Magnetic and vertical shear instabilities in accretion discs

Protoplanetary discs should exhibit a weak vertical variation in their rotation profiles. Typically, this ‘vertical shear’ issues from a baroclinic effect driven by the central star’s radiation field, but it might also arise during the launching of a magnetocentrifugal wind. As a consequence, protoplanetary discs are subject to a hydrodynamical instability, the ‘vertical shear instability’ (VSI), whose breakdown into turbulence could transport a moderate amount of angular momentum and facilitate, or interfere with, the process of planet formation. Magnetic fields may suppress the VSI, however, either directly via magnetic tension or indirectly through magnetorotational turbulence. On the other hand, protoplanetary discs exhibit notoriously low ionization fractions, and non-ideal effects, if sufficiently dominant, may come to the VSI’s rescue. In this paper, we develop a local linear theory that explores how non-ideal magnetohydrodynamics influences the VSI, while exciting additional diffusive shear instabilities. We derive a set of analytical criteria that establish when the VSI prevails, and then show how it can be applied to a representative global model of a protoplanetary disc. Our calculations suggest that within ∼10 au the VSI should have little trouble emerging in the main body of the disc, but beyond that, and in the upper regions of the disc, its onset depends sensitively on the size of the preponderant dust grains.

Context. Planets are thought to form inside weakly ionized regions of protoplanetary disks, where turbulence creates ideal conditions for solid growth. However, the nature of this turbulence is still uncertain. In fast cooling parts of this zone the vertical shear instability (VSI) can operate, inducing a low level of gas turbulence and large-scale gas motions. Resolving the kinematic signatures of active VSI could reveal the origin of turbulence in planet-forming disk regions. However, an exploration of kinematic signatures of the interplay between VSI and forming planets is needed for a correct interpretation of radio interferometric observations. A robust detection of VSI would lead the way to a deeper understanding of the impact of gas turbulence on planet formation. Aims. The objective of this study is to explore the effect of VSI on the disk substructures triggered by an embedded fairly massive planet. We focus on the impact of this interplay on CO kinematic observations with the ALMA interferometer. Methods. We conducted global 3D hydrodynamical simulations of VSI-unstable disks with and without embedded massive planets, exploring Saturn- and Jupiter-mass cases. We studied the effect of planets on the VSI gas dynamics, and made a comparison with viscous disks. Post-processing the simulations with a radiative transfer code, we examined the kinematic signatures expected in CO molecular line emission, varying disk inclination. Further, we simulated deep ALMA high-resolution observations of our synthetic images, to test the observability of VSI and planetary signatures. Results. The embedded planet produces a damping of the VSI along a radial region, most effective at the disk midplane. For the Saturn case, the VSI modes are distorted by the planet’s spirals producing mixed kinematic signatures. For the Jupiter case, the planet’s influence dominates the overall disk gas kinematics. Conclusions. The presence of massive planets embedded in the disk can weaken the VSI large-scale gas flows, limiting its observability in CO kinematic observations with ALMA.

The vertical shear instability (VSI) offers a potential hydrodynamic mechanism for angular momentum transport in protoplanetary disks (PPDs). The VSI is driven by a weak vertical gradient in the disk's orbital motion, but must overcome vertical buoyancy, a strongly stabilizing influence in cold disks, where heating is dominated by external irradiation. Rapid radiative cooling reduces the effective buoyancy and allows the VSI to operate. We quantify the cooling timescale tc needed for efficient VSI growth, through a linear analysis of the VSI with cooling in vertically global, radially local disk models. We find the VSI is most vigorous for rapid cooling with in terms of the Keplerian orbital frequency, , the disk's aspect-ratio, , the radial power-law temperature gradient, q, and the adiabatic index, γ. For longer tc, the VSI is much less effective because growth slows and shifts to smaller length scales, which are more prone to viscous or turbulent decay. We apply our results to PPD models where tc is determined by the opacity of dust grains. We find that the VSI is most effective at intermediate radii, from ∼5 to ∼50 AU with a characteristic growth time of ∼30 local orbital periods. Growth is suppressed by long cooling times both in the opaque inner disk and the optically thin outer disk. Reducing the dust opacity by a factor of 10 increases cooling times enough to quench the VSI at all disk radii. Thus the formation of solid protoplanets, a sink for dust grains, can impede the VSI.

