Turbulence In Accretion Disks Research Articles

Keplerian turbulence in Taylor–Couette flow of dilute polymeric solutions

Turbulent flow induced by elastorotational instability in viscoelastic Taylor–Couette flow (TCF) with Keplerian rotation is analogous to a turbulent accretion disk destabilized by magnetorotational instability. We examine this novel viscoelastic Keplerian turbulence via direct numerical simulations (DNS) for the shear Reynolds number ( $Re$ ) ranging from $10^2$ to $10^4$ . The observed characteristic flow structure consists of penetrating streamwise vortices with axial length scales much smaller than the gap width, distinct from the classic centrifugally induced Taylor vortices, which have axial lengths of the gap width. These intriguing vortices persist for the wide $Re$ range considered and give rise to intriguing scaling behaviour in key flow quantities. Specifically, the characteristic axial length of the penetrating vortices is shown to scale as $Re^{-0.22}$ ; the angular momentum transport scales as $Re^{0.42}$ ; the kinetic and elastic boundary-layer thicknesses based on angular velocity and hoop stress near the inner cylinder wall scale as $Re^{-0.48}$ and $Re^{-0.49}$ , respectively. This implies that the viscoelastic Keplerian turbulence belongs to the classical turbulent regime of TCF with the Prandtl–Blasius-type boundary layer. Furthermore, we present an analytical relation between the viscous and elastic dissipation rates of kinetic energy and the angular momentum transport and in turn demonstrate its validity using our DNS data. This study has paved the way for future research to explore astrophysics-related Keplerian turbulence and angular momentum transport via the scaling relations of the analogous TCF of dilute polymeric solutions.

One-winged butterflies: mode selection for azimuthal magnetorotational instability by thermal convection

The effects of thermal convection on turbulence in accretion discs, and particularly its interplay with the magnetorotational instability (MRI), are of significant astrophysical interest. Despite extensive theoretical and numerical studies, such an interplay has not been explored experimentally. We conduct linear analysis of the azimuthal version of MRI (AMRI) in the presence of thermal convection and compare the results with our experimental data published before. We show that the critical Hartmann number ( $Ha$ ) for the onset of AMRI is reduced by convection. Importantly, convection breaks symmetry between $m = \pm 1$ instability modes ( $m$ is the azimuthal wavenumber). This preference for one mode over the other makes the AMRI wave appear as a ‘one-winged butterfly’.

Plasmoid identification and statistics in two-dimensional Harris sheet and GRMHD simulations

ABSTRACT Magnetic reconnection is a ubiquitous phenomenon for magnetized plasma and leads to the rapid reconfiguration of magnetic field lines. During reconnection events, plasma is heated and accelerated until the magnetic field lines enclose and capture the plasma within a circular configuration. These so-called plasmoids could therefore observationally manifest themselves as hotspots, which are associated with flaring behaviour in supermassive black hole systems, such as Sagittarius A*. We have developed a novel algorithm for identifying plasmoid structures, which incorporates watershed and custom closed contouring steps. From the identified structures, we determine the plasmoids’ plasma characteristics and energetics in magnetohydrodynamical simulations. The algorithm’s performance is showcased for a high-resolution suite of axisymmetric ideal and resistive magnetohydrodynamical simulations of turbulent accretion discs surrounding a supermassive black hole. For validation purposes, we also evaluate several Harris current sheets that are well-investigated in the literature. We recover the characteristic power-law distribution of plasmoid sizes for both the black hole and Harris sheet simulations. This indicates that while the dynamics are vastly different, with different dominant plasma instabilities, the plasmoid creation characteristics are similar. Plasmoid formation rates for resistive general relativistic magnetohydrodynamical simulations are significantly higher than for their ideal counterpart. Moreover, the largest identified plasmoids are consistent with sizes typically assumed for semi-analytical interpretation of observations. We recover a positive correlation between the plasmoid formation rate and decreases in black-hole-horizon-penetrating magnetic flux, during which the accretion flow is temporarily halted. These results demonstrate the efficacy of the newly developed algorithm which has enabled an extensive quantitative analysis of plasmoid formation in black hole accretion simulations.

