Viscosity Prescription Research Articles

VISCOSITY PRESCRIPTION FOR GRAVITATIONALLY UNSTABLE ACCRETION DISKS

Gravitationally unstable accretion disks emerge in a variety of astrophysical contexts - giant planet formation, FU Orioni outbursts, feeding of AGNs, and the origin of Pop III stars. When a gravitationally unstable disk is unable to cool rapidly it settles into a quasi-stationary, fluctuating gravitoturbulent state, in which its Toomre Q remains close to a constant value Q_0~1. Here we develop an analytical formalism describing the evolution of such a disk, which is based on the assumptions of Q=Q_0 and local thermal equilibrium. Our approach works in the presence of additional sources of angular momentum transport (e.g. MRI), as well as external irradiation. Thermal balance dictates a unique value of the gravitoturbulent stress \alpha_{gt} driving disk evolution, which is a function of the local surface density and angular frequency. We compare this approach with other commonly used gravitoturbulent viscosity prescriptions, which specify the explicit dependence of stress \alpha_{gt} on Toomre Q in an ad hoc fashion, and identify the ones that provide consistent results. We nevertheless argue that our Q=Q_0 approach is more flexible, robust, and straightforward, and should be given preference in applications. We illustrate this with a couple of analytical calculations - locations of the snow line and of the outer edge of the dead zone in a gravitoturbulent protoplanetary disk - which clearly show the simplicity and versatility of the Q=Q_0 approach.

Self-similar evolution of self-gravitating viscous accretion discs

A new one-dimensional, dynamical model is proposed for geometrically thin, self-gravitating viscous accretion discs. The vertically integrated equations are simplified using the slow accretion limit and the monopole approximation with a time-dependent central point mass to account for self-gravity and accretion. It is shown that the system of partial differential equations can be reduced to a single non-linear advection diffusion equation which describes the time evolution of angular velocity. In order to solve the equation three different turbulent viscosity prescriptions are considered. It is shown that for these parametrizations the differential equation allows for similarity transformations depending only on a single non-dimensional parameter. A detailed analysis of the similarity solutions reveals that this parameter is the initial power law exponent of the angular velocity distribution at large radii. The radial dependence of the self-similar solutions is in most cases given by broken power laws. At small radii the rotation law always becomes Keplerian with respect to the current central point mass. In the outer regions the power law exponent of the rotation law deviates from the Keplerian value and approaches asymptotically the value determined by the initial condition. It is shown that accretion discs with flatter rotation laws at large radii yield higher accretion rates. The methods are applied to self-gravitating accretion discs in active galactic nuclei. Fully self-gravitating discs are found to evolve faster than nearly Keplerian discs. The implications on supermassive black hole formation and Quasar evolution are discussed.

Recurrent Outbursts and Jet Ejections Expected in Swift J1644+57: Limit-Cycle Activities in a Supermassive Black Hole

Abstract A tidal disruption event by a supermassive black hole in Swift J1644$+$57 can trigger limit-cycle oscillations between a supercritically accreting X-ray bright state and a subcritically accreting X-ray dim state. The time evolution of debris gas around a black hole with mass $M = 10^{6} M_\odot$ is studied by performing axisymmetric, two-dimensional radiation hydrodynamic simulations. We assume the $\alpha$-prescription of viscosity, in which the viscous stress is proportional to the total pressure. The mass supply rate from the outer boundary was assumed to be ${\dot M}_{\rm supply} = 100 L_{\rm Edd}/c^2$, where $L_{\rm Edd}$ is the Eddington luminosity, and $c$ is the light speed. Since the mass accretion rate decreases inward by outflows driven by radiation pressure, the state transition from a supercritically accreting slim disk state to a subcritically accreting Shakura–Sunyaev disk starts from the inner disk, and propagates outward on a timescale of one day. The sudden drop of the X-ray flux observed in Swift J1644$+$57 in 2012 August can be explained by this transition. As long as ${\dot M}_{\rm supply}$ exceeds the threshold for the existence of a radiation pressure dominant disk, the accumulation of accreting gas in the subcritically accreting region triggers the transition from a gas pressure dominant Shakura–Sunyaev disk to a slim disk. This transition takes place at $t {\sim} 50 /$ (${\alpha}/$ 0.1) d after the X-ray darkening. We expect that if $\alpha \gt $ 0.01, X-ray emission with luminosity $\gtrsim 10^{44}$ erg s$^{-1}$ and jet ejection will revive in Swift J1644$+$57 in 2013–2014.

