Transport In Accretion Discs Research Articles

Unified treatment of mean-field dynamo and angular-momentum transport in magnetorotational instability-driven turbulence.

Magnetorotational instability-driven (MRI-driven) turbulence and dynamo phenomena are analyzed using direct statistical simulations. Our approach begins by developing a unified mean-field model that combines the traditionally decoupled problems of the large-scale dynamo and angular momentum transport in accretion disks. The model consists of a hierarchical set of equations, capturing up to the second-order correlators, while a statistical closure approximation is employed to model the three-point correlators. We highlight the web of interactions that connect different components of stress tensors-Maxwell, Reynolds, and Faraday-through shear, rotation, correlators associated with mean fields, and nonlinear terms. We determine the dominant interactions crucial for the development and sustenance of MRI turbulence. Our general mean-field model for the MRI-driven system allows for a self-consistent construction of the electromotive force, inclusive of inhomogeneities and anisotropies. Within the realm of large-scale magnetic field dynamo, we identify two key mechanisms-the rotation-shear-current effect and the rotation-shear-vorticity effect-that are responsible for generating the radial and vertical magnetic fields, respectively. We provide the explicit (nonperturbative) form of the transport coefficients associated with each of these dynamo effects. Notably, both of these mechanisms rely on the intrinsic presence of large-scale vorticity dynamo within MRI turbulence.

Saturation of the magnetorotational instability and the origin of magnetically elevated accretion discs

ABSTRACT We propose that the strength of angular momentum transport in accretion discs threaded by net vertical magnetic field is determined by a self-regulation mechanism: the magnetorotational instability (MRI) grows until its own turbulent resistivity damps the fastest growing mode on the scale of the disc thickness. Given weak assumptions as to the structure of MRI-derived turbulence, supported by prior simulation evidence, the proposed mechanism reproduces the known scaling of the viscous α-parameter, $\alpha \propto \beta _z^{-1/2}$. Here, $\beta _z = 8\pi p_{\rm g}/B_{z0}^2$ is the initial plasma β-parameter on the disc mid-plane, Bz0 is the net field, and pg is the mid-plane gas pressure. We generalize the argument to discs with strong suprathermal toroidal magnetic fields, where the MRI growth rate is modified from the weak-field limit. Additional sources of turbulence are required if such discs are to become magnetically elevated, with the increased scale heights near the mid-plane that are seen in simulations. We speculate that tearing modes, associated with current sheets broadened by the effective resistivity, are a possible source of enhanced turbulence in elevated discs.

A Nonlocal Magneto-curvature Instability in a Differentially Rotating Disk

A global mode is shown to be unstable to nonaxisymmetric perturbations in a differentially rotating Keplerian disk containing either vertical or azimuthal magnetic fields. In an unstratified cylindrical disk model, using both global eigenvalue stability analysis and linear global initial-value simulations, it is demonstrated that this instability dominates at strong magnetic fields where local standard magnetorotational instability (MRI) becomes stable. Unlike the standard MRI mode, which is concentrated in the high flow shear region, these distinct global modes (with low azimuthal mode numbers) are extended in the global domain and are Alfvén-continuum-driven unstable modes. As its mode structure and relative dominance over MRI are inherently determined by the global spatial curvature as well as the flow shear in the presence of a magnetic field, we call it the magneto-curvature (magneto-spatial-curvature) instability. Consistent with the linear analysis, as the field strength is increased in the nonlinear simulations, a transition from MRI-driven turbulence to a state dominated by global nonaxisymmetric modes is obtained. This global instability could therefore be a source of nonlinear transport in accretion disks at a higher magnetic field than predicted by local models.