The persistence of magneto-rotational turbulence in gravitationally turbulent accretion disks

Aims. Our main goal is to probe the persistence of turbulence originating from the magneto-rotational instability (MRI) in gravito-turbulent disks. This state is referred to here as GI-MRI coexistence, with GI standing for gravitational instability. We test the influence of GI strength, controlled by the cooling law, and the impact of Ohmic resistivity. Methods. Our starting point was three-dimensional, ideal, magnetohydrodynamic (MHD) simulations of gravitational turbulence in the local shearing-box approximation using the code Athena. We introduced a zero-net-flux magnetic seed field in a GI-turbulent state and investigated the nonlinear evolution. The GI strength was varied by modifying the cooling parameters. We tested the cooling times τcΩ0 = 10, τcΩ0 = 20, and τcΩ0 = 10, with additional background heating. For some resistive cases, ideal-MHD simulations, which had already developed GI-MRI coexistence, were restarted with a finite Ohmic resistivity enabled at the moment of restart. Results. It appears that there are two possible saturated dynamo states in the ideal-MHD regime: a state of GI-MRI coexistence (for low GI activity) and a strong-GI dynamo. The cases with lower GI activity eventually develop a clearly visible butterfly pattern. For the case with the highest GI activity (τcΩ0 = 10, no heating), a clearly visible butterfly pattern is absent, though more chaotic field reversals are observed above (and below) the mid-plane. We were also able to reproduce the results of previous simulations. With Ohmic resistivity, the simulation outcome can be substantially different. There exists a critical magnetic Reynolds number, ⟨Rm⟩ ∼ 500, below which the ideal-MHD outcome is replaced by a new dynamo state. For larger Reynolds numbers, one recovers turbulent states that are more reminiscent of the ideal-MHD states, and especially the strong-GI case. This new state leads to oscillations, which are caused by a significant heat production due to the resistive dissipation of magnetic energy. The additional heat periodically quenches GI, and the quenching events correspond to maxima of the Toomre value, Q.

Taylor-Couette flow for astrophysical purposes.

A concise review is given of astrophysically motivated experimental and theoretical research on Taylor-Couette flow. The flows of interest rotate differentially with the inner cylinder faster than the outer, but are linearly stable against Rayleigh's inviscid centrifugal instability. At shear Reynolds numbers as large as [Formula: see text], hydrodynamic flows of this type (quasi-Keplerian) appear to be nonlinearly stable: no turbulence is seen that cannot be attributed to interaction with the axial boundaries, rather than the radial shear itself. Direct numerical simulations agree, although they cannot yet reach such high Reynolds numbers. This result indicates that accretion-disc turbulence is not purely hydrodynamic in origin, at least insofar as it is driven by radial shear. Theory, however, predicts linear magnetohydrodynamic (MHD) instabilities in astrophysical discs: in particular, the standard magnetorotational instability (SMRI). MHD Taylor-Couette experiments aimed at SMRI are challenged by the low magnetic Prandtl numbers of liquid metals. High fluid Reynolds numbers and careful control of the axial boundaries are required. The quest for laboratory SMRI has been rewarded with the discovery of some interesting inductionless cousins of SMRI, and with the recently reported success in demonstrating SMRI itself using conducting axial boundaries. Some outstanding questions and near-future prospects are discussed, especially in connection with astrophysics. This article is part of the theme issue 'Taylor-Couette and related flows on the centennial of Taylor's seminal Philosophical Transactions paper (Part 2)'.

Deciphering the radio–star formation correlation on kpc scales

Given the multiple energy-loss mechanisms of cosmic-ray (CR) electrons in galaxies, the tightness of the infrared (IR)–radio continuum correlation is surprising. As the radio continuum emission at GHz frequencies is optically thin, this offers the opportunity to obtain unbiased star formation rates (SFRs) from radio-continuum flux-density measurements. The calorimeter theory can naturally explain the tightness of the far-infrared (FIR)–radio correlation but makes predictions that do not agree with observations. Noncalorimeter models often have to involve a conspiracy to maintain the tightness of the FIR–radio correlation. We extended a published analytical model of galactic disks by including a simplified prescription for the synchrotron emissivity. The galactic gas disks of local spiral galaxies, low-z starburst galaxies, high-z main sequence star-forming galaxies, and high-z starburst galaxies are treated as turbulent clumpy accretion disks. The magnetic field strength is determined by the equipartition between the turbulent kinetic and the magnetic energy densities. Our fiducial model, which includes neither galactic winds nor CR electron secondaries, reproduces the observed radio continuum spectral energy distributions of most (∼70%) of the galaxies. Except for the local spiral galaxies, fast galactic winds can potentially make the conflicting models agree with observations. The observed IR–radio correlations are reproduced by the model within 2σ of the joint uncertainty of model and data for all datasets. The model agrees with the observed SFR–radio correlations within ∼4σ. Energy equipartition between the CR particles and the magnetic field only approximately holds in our models of main sequence star-forming galaxies. If a CR electron calorimeter is assumed, the slope of the IR–radio correlation flattens significantly. Inverse Compton losses are not dominant in the starburst galaxies because in these galaxies not only the gas density but also the turbulent velocity dispersion is higher than in normal star-forming galaxies. Equipartition between the turbulent kinetic and magnetic field energy densities then leads to very high magnetic field strengths and very short synchrotron timescales. The exponents of our model SFR–radio correlations at 150 MHz and 1.4 GHz are very close to one.