Characterization of global flow and local fluctuations in 3D SPH simulations of protoplanetary discs

A complete and detailed knowledge of the structure of the gaseous component in protoplanetary discs is essential to the study of dust evolution during the early phases of pre-planetesimal formation. The aim of this paper is to determine if three-dimensional accretion discs simulated by the smoothed particle hydrodynamics (SPH) method can reproduce the observational data now available and the expected turbulent nature of protoplanetary discs. The investigation is carried out by setting up a suite of diagnostic tools specifically designed to characterize both the global flow and the fluctuations of the gaseous disc. The main result concerns the role of the artificial viscosity implementation in the SPH method: in addition to the already known ability of SPH artificial viscosity to mimic a physical-like viscosity under specific conditions, we show how the same artificial viscosity prescription behaves like an implicit turbulence model. In fact, we identify a threshold for the parameters in the standard artificial viscosity above which SPH disc models present a cascade in the power spectrum of velocity fluctuations, turbulent diffusion and a mass accretion rate of the same order of magnitude as measured in observations. Furthermore, the turbulence properties observed locally in SPH disc models are accompanied by meridional circulation in the global flow of the gas, proving that the two mechanisms can coexist.

VISCOUS ACCRETION OF A POLYTROPIC SELF-GRAVITATING DISK IN THE PRESENCE OF WIND

Self-similar and semi-analytical solutions are found for the height-averaged equations govern the dynamical behavior of a polytropic, self-gravitating disk under the effects of winds, around the nascent object. In order to describe time evolution of the system, we adopt a radius dependent mass loss rate, then highlight its importance on both the traditional $\alpha$ and innovative $\beta$ models of viscosity prescription. In agreement with some other studies, our solutions represent that Toomre parameter is less than one in most regions on the $\beta$-disk which indicate that in such disks gravitational instabilities can occur in various distances from the central accretor and so the $\beta$-disk model might provide a good explanation of how the planetary systems form. The purpose of the present work is twofold. First, examining the structure of disk with wind in comparison to no-wind solution; and second, to see if the adopted viscosity prescription affects significantly the dynamical behavior of the disk-wind system. We also considered the temperature distribution in our disk by a polytropic condition. The solutions imply that, under our boundary conditions, the radial velocity is larger for $\alpha$-disks and increases as wind becomes stronger in both viscosity models. Also, we noticed that the disk thickness increases by amplifying the wind or adopting larger values for polytropic exponent $\gamma$. It also may globally decrease if one prescribe $\beta$-model for the viscosity. Moreover, in both viscosity models, surface density and mass accretion rate reduce as wind gets stronger or $\gamma$ increases.

Influence of viscosity and the adiabatic index on planetary migration

The strength and direction of migration of low mass embedded planets depends on the disk's thermodynamic state, where the internal dissipation is balanced by radiative transport, and the migration can be directed outwards, a process which extends the lifetime of growing embryos. Very important parameters determining the structure of disks, and hence the direction of migration, are the viscosity and the adiabatic index. In this paper we investigate the influence of different viscosity prescriptions (alpha-type and constant) and adiabatic indices on disk structures and how this affects the migration rate of planets embedded in such disks. We perform 3D numerical simulations of accretion disks with embedded planets. We use the explicit/implicit hydrodynamical code NIRVANA that includes full tensor viscosity and radiation transport in the flux-limited diffusion approximation, as well as a proper equation of state for molecular hydrogen. The migration of embedded 20Earthmass planets is studied. Low-viscosity disks have cooler temperatures and the migration rates of embedded planets tend toward the isothermal limit. In these disks, planets migrate inwards even in the fully radiative case. The effect of outward migration can only be sustained if the viscosity in the disk is large. Overall, the differences between the treatments for the equation of state seem to play a more important role in disks with higher viscosity. A change in the adiabatic index and in the viscosity changes the zero-torque radius that separates inward from outward migration. For larger viscosities, temperatures in the disk become higher and the zero-torque radius moves to larger radii, allowing outward migration of a 20 Earth-mass planet to persist over an extended radial range. In combination with large disk masses, this may allow for an extended period of the outward migration of growing protoplanetary cores.