Modelling hystereses observed during dwarf nova outbursts

Context. Although the disc instability model is widely accepted as the explanation for dwarf nova outbursts, it is still necessary to compare its predictions to observations because many of the constraints on angular momentum transport in accretion discs are derived from the application of this model to real systems. Aims. We test the predictions of the model concerning the multicolour time evolution of outbursts for two well-observed systems, SS Cyg and VW Hyi. Methods. We calculate the multicolour evolution of dwarf nova outbursts using the disc instability model and taking into account the contribution from the irradiated secondary, the white dwarf and the hot spot. Results. Observations definitely show the existence of a hysteresis in the optical colour–magnitude diagram during the evolution of dwarf nova outbursts. We find that the disc instability model naturally explains the existence and the orientation of this hysteresis. For the specific cases of SS Cyg and VW Hyi, the colour and magnitude ranges covered during the evolution of the system are in reasonable agreement with observations. However, the observed colours are bluer than observed near the peak of the outbursts, as in steady systems, and the amplitude of the hysteresis cycle is smaller than observed. The predicted colours significantly depend on the assumptions made for calculating the disc spectrum during rise, and on the magnitude of the secondary irradiation for the decaying part of the outburst. Conclusions. Improvements in the spectral disc models are strongly needed if the system evolution in the UV is to be addressed.

Angular momentum transport in accretion disks: a hydrodynamical perspective

The radial transport of angular momentum in accretion disk is a fundamental process in the universe. It governs the dynamical evolution of accretion disks and has implications for various issues ranging from the formation of planets to the growth of supermassive black holes. While the importance of magnetic fields for this problem has long been demonstrated, the existence of a source of transport solely hydrodynamical in nature has proven more difficult to establish and to quantify. In recent years, a combination of results coming from experiments, theoretical work and numerical simulations has dramatically improved our understanding of hydrodynamically mediated angular momentum transport in accretion disk. Here, based on these recent developments, we review the hydrodynamical processes that might contribute to transporting angular momentum radially in accretion disks and highlight the many questions that are still to be answered.

Supertransient magnetohydrodynamic turbulence in Keplerian shear flows: The role of the Hall effect

Space and astrophysical plasmas are frequently found in the regime of differential rotation, where the presence of a magnetic field can result in the magnetorotational instability, directly responsible for important phenomena such as turbulent angular-momentum transport in accretion disks. In the absence of an imposed magnetic field, a nonlinear dynamo is necessary for this transport mechanism to take place. In protoplanetary disks there are regions with high density and very low temperatures, which are two necessary conditions for the Hall effect to operate, affecting the development of the dynamo and the associated turbulence. In this work we perform local magnetohydrodynamic (MHD) simulations to study transition to weak turbulence in Keplerian shear flows with Hall effect. The Hall effect is shown to lead the system to long-lived turbulent transients whose decay time follows an exponential dependence on the magnetic Reynolds number and the Hall parameter.

Magnetic Field Transport in Accretion Disks

Abstract The leading models for launching astrophysical jets rely on strong poloidal magnetic fields threading the central parts of their host accretion disks. Numerical simulations of magneto-rotationally turbulent disks suggest that such fields are actually advected from the environment by the accreting matter rather than generated by internal dynamos. This is puzzling from a theoretical point of view, since the reconnection of the radial field across the midplane should cause an outward drift on timescales much shorter than the accretion time. We suggest that a combination of effects are responsible for reducing the radial field near the midplane, causing efficient inward advection of the poloidal field. Magnetic buoyancy in subsonic turbulence pushes the field lines away from the midplane, decreasing the large-scale radial field in the main body of the disk. In magneto-rotationally driven turbulence, magnetic buoyancy dominates over the effects of turbulent pumping, which works against it, and turbulent diamagnetism, which works with it, in determining the vertical drift of the magnetic field. Balancing buoyancy with diffusion implies that the bending angle of the large-scale poloidal field can be very large near the surface, as required for outflows, but vanishes near the midplane, which impedes turbulent reconnection and outward diffusion. This effect becomes less efficient as the poloidal flux increases. This suggests that accretion disks are less likely to form jets if they have a modest ratio of outer to inner radii or if the ambient field is very weak. The former effect is probably responsible for the scarcity of jets in cataclysmic variable systems.

