A non-linear theory of vertical resonances in accretion discs

An important and widely neglected aspect of the interaction between an accretion disc and a massive companion with a coplanar orbit is the vertical component of the tidal force. As shown by Lubow, the response of the disc to vertical forcing is resonant at certain radii, at which a localized torque is exerted, and from which a compressive wave (p mode) may be emitted. Although these vertical resonances are weaker than the corresponding Lindblad resonances, the   m =2  inner vertical resonance in a binary star is typically located within the tidal truncation radius of a circumstellar disc.
In this paper I develop a general theory of vertical resonances, allowing for non-linearity of the response, and dissipation by radiative damping and turbulent viscosity. The problem is reduced to a universal, non-linear ordinary differential equation with two real parameters. Solutions of the complex non-linear Airy equation are presented to illustrate the non-linear saturation of the resonance and the effects of dissipation. It is argued that the   m =2  inner vertical resonance is unlikely to truncate the disc in cataclysmic variable stars, but contributes to angular momentum transport and produces a potentially observable non-axisymmetric structure.     

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.     

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.     

The formation of an eccentric gap in a gas disc by a planet in an eccentric orbit

We investigate the effect of a planet on an eccentric orbit on a two-dimensional low-mass gaseous disc. At a planet eccentricity above the planet's Hill radius divided by its semimajor axis, we find that the disc morphology differs from that exhibited by a disc containing a planet in a circular orbit. An eccentric gap is created with eccentricity that can exceed the planet's eccentricity and precesses with respect to the planet's orbit. We find that a more massive planet is required to open a gap when the planet is on an eccentric orbit. We attribute this behaviour to spiral density waves excited at corotation resonances by the eccentric planet. These act to increase the disc's eccentricity and exert a torque opposite in sign to that exerted by the Lindblad resonances. The reduced torque makes it more difficult for waves driven by the planet to overcome viscous inflow in the disc.     

Eccentric discs in binaries with intermediate mass ratios: superhumps in the VY Sculptoris stars

We investigate the role of the eccentric disc resonance in systems with mass ratios q ≳1/4, and demonstrate the effects that changes in the mass flux from the secondary star have upon the disc radius and structure. The addition of material with low specific angular momentum to the outer edge of a disc restricts that disc radially. Should the mass flux from the secondary be reduced, it is possible for the disc in a system with mass ratio as large as 1/3 to expand to the 3:1 eccentric inner Lindblad resonance and for superhumps to be excited.     

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.     

Coupled galactic density-wave modes in a composite system of thin stellar and gaseous discs

A rotating disc galaxy is modelled as a composite system consisting of thin stellar and gaseous discs, which are described by a two-fluid modal formalism. The composite disc system is assumed to retain axisymmetry in the background equilibrium. General density-wave perturbations in the two discs are coupled through the mutual gravitational interaction. We study the basic properties of open and tight spiral density-wave modes in such a composite disc system. Within the Lindblad resonances, perturbation enhancements of surface mass density in stellar and gaseous discs are in phase; this is also true during the initial growth phase of density-wave perturbations. Outside the Lindblad resonances, there exists a possible spiral density-wave branch for which perturbation enhancements of surface mass density in stellar and gaseous discs are out of phase. We discuss implications of these results on the critical parameters for global star formation in barred and normal spiral galaxies and on magnetohydrodynamic density waves within the Lindblad resonances.     

Oscillation modes of relativistic slender tori

Accretion flows with pressure gradients permit the existence of standing waves which may be responsible for observed quasi-periodic oscillations (QPO's) in X-ray binaries. We present a comprehensive treatment of the linear modes of a hydrodynamic, non-self-gravitating, polytropic slender torus, with arbitrary specific angular momentum distribution, orbiting in an arbitrary axisymmetric space–time with reflection symmetry. We discuss the physical nature of the modes, present general analytic expressions and illustrations for those which are low order, and show that they can be excited in numerical simulations of relativistic tori. The mode oscillation spectrum simplifies dramatically for near Keplerian angular momentum distributions, which appear to be generic in global simulations of the magnetorotational instability. We discuss our results in light of observations of high frequency QPO's, and point out the existence of a new pair of modes which can be in an approximate 3:2 ratio for arbitrary black hole spins and angular momentum distributions, provided the torus is radiation pressure dominated. This mode pair consists of the axisymmetric vertical epicyclic mode and the lowest order axisymmetric breathing mode.     

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.     