MRI turbulence in accretion discs at large magnetic Prandtl numbers

ABSTRACT The effect of large magnetic Prandtl number Pm (the ratio of viscosity to resistivity) on the turbulent transport and energetics of the magnetorotational instability (MRI) is poorly understood, despite the realization of this regime in astrophysical environments as disparate as discs from binary neutron star (BNS) mergers, the inner regions of low-mass X-ray binaries and active galactic nuclei, and the interiors of protoneutron stars. We investigate the MRI dynamo and associated turbulence in the regime Pm > 1 by carrying out fully compressible, 3D MHD-shearing box simulations using the finite-volume code pluto, focusing mostly on the case of Keplerian shear relevant to accretion discs. We find that when the magnetic Reynolds number is kept fixed, the turbulent transport (as parameterized by α, the ratio of stress to thermal pressure) scales with the magnetic Prandtl number as α ∼ Pmδ, with δ ∼ 0.5−0.7 up to Pm ∼ 128. However, this scaling weakens as the magnetic Reynolds number is increased. Importantly, compared to previous studies, we find a new effect at very large Pm – the turbulent energy and stress begin to plateau, no longer depending on Pm. To understand these results we have carried out a detailed analysis of the turbulent dynamics in Fourier space, focusing on the effect of increasing Pm on the transverse cascade – a key non-linear process induced by the disc shear flow that is responsible for the sustenance of MRI turbulence. Finally, we find that α–Pm scaling is sensitive to the box vertical-to-radial aspect ratio, as well as to the background shear.

Evolution of a Mode of Oscillation within Turbulent Accretion Disks

Abstract We investigate the effects of subsonic turbulence on a normal mode of oscillation (a possible origin of the high-frequency quasi-periodic oscillations (HFQPOs) within some black hole accretion disks). We consider perturbations of a time-dependent background (steady-state disk plus turbulence), obtaining an oscillator equation with stochastic damping, (mildly) nonlinear restoring, and stochastic driving forces. The (long-term) mean values of our turbulent functions vanish. In particular, turbulence does not damp the oscillation modes, so “turbulent viscosity” is not operative. However, the frequency components of the turbulent driving force near that of the mode can produce significant changes in the amplitude of the mode. Even with an additional (phenomenological constant) source of damping, this leads to an eventual “blowout” (onset of effects of nonlinearity) if the turbulence is sufficiently strong or the damping constant is sufficiently small. The infrequent large increases in the energy of the mode could be related to the observed low duty cycles of the HFQPOs. The width of the peak in the power spectral density (PSD) is proportional to the amount of nonlinearity. A comparison with observed continuum PSDs indicates the conditions required for the visibility of the mode.

More Variable Quasars Have Stronger Emission Lines

Abstract UV/optical variation, likely driven by accretion disk turbulence, is a defining characteristic of type 1 active galactic nuclei and quasars. In this work we investigate an interesting consequence of such turbulence using quasars in Sloan Digital Sky Survey Stripe 82 for which the measurements of the UV/optical variability amplitude are available from ∼10 yr long light curves. We discover positive correlations between the UV/optical variability amplitude σ rms and equivalent widths of C iv, Mg ii, and [O iii]5007 emission lines. Such correlations remain statistically robust through partial correlation analyses, i.e., after controlling the effects of other variables including bolometric luminosity, central supermassive black hole mass, Eddington ratio, and redshift. This, for the first time, indicates a causal link between disk turbulence and emission line production. We propose two potential underlying mechanisms, both of which may be involved: (1) quasars with stronger disk turbulence have on average a bluer/harder broadband spectral energy distribution, an expected effect of the disk thermal fluctuation model; (2) stronger disk turbulence could lead to the launch of emission line regions with larger covering factors.