Magnetic field gradient effects on the magnetorotational instability

Magnetorotational instability (MRI), also known as the Balbus–Hawley instability, is thought to have an important role on the initiation of turbulence and angular momentum transport in accretion disks. In this work, we investigate the effect of the magnetic field gradient in the azimuthal direction on MRI. We solve the magnetohydrodynamic equations by including the azimuthal component of the field gradient. We find the dispersion relation and calculate the growth rates of the instability numerically. The inclusion of the azimuthal magnetic field gradient produces a new unstable region in the wavenumber space. It also modifies the growth rate and the wavelength range of the unstable mode: the higher the magnitude of the field gradient, the greater the growth rate and the wider the unstable wavenumber range. Such a gradient in the magnetic field may be important in T Tauri disks, where the stellar magnetic field has an axis that is misaligned with respect to the rotation axis of the disk.

Erratum: On characterizing non-locality and anisotropy for the magnetorotational instability

The extent to which angular momentum transport in accretion discs is primarily local or nonlocal and what determines this is an important avenue of study for understanding accretion engines. Taking a step along this path, we analyse simulations of the magnetorotational instability (MRI) by calculating energy and stress power spectra in stratified isothermal shearing box simulations in several new ways. We divide our boxes in two regions, disc and corona wherethediscistheMRIunstableregionandcoronaisthemagneticallydominatedregion.We calculate the fractional power in different quantities, including magnetic energy and Maxwell stresses and find that they are dominated by contributions from the lowest wave numbers. This is even more dramatic for the corona than the disc, suggesting that transport in the corona regionisdominatedbylargerstructuresthanthedisc.Bycalculatingaveragedpowerspectrain one direction of k space at a time, we also show that the MRI turbulence is strongly anisotropic on large scales when analysed by this method, but isotropic on small scales. Although the shearing box itself is meant to represent a local section of an accretion disc, the fact that the stress and energy are dominated by the largest scales highlights that the locality is not captured within the box. This helps to quantify the intuitive importance of global simulations for addressing the question of locality of transport, for which similar analyses can be performed.

Angular momentum transport and large eddy simulations in magnetorotational turbulence: the small Pm limit

Angular momentum transport in accretion discs is often believed to be due to magnetohydrodynamic turbulence mediated by the magnetorotational instability. Despite an abundant literature on the MRI, the parameters governing the saturation amplitude of the turbulence are poorly understood and the existence of an asymptotic behavior in the Ohmic diffusion regime is not clearly established. We investigate the properties of the turbulent state in the small magnetic Prandtl number limit. Since this is extremely computationally expensive, we also study the relevance and range of applicability of the most common subgrid scale models for this problem. Unstratified shearing boxes simulations are performed both in the compressible and incompressible limits, with a resolution up to 800 cells per disc scale height. The latter constitutes the largest resolution ever attained for a simulation of MRI turbulence. In the presence of a mean magnetic field threading the domain, angular momentum transport converges to a finite value in the small Pm limit. When the mean vertical field amplitude is such that {\beta}, the ratio between the thermal and magnetic pressure, equals 1000, we find {\alpha}~0.032 when Pm approaches zero. In the case of a mean toroidal field for which {\beta}=100, we find {\alpha}~0.018 in the same limit. Both implicit LES and Chollet-Lesieur closure model reproduces these results for the {\alpha} parameter and the power spectra. A reduction in computational cost of a factor at least 16 (and up to 256) is achieved when using such methods. MRI turbulence operates efficiently in the small Pm limit provided there is a mean magnetic field. Implicit LES offers a practical and efficient mean of investigation of this regime but should be used with care, particularly in the case of a vertical field. Chollet-Lesieur closure model is perfectly suited for simulations done with a spectral code.