The stability of accretion discs with inflow driven purely by magnetic winds

A model is presented for an accretion disc in which the inflow is driven purely by the angular momentum removed in a centrifugally accelerated magnetic wind. Turbulent discs around compact stars are considered, with the required magnetic field being generated in the disc by a simple dynamo. The turbulent magnetic Prandtl number, N _p, measures the ratio of turbulent viscosity to turbulent magnetic diffusivity. Formally, the hypothetical limit   N _p→ 0  corresponds to the magnetic wind torque dominating the viscous torque, but in practice the inflow is magnetically controlled for   N _p≲ 0.1  .
The suggestion by previous authors that purely magnetic wind-driven discs may be unstable is investigated. A detailed steady solution is found which allows perturbations to the thermal balance and vertical equilibrium to be calculated, and hence the effect of perturbations to the magnetic diffusivity, η, to be assessed. For a standard parametrized form of η, the wind-driven angular momentum balance is found to be linearly unstable. An increase in the inflow rate leads to increased bending of the poloidal magnetic field and an enhanced wind mass loss rate. This increases the angular momentum loss rate which drives further inflow. There is a resultant increase in η, due to the temperature perturbation, but this does not relieve field bending sufficiently to prevent the instability.     

Disc structure around strongly magnetic accretors: a full disc solution with turbulent diffusivity

The problem of the effect of a strongly magnetic star on a surrounding accretion disc is considered. For stellar rotation periods greater than a critical value, a numerical solution is found for a steady disc with turbulent magnetic diffusion, including electron scattering opacity and radiation pressure. Inside the corotation radius, the extraction of disc angular momentum by magnetic coupling to the star becomes strong and this leads to enhanced viscous stress and dissipation. The resulting elevated temperature causes electron scattering opacity and radiation pressure to become significant further from the star than in the absence of its magnetic field. The disc ends as its height increases rapidly due to the large central pressure, its density decreases and magnetically induced viscous instability occurs.     

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.     

Radial heating of a galactic disc by multiple spiral density waves

We consider a differentially rotating, 2D stellar disc perturbed by two steady-state spiral density waves moving at different pattern speeds. Our investigation is based on direct numerical integration of initially circular test-particle orbits. We examine a range of spiral strengths and spiral speeds and show that stars in this time-dependent gravitational field can be heated (their random motions increased). This is particularly noticeable in the simultaneous propagation of a two-armed spiral density wave near the corotation resonance (CR), and a weak four-armed one near the inner and outer 4:1 Lindblad resonances. In simulations with two spiral waves moving at different pattern speeds, we find: (i) the variance of the radial velocity,  σ² _R   , exceeds the sum of the variances measured from simulations with each individual pattern; (ii)  σ² _R   can grow with time throughout the entire simulation; (iii)  σ² _R   is increased over a wider range of radii compared to that seen with one spiral pattern; and (iv) particles diffuse radially in real space, whereas they do not when only one spiral density wave is present. Near the CR with the stronger, two-armed pattern, test-particles are observed to migrate radially. These effects take place at or near resonances of both spirals, so we interpret them as the result of stochastic motions. This provides a possible new mechanism for increasing the stellar velocity dispersion in galactic discs. If multiple spiral patterns are present in the Galaxy, we predict that there should be large variations in the stellar velocity dispersion as a function of radius.     

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.     

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.     

Confined chaotic motion in three-body resonances: trapping of trans-Neptunian material induced by gas-drag

Jiang & Yeh proposed gas-drag-induced resonant capture as a mechanism able to explain the dominant 3:2 resonance observed in the trans-Neptunian belt. Using a model of a disc–star–planet system they concluded that gaseous drag in a protoplanetary disc can trap trans-Neptunian object (TNO) embryos into the 3:2 resonance rather easily although it could not trap objects into the 2:1 resonance. Here we further investigate this scenario using numerical simulations within the context of the planar restricted four-body problem by including both present-day Uranus and Neptune. Our results show that mean motion and corotation resonances are possible and trapping into both the 3:2 and 2:1 resonances as well as other resonances is observed. The associated corotation centres may easily form larger planetesimals from smaller ones. Corotation resonances evolve into pure Lindblad resonances in a time-scale of 0.5 Myr. The non-linear corotation and mean motion resonances produced are very size selective. The 3:2 resonance is dominant for submetric particles but for larger particles the 2:1 resonance is stronger. In summary, our calculations show that confined chaotic motion around the resonances not only increases trapping efficiency but also the orbital eccentricities of the trapped material, modifying the relative abundance of trapped particles in different resonances. If we assume a more compact planetary system, instead of using the present-day values of the orbital elements of Uranus and Neptune, our results remain largely unchanged.     

Low T/|W| dynamical instability in differentially rotating stars: diagnosis with canonical angular momentum

We study the nature of non-axisymmetric dynamical instabilities in differentially rotating stars with both linear eigenmode analysis and hydrodynamic simulations in Newtonian gravity. We especially investigate the following three types of instability; the one-armed spiral instability, the low   T /| W |  bar instability, and the high   T /| W |  bar instability, where T is the rotational kinetic energy and W is the gravitational potential energy. The nature of the dynamical instabilities is clarified by using a canonical angular momentum as a diagnostic. We find that the one-armed spiral and the low   T /| W |  bar instabilities occur around the corotation radius, and they grow through the inflow of canonical angular momentum around the corotation radius. The result is a clear contrast to that of a classical dynamical bar instability in high   T /| W |  . We also discuss the feature of gravitational waves generated from these three types of instability.     

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.