\n The nonlinear evolution of the vertical shear instability in\naccretion discs is investigated using three-dimensional hydrodynamic\nsimulations. A vertical dependence of the angular velocity destabilizes the \ndisc and leads to the generation of velocity fluctuations enhancing\nthe angular momentum transport. The instability emerges\nin the numerical models for large radial perturbation wave numbers.\nThe growth time is a few tens of orbital revolutions.\n\n\n

The outer protoplanetary discs (PPDs) can be subject to the magnetorotational instability (MRI) and the vertical shear instability (VSI). While both processes can drive turbulence in the disc, existing numerical simulations have studied them separately. In this paper, we conduct global 3D non-ideal magnetohydrodynamic (MHD) simulations for outer PPDs, with ambipolar diffusion and instantaneous cooling, and hence conductive to both instabilities. Given the range of ambipolar Elsässer numbers (Am) explored, it is found that the VSI turbulence dominates over the MRI when ambipolar diffusion is strong (Am = 0.1); the VSI and MRI can co-exist for Am = 1; and the VSI is overwhelmed by the MRI when ambipolar diffusion is weak (Am = 10). Angular momentum transport process is primarily driven by MHD winds, while viscous accretion due to MRI and/or VSI turbulence makes a moderate contribution in most cases. Spontaneous magnetic flux concentration and formation of annular substructures remain robust in strong ambipolar diffusion-dominated discs (Am ≤ 1) with the presence of the VSI. Ambipolar diffusion is the major contributor to the magnetic flux concentration phenomenon rather than advection.

The global nonlinear time dependent evolution of thermal and viscous instabilities in Keplerian accretion disks in x‐ray binary sources has been investigated to understand the low frequency (∼0.04 Hz) quasi‐periodic oscillations (QPOs) seen recently in some black hole candidates (Cyg X–1 and GRO J0422+32) and the Rapid Burster MXB 1730–335. Within the framework of the α‐viscosity model, we assume the viscous stress scales with gas pressure only but that the α parameter is formulated as a function of the local scale height, h. Specifically, α=min[α0(h/r)n, αmax], where r is the distance from the compact object, and n, α0, and αmax are constants. It is found that nonsteady behavior may arise for sufficiently large n (∼1.2). We show that, the low frequency QPOs may be explicable in terms of a thermal‐viscous instability in accretion disks, which results in periodic or quasi‐periodic oscillations in the mass accretion rate. These oscillations are globally coherent in the unstable regions of the disk and the d...

view Abstract Citations (49) References (30) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Local Magnetohydrodynamic Instabilities and the Wave-driven Dynamo in Accretion Disks Vishniac, Ethan T. ; Diamond, Patrick Abstract We consider the consequences of magnetic buoyancy and the magnetic shearing instability (MSI) on the strength and organization of the magnetic field in a thin accretion disk. We discuss a model in which the wave-driven dynamo growth rate is balanced by the dissipative effects of the MSI. As in earlier work, the net helicity is due to small advective motions driven by nonlinear interactions between internal waves. Assuming a simple model of the internal wave spectrum generated from the primary m = 1 internal waves, we find that the magnetic energy density saturates at about (H/r) exp 4/3 times the local pressure (where H is the disk thickness and r is its radius). On very small scales the shearing instability will produce an isotropic fluctuating field. For a stationary disk this is equivalent to a dimensionless 'viscosity' of about (H/r) exp 4/3. The vertical and radial diffusion coefficients will be comparable to each other. Magnetic buoyancy will be largely suppressed by the turbulence due to the MSI. We present a rough estimate of its effects and find that it removes magnetic flux from the disk at a rate comparable to that caused by turbulent diffusion. Publication: The Astrophysical Journal Pub Date: October 1992 DOI: 10.1086/171881 Bibcode: 1992ApJ...398..561V Keywords: Accretion Disks; Dynamo Theory; Magnetohydrodynamic Stability; Stellar Physics; Buoyancy; Computational Astrophysics; Interstellar Magnetic Fields; Magnetohydrodynamic Turbulence; Propagation Modes; Viscosity; Astrophysics; ACCRETION; ACCRETION DISKS; INSTABILITIES; MAGNETOHYDRODYNAMICS: MHD full text sources ADS |