Accretion disc viscosity: a limit on the anisotropy

Observations of warped discs can give insight into the nature of angular momentum transport in accretion discs. Only a few objects are known to show strong periodicity on long timescales, but when such periodicity is present it is often attributed to precession of the accretion disc. The X-ray binary Hercules X-1/HZ Herculis (Her X-1) is one of the best examples of such periodicity and has been linked to disc precession since it was first observed. By using the current best-fitting models to Her X-1, which invoke precession driven by radiation warping, I place a constraint on the effective viscosities that act in a warped disc. These effective viscosities almost certainly arise due to turbulence induced by the magneto-rotational instability. The constraints derived here are in agreement with analytical and numerical investigations into the nature of magneto-hydrodynamic disc turbulence, but at odds with some recent global simulations.

MAGNETIC FLUX CONCENTRATION AND ZONAL FLOWS IN MAGNETOROTATIONAL INSTABILITY TURBULENCE

Accretion disks are likely threaded by external vertical magnetic flux, which enhances the level of turbulence via the magnetorotational instability (MRI). Using shearing-box simulations, we find that such external magnetic flux also strongly enhances the amplitude of banded radial density variations known as zonal flows. Moreover, we report that vertical magnetic flux is strongly concentrated toward low-density regions of the zonal flow. Mean vertical magnetic field can be more than doubled in low-density regions, and reduced to nearly zero in high density regions in some cases. In ideal MHD, the scale on which magnetic flux concentrates can reach a few disk scale heights. In the non-ideal MHD regime with strong ambipolar diffusion, magnetic flux is concentrated into thin axisymmetric shells at some enhanced level, whose size is typically less than half a scale height. We show that magnetic flux concentration is closely related to the fact that the magnetic diffusivity of the MRI turbulence is anisotropic. In addition to a conventional Ohmic-like turbulent resistivity, we find that there is a correlation between the vertical velocity and horizontal magnetic field fluctuations that produces a mean electric field that acts to anti-diffuse the vertical magnetic flux. The anisotropic turbulent diffusivity has analogies to the Hall effect, and may have important implications for magnetic flux transport in accretion disks. The physical origin of magnetic flux concentration may be related to the development of channel flows followed by magnetic reconnection, which acts to decrease the mass-to-flux ratio in localized regions. The association of enhanced zonal flows with magnetic flux concentration may lead to global pressure bumps in protoplanetary disks that helps trap dust particles and facilitates planet formation.

SNOW LINES AS PROBES OF TURBULENT DIFFUSION IN PROTOPLANETARY DISKS

Sharp chemical discontinuities can occur in protoplanetary discs, particularly at `snow-lines' where a gas-phase species freezes out to form ice grains. Such sharp discontinuities will diffuse out due to the turbulence suspected to drive angular momentum transport in accretion discs. We demonstrate that the concentration gradient - in the vicinity of the snow-line - of a species present outside a snow-line but destroyed inside is strongly sensitive to the level of turbulent diffusion (provided the chemical and transport time-scales are decoupled) and provides a direct measurement of the radial `Schmidt number' (the ratio of the angular momentum transport to radial turbulent diffusion). Taking as an example the tracer species N$_2$H$^+$, which is expected to be destroyed inside the CO snow-line (as recently observed in TW Hya) we show that ALMA observations possess significant angular resolution to constrain the Schmidt number. Since different turbulent driving mechanisms predict different Schmidt numbers, a direct measurement of the Schmidt number in accretion discs would allow inferences about the nature of the turbulence to be made.