Magnetic outflows from turbulent accretion disks

Context. Astrophysical disks are likely embedded in an ambient vertical magnetic field generated by its environment. This ambient field is known to drive magneto-rotational turbulence in the disk bulk, but it is also responsible for launching magnetised outflows at the origin of astrophysical jets. Yet, the interplay between turbulence and outflows is not understood. In particular, the vertical structure and long-term (secular) evolution of such a system lack quantitative predictions. It is, nevertheless, this secular evolution which is proposed to explain time variability in many accreting systems such as FuOr, X-ray binaries, and novae like systems. Aims. We seek to constraint the structure and long-term evolution of turbulent astrophysical disks subject to magnetised outflows in the non-relativistic regime. More specifically we aim to characterise the mechanism driving accretion, the dynamics of the disk atmosphere, the role played by the outflow, and the long-term evolution of mass and magnetic flux distributions. Methods. We computed and analysed global 3D ideal magnetohydrynamic (MHD) simulations of an accretion disk threaded by a large-scale magnetic field. We measured the turbulent state of the system by Reynolds averaging the ideal MHD equations and evaluate the role of the turbulent terms in the equilibrium of the system. We then computed the transport of mass, angular momentum, and magnetic fields in the disk to characterise its secular evolution. Finally, we performed a parameter exploration survey in order to characterise how the transport properties depend on the disk properties. Results. We find that weakly magnetised disks drive jets that carry a small fraction of the disk angular momentum away. The mass-weighted accretion speed remains subsonic, although there is always an upper turbulent atmospheric region where transsonic accretion takes place. We show that this turbulence is driven by a strongly magnetised version of the magneto-rotational instability. The internal disk structure therefore appears drastically different from the conventional hydrostatic picture. We expect that the turbulent atmosphere region will lead to non-thermal features in the emission spectra from compact objects. In addition, we show that the disk is subject to a secular viscous-type instability, which leads to the formation of long-lived ring-like structures in the disk surface density distribution. This instability is likely connected to the magnetic field transport. Finally, we show that for all of the parameters explored, the ambient magnetic field is always dragged inward in the disk at a velocity which increases with the disk magnetisation. Beyond a threshold on the latter, the disk undergoes a profound radial readjustment. It leads to the formation of an inner accretion-ejection region with a supersonic mass-weighted accretion speed and where the magnetic field distribution becomes steady and reaches a magnitude near equipartition with the thermal pressure. This inner structure shares many properties with the jet emitting disk model. Overall, these results pave the way for quantitative self-consistent secular disk models.

Planetesimal formation around the snow line

Context.The formation of rocky planetesimals is a long-standing problem in planet formation theory. One of the possibilities is that it results from gravitational instability as a result of the pile-up of small silicate dust particles released from sublimating icy pebbles that pass the snow line.Aims.We want to understand and quantify the role of the water snow line for the formation of rock-rich and ice-rich planetesimals. In this paper, we focus on the formation of rock-rich planetesimals. A companion paper examines the combined formation of both rock-rich and ice-rich planetesimals.Methods.We developed a new Monte Carlo code to calculate the radial evolution of silicate particles in a turbulent accretion disk, accounting for the back reaction (i.e., inertia) of the particles on their radial drift velocity and diffusion. Results depend in particular on the particle injection width (determined from the radial sublimation width of icy pebbles), the pebble scale height, and the pebble mass flux through the disk. The scale height evolution of the silicate particles, which is the most important factor for the runaway pile-up, is calculated automatically in this Lagrange method.Results.From the numerical results, we derive semi-analytical relations for the scale height of the silicate dust particles and the particle-to-gas density ratio at the midplane, as functions of a pebble-to-gas mass flux ratio and theαparameters for disk gas accretion and vertical/radial diffusion We find that the runaway pile-up of the silicate particles (formation of rocky planetesimals) occurs if the pebble-to-gas mass flux ratio is ≳ [(αDz/αacc)/3 × 10−2]1/2, whereαDzandαaccare theαparameters for vertical turbulent diffusion and disk gas accretion.

On twin peak quasi-periodic oscillations resulting from the interaction between discoseismic modes and turbulence in accretion discs around black holes

ABSTRACT Given the peculiar and (in spite of many efforts) unexplained quasi-periodic oscillation twin peak phenomena in accretion disc power spectral density observations, the present exploratory analytical article tries to inquire deeper into the relationship between discoseismic modes and the underlying driving turbulence in order to assess its importance. We employ a toy model in the form of a Gaussian white noise-driven damped harmonic oscillator with stochastic frequency. This oscillator represents the discoseismic mode. (Stochastic damping was also considered, but interestingly it was found to be less relevant for the case at hand.) In the context of this model, we find that turbulence interacts with disc oscillations in interesting ways. In particular, the stochastic part in the oscillator frequency behaves as a separate driving agent. This gives rise to 3:2 twin peaks for some values of the physical parameters, which we find. We conclude with the suggestion that the study of turbulence be brought to the forefront of disc oscillation dynamics, as opposed to being a mere background feature. This change of perspective carries immediate observable consequences, such as considerably shifting the values of the (discoseismic) oscillator frequencies.