Theoretical models of the ionization state in protoplanetary disks suggest the existence of large areas with low ionization and weak coupling between the gas and magnetic fields. In this regime hydrodynamical instabilities may become important. In this work we investigate the gas and dust structure and dynamics for a typical T Tauri system under the influence of the vertical shear instability (VSI). We use global 3D radiation hydrodynamics simulations covering all 360° of azimuth with embedded particles of 0.1 and 1 mm size, evolved for 400 orbits. Stellar irradiation heating is included with opacities for 0.1–10 μm sized dust. Saturated VSI turbulence produces a stress-to-pressure ratio of . The value of α is lowest within 30 au of the star, where thermal relaxation is slower relative to the orbital period and approaches the rate below which VSI is cut off. The rise in α from 20 to 30 au causes a dip in the surface density near 35 au, leading to Rossby wave instability and the generation of a stationary, long-lived vortex spanning about 4 au in radius and 40 au in azimuth. Our results confirm previous findings that millimeter-sized grains are strongly vertically mixed by the VSI. The scale height aspect ratio for 1 mm grains is determined to be 0.037, much higher than the value H/r = 0.007 obtained from millimeter-wave observations of the HL Tau system. The measured aspect ratio is better fit by nonideal MHD models. In our VSI turbulence model, the millimeter grains drift radially inwards and many are trapped and concentrated inside the vortex. The turbulence induces a velocity dispersion of ∼12 m s−1 for the millimeter grains, indicating that grain–grain collisions could lead to fragmentation.

Erratum: Linear and non-linear evolution of the vertical shear instability in accretion discs

The 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.

A high-resolution fourth-order Padé scheme is used to simulate locally isothermal 3D disk turbulence driven by the vertical shear instability (VSI) using 268.4 M points. In the early nonlinear period of axisymmetric VSI, angular momentum transport by vertical jets creates correlated N-shaped radial profiles of perturbation vertical and azimuthal velocity. This implies dominance of positive perturbation vertical vorticity layers and a recently discovered angular momentum staircase with respect to radius (r). These features are present in 3D in a weaker form. The 3D flow consists of vertically and azimuthally coherent turbulent shear layers containing small vortices with all three vorticity components active. Previously observed large persistent vortices in the interior of the domain driven by the Rossby wave instability are absent. We speculate that this is due to a weaker angular momentum staircase in 3D in the present simulations compared to a previous simulation. The turbulent viscosity parameter α(r) increases linearly with r. At intermediate resolution, the value of α(r) at midradius is close to that of a previous simulation. The specific kinetic energy spectrum with respect to radial wavenumber has a power-law region with exponent −1.84, close to the value −2 expected for shear layers. The spectrum with respect to azimuthal wavenumber has a −5/3 region and lacks a −5 region reported in an earlier study. Finally, it is found that axisymmetric VSI has artifacts at late times, including a very strong angular momentum staircase, which in 3D is present weakly in the disk’s upper layers.

Massive eccentric disks (gaseous or particulate) orbiting a dominant central mass appear in many astrophysical systems, including planetary rings, protoplanetary and accretion disks in binaries, and nuclear stellar disks around supermassive black holes in galactic centers. We present an analytical framework for treating the nearly Keplerian secular dynamics of test particles driven by the gravity of an eccentric, apsidally aligned, zero-thickness disk with arbitrary surface density and eccentricity profiles. We derive a disturbing function describing the secular evolution of coplanar objects, which is explicitly related (via one-dimensional, convergent integrals) to the disk surface density and eccentricity profiles without using any ad hoc softening of the potential. Our analytical framework is verified via direct orbit integrations, which show it to be accurate in the low-eccentricity limit for a variety of disk models (for disk eccentricity ≲0.1–0.2). We find that free precession in the potential of a disk with a smooth surface density distribution can naturally change from prograde to retrograde within the disk. Sharp disk features—edges and gaps—are the locations where this tendency is naturally enhanced, while the precession becomes very fast. Radii where free precession changes sign are the locations where substantial (formally singular) growth of the forced eccentricity of the orbiting objects occurs. Based on our results, we formulate a self-consistent analytical framework for computing an eccentricity profile for an aligned, eccentric disk (with a prescribed surface density profile) capable of precessing as a solid body under its own self-gravity.