LINEAR STABILITY OF MAGNETIZED MASSIVE PROTOPLANETARY DISKS

Magneto-rotational instability (MRI) and gravitational instability (GI) are the two principle routes to turbulent angular momentum transport in accretion disks. Protoplanetary disks may develop both. This paper aims to reinvigorate interest in the study of magnetized massive protoplanetary disks, starting from the basic issue of stability. The local linear stability of a self-gravitating, uniformly magnetized, differentially rotating, three-dimensional stratified disk subject to axisymmetric perturbations is calculated numerically. The formulation includes resistivity. It is found that the reduction in the disk thickness by self-gravity can decrease MRI growth rates; the MRI becomes global in the vertical direction, and MRI modes with small radial length scales are stabilized. The maximum vertical field strength that permits the MRI in a strongly self-gravitating polytropic disk with polytropic index $\Gamma=1$ is estimated to be $B_{z,\mathrm{max}} \simeq c_{s0}\Omega\sqrt{\mu_0/16\pi G} $, where $c_{s0}$ is the midplane sound speed and $\Omega$ is the angular velocity. In massive disks with layered resistivity, the MRI is not well-localized to regions where the Elsasser number exceeds unity. For MRI modes with radial length scales on the order of the disk thickness, self-gravity can enhance density perturbations, an effect that becomes significant in the presence of a strong toroidal field, and which depends on the symmetry of the underlying MRI mode. In gravitationally unstable disks where GI and MRI growth rates are comparable, the character of unstable modes can transition smoothly between MRI and GI. Implications for non-linear simulations are discussed briefly.

Global simulations of magnetorotational turbulence - III. Influence of field configuration and mass injection

The stresses produced by magnetorotational turbulence can provide effective angular momentum transport in accretion disks. However, questions remain about the ability of simulated disks to reproduce observationally inferred stress-to-gas-pressure ratios. In this paper we present a set of high resolution global magnetohydrodynamic disk simulations which are initialised with different field configurations: purely toroidal, vertical field lines, and nested poloidal loops. A mass source term is included which allows the total disk mass to equilibrate in simulations with long run times, and also enables the impact of rapid mass injection to be explored. Notably different levels of angular momentum transport are observed during the early-time transient disk evolution. However, given sufficient time to relax, the different models evolve to a statistically similar quasi-steady state with a stress-to-gas-pressure ratio, $\alpha \sim 0.032-0.036$. The indication from our results is that {\it steady, isolated} disks may be unable to maintain a large-scale magnetic field or produce values for the stress-to-gas-pressure ratio implied by some observations. Supplementary simulations exploring the influence of trapping magnetic field, injecting vertical field, and rapidly injecting additional mass into the disk show that large stresses ($\alpha \sim 0.1-0.25$) can be induced by these mechanisms. The simulations highlight the common late-time evolution and characteristics of turbulent disks for which the magnetic field is allowed to evolve freely. If the boundaries of the disk, the rate of injection of magnetic field, or the rate of mass replenishment are modified to mimic astrophysical disks, markedly different disk evolution occurs.

On characterizing non-locality and anisotropy for the magnetorotational instability

The extent to which angular momentum transport in accretion discs is primarily local or non-local and what determines this is an important avenue of study for understanding accretion engines. Taking a step along this path, we analyse simulations of the magnetorotational instability (MRI) by calculating energy and stress power spectra in stratified isothermal shearing box simulations in several new ways. We divide our boxes in two regions, disc and corona where the disc is the MRI unstable region and corona is the magnetically dominated region. We calculate the fractional power in different quantities, including magnetic energy and Maxwell stresses and find that they are dominated by contributions from the lowest wave numbers. This is even more dramatic for the corona than the disc, suggesting that transport in the corona region is dominated by larger structures than the disc. By calculating averaged power spectra in one direction of k space at a time, we also show that the MRI turbulence is strongly anisotropic on large scales when analysed by this method, but isotropic on small scales. Although the shearing box itself is meant to represent a local section of an accretion disc, the fact that the stress and energy are dominated by the largest scales highlights that the locality is not captured within the box. This helps to quantify the intuitive importance of global simulations for addressing the question of locality of transport, for which similar analyses can be performed.

