amplitude propagating in opposite directions are produced, both of which produce an outward angular momentum flux that we give expressions for as functions of the disc parameters and azimuthal wavelength.
By solving the wave amplitude equations numerically, we justify the WKBJ approach for a Keplerian rotation law for all parameter regimes of interest. In order to quantify the wave excitation completely, the important wave source terms need to be specified. Assuming conditions of weak non-linearity, these can be identified and are associated with a quantity related to the potential vorticity, being the only survivors in the linear regime. Under the additional assumption that the source has a flat power spectrum at long azimuthal wavelengths, the optimal azimuthal wavelength produced is found to be determined solely by the WKBJ response and is estimated to be  2π H   , with H being the nominal disc scaleheight. In a following paper by Heinemann & Papaloizou, we perform direct three-dimensional simulations and compare results manifesting the wave excitation process and its source with the assumptions made and the theory developed here in detail, finding excellent agreement.     

Pseudo-viscous modelling of self-gravitating discs and the formation of low mass ratio binaries

We present analytic models for the local structure of self-regulated self-gravitating accretion discs that are subject to realistic cooling. Such an approach can be used to predict the secular evolution of self-gravitating discs (which can usefully be compared with future radiation hydrodynamical simulations) and to define various physical regimes as a function of radius and equivalent steady state accretion rate. We show that fragmentation is inevitable, given realistic rates of infall into the disc, once the disc extends to radii >70 au (in the case of a solar mass central object). Owing to the outward redistribution of disc material by gravitational torques, we also predict fragmentation at >70 au even in the case of low angular momentum cores which initially collapse to a much smaller radius. We point out that 70 au is close to the median binary separation and propose that such delayed fragmentation, at the point that the disc expands to >70 au, ensures the creation of low mass ratio companions that can avoid substantial further growth and consequent evolution towards unit mass ratio. We thus propose this as a promising mechanism for producing low mass ratio binaries, which, while abundant observationally, are severely underproduced in hydrodynamical models.     

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.     

Radiatively heated, protoplanetary discs with dead zones – I. Dust settling and thermal structure of discs around M stars

The irradiation of protoplanetary discs by central stars is the main heating mechanism for discs, resulting in their flared geometric structure. In a series of papers, we investigate the deep links between two-dimensional self-consistent disc structure and planetary migration in irradiated discs, focusing particularly on those around M stars. In this first paper, we analyse the thermal structure of discs that are irradiated by an M star by solving the radiative transfer equation by means of a Monte Carlo code. Our simulations of irradiated hydrostatic discs are realistic and self-consistent in that they include dust settling with multiple grain sizes  ( N = 15)  , the gravitational force of an embedded planet on the disc and the presence of a dead zone (a region with very low levels of turbulence) within it. We show that dust settling drives the temperature of the mid-plane from an   r ^−3/5  distribution (well mixed dust models) towards an   r ^−3/4  . The dead zone, meanwhile, leaves a dusty wall at its outer edge because dust settling in this region is enhanced compared to the active turbulent disc at larger disc radii. The disc heating produced by this irradiated wall provides a positive gradient region of the temperature in the dead zone in front of the wall. This is crucially important for slowing planetary migration because Lindblad torques are inversely proportional to the disc temperature. Furthermore, we show that low turbulence of the dead zone is self-consistently induced by dust settling, resulting in the Kelvin–Helmholtz instability (KHI). We show that the strength of turbulence arising from the KHI in the dead zone is  α= 10⁻⁵  .     

Simulations of the formation of stellar discs in the Galactic Centre via cloud–cloud collisions

Young massive stars in the central parsec of our Galaxy are best explained by star formation within at least one, and possibly two, massive self-gravitating gaseous discs. With help of numerical simulations, we here consider whether the observed population of young stars could have originated from a large angle collision of two massive gaseous clouds at   R ≃ 1 pc  from Sgr A*. In all the simulations performed, the post-collision gas flow forms an inner, nearly circular gaseous disc and one or two eccentric outer filaments, consistent with the observations. Furthermore, the radial stellar mass distribution is always very steep,  Σ_*∝ R ⁻²  , again consistent with the observations. All of our simulations produce discs that are warped by between 30° and 60°, in accordance with the most recent observations. The three-dimensional velocity structure of the stellar distribution is sensitive to initial conditions (e.g. the impact parameter of the clouds) and gas cooling details. For example, the runs in which the inner disc is fed intermittently with material possessing fluctuating angular momentum result in multiple stellar discs with different orbital orientations, contradicting the observed data. In all the cases the amount of gas accreted by our inner boundary condition is large, enough to allow Sgr A* to radiate near its Eddington limit over ∼10⁵ yr. This suggests that a refined model would have physically larger clouds (or a cloud and a disc such as the circumnuclear disc) colliding at a distance of a few parsecs rather than 1 pc as in our simulations.     

The dependence on system parameters of strange-mode instabilities in accretion discs

A systematic study of the dependence on disc parameters and input physics, such as opacity and the treatment of convection, of strange-mode instabilities in thin accretion discs, which have been discovered recently, is presented. The instabilities are found to exist for a wide range of parameters, are partly very robust, and their growth rates can reach the dynamical range. Even discs on galactic scales around massive black holes are affected by them. Two groups of instabilities can be distinguished, the first of which is related to the radiation-pressure-dominated part of the disc, and the second to helium/hydrogen ionization. By application of the NAR approximation, both of them can be shown to be of mechanical origin, and the classical κ -mechanism can be excluded as the instability mechanism. A heuristic model for strange-mode instabilities proposed in the context of stellar strange-mode instabilities in luminous stars seems to be applicable to the group associated with dominant radiation pressure.     

Simulations of superhumps and superoutbursts

We numerically study the tidal instability of accretion discs in close binary systems using a two-dimensional SPH code. We find that the precession rate of tidally unstable, eccentric discs does not only depend upon the binary mass ratio q . Although the (prograde) disc precession rate increases with the strength of the tidal potential, we find that increasing the shear viscosity ν also has a significant prograde effect. Increasing the disc temperature has a retrograde impact upon the precession rate.   We find that motion relative to the binary potential results in superhump-like, periodic luminosity variations in the outer reaches of an eccentric disc. The nature and location of the luminosity modulation are functions of ν. Light curves most similar to observations are obtained for ν values appropriate for a dwarf nova in outburst.   We investigate the thermal–tidal instability model for superoutburst. A dwarf nova outburst is simulated by instantaneously increasing ν, which causes a rapid radial expansion of the disc. Should the disc encounter the 3: 1 eccentric inner Lindblad resonance and become tidally unstable, then tidal torques become much more efficient at removing angular momentum from the disc. The disc then shrinks and M _d increases. The resulting increase in disc luminosity is found to be consistent with the excess luminosity of a superoutburst.     

An unconventional accretion disc dynamo

In the light of recent results from numerical simulations of accretion disc MHD turbulence, we revisit the problem of the configuration of large-scale magnetic fields resulting from an α Ω dynamo operating in a thin accretion disc. In particular, we analyse the consequences of the peculiar sign of the α -effect suggested by numerical simulations . We determine the symmetry of the fastest-growing modes in the kinematic dynamo approximation and, in the framework of an ' α -quenched' dynamo model, study the evolution of the magnetic field. We find that the resulting field for this negative polarity of the α -effect generally has dipole symmetry with respect to the disc midplane, although the existence of an equilibrium configuration depends on the properties of the turbulence. The role of magnetic field dragging is discussed and, finally, the presence of an external uniform magnetic field is included to address the issue of magneto centrifugal wind launching from accretion discs.     

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.