Vortices have long been speculated to play a role in planet formation, via the collection of dust in the pressure maxima that arise at the cores of vortices in protoplanetary discs (PPDs). The question remains, however: as dust collects in the core of a vortex, when does that vortex remain stable and able to collect further dust, and when and why does it break up? We study this question by running high-resolution 2D simulations of dust-laden vortices. By using the terminal velocity approximation in a local shearing box, it was possible to efficiently run simulations of back-reacting dust in a gas at high resolution. Our results show how the stability of 2D dust-laden vortices in PPDs depends on their size relative to the disc scale height, as well as the dust coupling. We find small vortices with semiminor axis much smaller than the scale height to be stable for the duration of the simulations (t > 2000 orbits). Larger vortices, with semiminor axis smaller than but of the order of scale height, exhibit a drag instability after undergoing a long period of contraction where the core becomes progressively more dust rich. The lifetime of these vortices depends on the dust size, with larger dust grains causing the instability to occur sooner. For the size ranges tested in this paper, micrometre- to millimetre-sized grains, vortices survived for several hundreds of orbits. The result implies that the stability of vortices formed by vertical shear instability and zombie vortex instability, or the breakup of larger vortices through hydrodynamic instabilities, is affected by the presence of dust in the disc. The lifetimes observed in this paper, while shortened by the presence of dust for larger vortices, were still long enough to lead to considerable dust enrichment in the vortex cores.

The growth process of protoplanets can be sped up by accreting a large number of solid, pebble-sized objects that are still present in the protoplanetary disc. It is still an open question on how efficient this process works in realistic turbulent discs. We investigate the accretion of pebbles in turbulent discs that are driven by the purely hydrodynamical vertical shear instability (VSI). For this purpose, we performed global 3D simulations of locally isothermal, VSI turbulent discs that have embedded protoplanetary cores from 5 to 100 M⊕, which are placed at 5.2 au distance from the star. In addition, we followed the evolution of a swarm of embedded pebbles of different sizes under the action of drag forces between gas and particles in this turbulent flow. Simultaneously, we performed a set of comparison simulations for laminar viscous discs where the particles experience stochastic kicks. For both cases, we measured the accretion rate onto the cores as a function of core mass and Stokes number (τs) of the particles and compared these values to recent magneto-rotational instability (MRI) turbulence simulations. Overall the dynamic is very similar for the particles in the VSI turbulent disc and the laminar case with stochastic kicks. For small mass planets (i.e. 5–10 M⊕), well-coupled particles with τs = 1, which have a size of about 1 m at this location, we find an accretion efficiency (rate of particles accreted over drifting inwards) of about 1.6−3%. For smaller and larger particles, this efficiency is higher. However, the fast inwards drift for τs = 1 particles makes them the most effective for rapid growth, leading to mass doubling times of about 20 000 yr. For masses between 10 and 30 M⊕ the core reaches the pebble isolation mass and the particles are trapped at the pressure maximum just outside of the planet, shutting off further particle accretion.

Our previous studies of large-scale vortical flows arising in shear flows of stellar accretion disks with Keplerian azimuthal velocity distributions as a result of the development of small perturbations are continued. The development of large-scale instability in an accretion disk is investigated via mathematical modeling. One result obtained is the change of the disk flow structure due to the formation of large vortices. In the limiting case, sufficiently long evolution leads to the formation of several asymmetric spiral structures of the flow of disk matter. The presence of large-scale structures leads to angular-momentum redistribution in the disk.