3D simulations of accretion onto a star: Fast funnel-wall accretion

AbstractWe show the results of global 3D magnetohydrodynamics simulations of an accretion disk with a rotating, weakly magnetized central star (Takasao et al. 2018). The disk is threaded by a weak large-scale poloidal magnetic field. The central star has no strong stellar magnetosphere initially and is only weakly magnetized. We investigate the structure of the accretion flows from a turbulent accretion disk onto the star. Our simulations reveal that fast accretion onto the star at high latitudes is established even without a stellar magnetosphere. We find that the failed disk wind becomes the fast, high-latitude accretion as a result of angular momentum exchange mediated by magnetic fields. The rapid angular momentum exchange occurs well above the disk, where the Lorentz force that decelerates the rotational motion of gas can be comparable to the centrifugal force. Unlike the classical magnetospheric accretion model, fast accretion streams are not guided by magnetic fields of the stellar magnetosphere. Nevertheless, the accretion velocity reaches the free-fall velocity at the stellar surface owing to the efficient angular momentum loss at a distant place from the star. Our model can be applied to Herbig Ae/Be stars whose magnetic fields are generally not strong enough to form magnetospheres, and also provides a possible explanation why Herbig Ae/Be stars show indications of fast accretion.

Low mass planet migration in Hall-affected disks

Recent developments in non-ideal magnetohydrodynamic simulations of protoplanetary disks suggest that instead of being traditional turbulent (viscous) accretion disks, they have a largely laminar flow with accretion driven by large-scale wind torques. These disks are possibly threaded by Hall-effect generated large-scale horizontal magnetic fields. We have examined the dynamics of the corotation region of a low mass planet embedded in such a disk and the evolution of the associated migration torque. These disks lack strong turbulence and associated turbulent diffusion, and the presence of a magnetic field and radial gas flow presents a situation outside the applicability of previous corotation torque theory. We summarize the analytical analysis of the corotation torque, give details on the numerical methods used, and in particular the relative merits of different numerical schemes for the inviscid problem.

A Three-dimensional Simulation of a Magnetized Accretion Disk: Fast Funnel Accretion onto a Weakly Magnetized Star

Abstract We present the results of a global, three-dimensional magnetohydrodynamics simulation of an accretion disk with a rotating, weakly magnetized central star. The disk is threaded by a weak, large-scale poloidal magnetic field, and the central star has no strong stellar magnetosphere initially. Our simulation investigates the structure of the accretion flows from a turbulent accretion disk onto the star. The simulation reveals that fast accretion onto the star at high latitudes occurs even without a stellar magnetosphere. We find that the failed disk wind becomes the fast, high-latitude accretion as a result of angular momentum exchange mediated by magnetic fields well above the disk, where the Lorentz force that decelerates the rotational motion of gas can be comparable to the centrifugal force. Unlike the classical magnetospheric accretion scenario, fast accretion streams are not guided by magnetic fields of the stellar magnetosphere. Nevertheless, the accretion velocity reaches the free-fall velocity at the stellar surface due to the efficient angular momentum loss at a distant place from the star. This study provides a possible explanation why Herbig Ae/Be stars whose magnetic fields are generally not strong enough to form magnetospheres also show indications of fast accretion. A magnetically driven jet is not formed from the disk in our model. The differential rotation cannot generate sufficiently strong magnetic fields for the jet acceleration because the Parker instability interrupts the field amplification.