Global simulations of magnetorotational turbulence – I. Convergence and the quasi-steady state

Magnetorotational turbulence provides a viable mechanism for angular momentum transport in accretion disks. We present global, three dimensional (3D), MHD accretion disk simulations that investigate the dependence of the turbulent stresses on resolution. Convergence in the time-and-volume-averaged stress-to-gas-pressure ratio, at a value of $\sim0.04$, is found for a model with radial, vertical, and azimuthal resolution of 12-51, 27, and 12.5 cells per scale-height (the simulation mesh is such that cells per scale-height varies in the radial direction). A control volume analysis is performed on the main body of the disk (|z|<2H) to examine the production and removal of magnetic energy. Maxwell stresses in combination with the mean disk rotation are mainly responsible for magnetic energy production, whereas turbulent dissipation (facilitated by numerical resistivity) predominantly removes magnetic energy from the disk. Re-casting the magnetic energy equation in terms of the power injected by Maxwell stresses on the boundaries of, and by Lorentz forces within, the control volume highlights the importance of the boundary conditions (of the control volume). The different convergence properties of shearing-box and global accretion disk simulations can be readily understood on the basis of choice of boundary conditions and the magnetic field configuration. Periodic boundary conditions restrict the establishment of large-scale gradients in the magnetic field, limiting the power that can be delivered to the disk by Lorentz forces and by stresses at the surfaces. The factor of three lower resolution required for convergence in turbulent stresses for our global disk models compared to stratified shearing-boxes is explained by this finding. (Abridged)

Extending the Range of the Inductionless Magnetorotational Instability

The magnetorotational instability (MRI) can destabilize hydrodynamically stable rotational flows, thereby allowing angular momentum transport in accretion disks. A notorious problem for the MRI is its questionable applicability in regions with low magnetic Reynolds number. Using the WKB method, we extend the range of applicability of the MRI by showing that the inductionless versions of the MRI, such as the helical MRI and the azimuthal MRI, can easily destabilize Keplerian profiles ∝r(-3/2) if the radial profile of the azimuthal magnetic field is only slightly modified from the current-free profile ∝r(-1). This way we further show how the formerly known lower Liu limit of the critical Rossby number Ro≈-0.828 connects naturally with the upper Liu limit Ro≈+4.828.

The viscosity parameterαand the properties of accretion disc outbursts in close binaries

The physical mechanisms driving angular momentum transport in accretion discs are still unknown. Although it is generally accepted that, in hot discs, the turbulence triggered by the magneto-rotational instability is at the origin of the accretion process in Keplerian discs, it has been found that the values of the stress-to-pressure ratio (the alpha "viscosity" parameter) deduced from observations of outbursting discs are an order of magnitude higher than those obtained in numerical simulations. We test the conclusion about the observation-deduced value of alpha using a new set of data and comparing the results with model outbursts. We analyse a set of observations of dwarf-nova and AM CVn star outbursts and from the measured decay times determine the hot-disc viscosity parameter alpha_h. We determine if and how this method is model dependent. From the dwarf-nova disc instability model we determine an amplitude vs recurrence-time relation and compare it to the empirical Kukarkin-Parenago relation between the same, but observed, quantities. We found that all methods we tried, including the one based on the amplitude vs recurrence-time relation, imply alpha_h ~ 0.1 - 0.2 and exclude values an order of magnitude lower. The serious discrepancy between the observed and the MRI-calculated values of the accretion disc viscosity parameter alpha is therefore real since there can be no doubt about the validity of the values deduced from observations of disc outbursts.

Wave modes from the magnetorotational instability in accretion discs

AbstractThe magnetorotational instability (MRI) is widely believed to be the source of turbulence in accretion discs. This turbulence is responsible for the anomalous angular momentum transport in accretion discs. The turbulence will affect other aspects of the dynamics of the disc as well, and we will concentrate on two such issues: a) what kind of oscillations can be excited by the turbulence itself, and b) how the turbulence is interacting with modes that have been excited by some other agent. This is of interest in understanding the quasi-periodic oscillations (QPOs) that have been observed in the X-ray light curves of accreting neutron star and black hole binaries. We carry out local three dimensional (3D) magnetohydrodynamic simulations of a keplerian differentially rotating accretion disc, using a shearing box configuration taking in account the effects of the vertical stratification.