Study of the Transition from MRI to Magnetic Turbulence via Parasitic Instability by a High-order MHD Simulation Code

Abstract Magnetic turbulence in accretion disks under ideal magnetohydrodynamic (MHD) conditions is expected to be driven by the magneto-rotational instability (MRI) followed by secondary parasitic instabilities. We develop a three-dimensional ideal MHD code that can accurately resolve turbulent structures, and carry out simulations with a net vertical magnetic field in a local shearing box disk model to investigate the role of parasitic instabilities in the formation process of magnetic turbulence. Our simulations reveal that a highly anisotropic Kelvin–Helmholtz (K–H) mode parasitic instability evolves just before the first peak in turbulent stress and then breaks large-scale shear flows created by MRI. The wavenumber of the enhanced parasitic instability is larger than the theoretical estimate, because the shear flow layers sometimes become thinner than those assumed in the linear analysis. We also find that interaction between antiparallel vortices caused by the K–H mode parasitic instability induces small-scale waves that break the shear flows. On the other hand, at repeated peaks in the nonlinear phase, anisotropic wavenumber spectra are observed only in the small wavenumber region and isotropic waves dominate at large wavenumbers unlike for the first peak. Restructured channel flows due to MRI at the peaks in nonlinear phase seem to be collapsed by the advection of small-scale shear structures into the restructured flow and resultant mixing.

Demonstration of a magnetic Prandtl number disc instability from first principles

Understanding what determines the strength of MHD turbulence in accretion discs is a question of fundamental theoretical and observational importance. In this work we investigate whether the dependence of the turbulent accretion disc stress ($\alpha$) on the magnetic Prandtl number (Pm) is sufficiently sensitive to induce thermal-viscous instability using 3D MHD simulations. We first investigate whether the $\alpha$-Pm dependence, found by many previous authors, has a physical or numerical origin by conducting a suite of local shearing-box simulations. We find that a definite $\alpha$-Pm dependence persists when simultaneously increasing numerical resolution and decreasing the absolute values of both the viscous and resistive dissipation coefficients. This points to a physical origin of the $\alpha$-Pm dependence. Using a further set of simulations which include realistic turbulent heating and radiative cooling, and by giving Pm a realistic physical dependence on the plasma temperature and density, we demonstrate that the $\alpha$-Pm dependence is sufficiently strong to lead to a local instability. We confirm that the instability manifests itself as an unstable limit cycle by mapping the local thermal-equilibrium curve of the disc. This is the first self-consistent MHD simulation demonstrating the Pm instability from first principles. This result is important because a physical Pm instability would lead to the global propagation of heating and cooling fronts and a transition between disc states on timescales compatible with the observed hard/soft state transitions in black hole binaries.

Excitation of turbulence in accretion disks of binary stars by non-linear perturbations

Accretion disks in binary systems can experience hydrodynamic impact at inner as well as outer edges. The first case is typical for protoplanetary disks around young T Tau stars. The second one is typical for circumstellar disks in close binaries. As a result of such an impact, perturbations with different scales and amplitudes are excited in the disk. We investigated the nonlinear evolution of perturbations of a finite, but small amplitude, at the background of sub-Keplerian flow. Nonlinear effects at the front of perturbations lead to the formation of a shock wave, namely the discontinuity of the density and radial velocity. At this, the tangential flow in the neighborhood of the shock becomes equivalent to the flow in in the boundary layer. Instability of the tangential flow further leads to turbulization of the disk. Characteristics of the turbulence depend on perturbation parameters, but alpha-parameter of Shakura-Sunyaev does not exceed ~0.1.

Hydrodynamic turbulence in quasi-Keplerian rotating flows

We report a direct-numerical-simulation study of the Taylor–Couette flow in the quasi-Keplerian regime at shear Reynolds numbers up to O(105). Quasi-Keplerian rotating flow has been investigated for decades as a simplified model system to study the origin of turbulence in accretion disks that is not fully understood. The flow in this study is axially periodic and thus the experimental end-wall effects on the stability of the flow are avoided. Using optimal linear perturbations as initial conditions, our simulations find no sustained turbulence: the strong initial perturbations distort the velocity profile and trigger turbulence that eventually decays.

Anisotropic hydrodynamic turbulence in accretion disks

Recently, the vertical shear instability (VSI) has become an attractive purely hydrodynamic candidate for the anomalous angular momentum transport required for weakly ionized accretion disks. In direct three-dimensional numerical simulations of VSI turbulence in disks, a meridional circulation pattern was observed that is opposite to the usual viscous flow behavior. Here, we investigate whether this feature can possibly be explained by an anisotropy of the VSI turbulence. Using three-dimensional hydrodynamical simulations, we calculate the turbulent Reynolds stresses relevant for angular momentum transport for a representative section of a disk. We find that the vertical stress is significantly stronger than the radial stress. Using our results in viscous disk simulations with different viscosity coefficients for the radial and vertical direction, we find good agreement with the VSI turbulence for the stresses and meridional flow; this provides additional evidence for the anisotropy. The results are important with respect to the transport of small embedded particles in disks.