Seismic diagnostics for transport of angular momentum in stars

Rotational splittings are currently measured for several main sequence stars and a large number of red giants with the space mission Kepler. This will provide stringent constraints on rotation profiles. Our aim is to obtain seismic constraints on the internal transport and surface loss of angular momentum of oscillating solar-like stars. To this end, we study the evolution of rotational splittings from the pre-main sequence to the red-giant branch for stochastically excited oscillation modes. We modified the evolutionary code CESAM2K to take rotationally induced transport in radiative zones into account. Linear rotational splittings were computed for a sequence of $1.3 M_{\odot}$ models. Rotation profiles were derived from our evolutionary models and eigenfunctions from linear adiabatic oscillation calculations. We find that transport by meridional circulation and shear turbulence yields far too high a core rotation rate for red-giant models compared with recent seismic observations. We discuss several uncertainties in the physical description of stars that could have an impact on the rotation profiles. For instance, we find that the Goldreich-Schubert-Fricke instability does not extract enough angular momentum from the core to account for the discrepancy. In contrast, an increase of the horizontal turbulent viscosity by 2 orders of magnitude is able to significantly decrease the central rotation rate on the red-giant branch. Our results indicate that it is possible that the prescription for the horizontal turbulent viscosity largely underestimates its actual value or else a mechanism not included in current stellar models of low mass stars is needed to slow down the rotation in the radiative core of red-giant stars.

Can solar wind viscous drag account for coronal mass ejection deceleration?

The forces acting on solar Coronal Mass Ejections (CMEs) in the interplanetary medium have been evaluated so far in terms of an empirical drag coefficient CD ∼ 1 that quantifies the role of the aerodynamic drag experienced by a typical CME due to its interaction with the ambient solar wind. We use a microphysical prescription for viscosity in the turbulent solar wind to obtain an analytical model for the drag coefficient CD. This is the first physical characterization of the aerodynamic drag experienced by CMEs. We use this physically motivated prescription for CD in a simple, 1D model for CME propagation to obtain velocity profiles and travel times that agree well with observations of deceleration experienced by fast CMEs.

Study on the instability of accretion disks with anomalous viscosity

On the basis of hydrodynamic equations and the new anomalous viscosity, we numerically calculate the steady state of geometrically thin accretion disk around a black hole, including surface density, temperature and radial velocity distribution. We found the density distribution, especially in the inner region, is greatly different from the one with conventional α-viscosity. The stability of the accretion disks with anomalous viscosity is also examined. The results show that the Outward–moving mode is unstable, and the viscous mode is stable throughout the disk. The instability of the Inward–moving mode depends on the wavelength. The results show that the prescription of viscosity has some influence on the stability properties of the disk.

Anomalous gravitomagnetic viscosity in accretion disks

We propose a viscosity prescription in a self-gravitating system based on the self-generated gravitomagnetic field. It has been proved that the self-generated gravitomagnetic field with very low frequency is modulational unstable, leading to a self-collapse, and resulting in the enhancement of perturbed matter density and gravitomagnetic field in a small region; the interactions between highly spatially intermittent gravitomagnetic structures and the macrofluid with differential rotation are responsible for the anomalous gravitomagnetic viscosity. The result indicates that, the gravitomagnetic viscosity would be nearly 108 stronger than the molecular viscosity, which maybe useful for investigating the structures and instabilities in the self-gravitating accretion disks; provide an explanation for the transportion of angular momentum and the formation of gravitomagnetic jets.

On the prescriptions of viscosity in an accretion disk model

We have investigated a model of an accretion disk around a supermassive black hole in active galactic nuclei (AGN). Three kinds of prescription of viscosity have been examined. It is found that the distinction among the constructed models with these prescriptions is realized when the self-gravity of the disk is dominant over the central gravity in the outer region, though they yield the same consequences for the non-self-gravitating disk. The Toomre instability parameter is less than unity in the outer region, and consequently the disk becomes gravitationally unstable and is broken up into many clumps. It is also found that the appropriate condition for the density and temperature for pumping luminous H2O masers is not satisfied in our disk model. The density turns out to be rather high at a distance of the sub-parsec scale from the central black hole.

Hydrodynamic simulations of viscous accretion flows around black holes

We study the time evolution of a rotating, axisymmetric, viscous accretion flow around black holes using a grid based finite difference method. We use the Shakura-Sunyaev viscosity prescription. However, we compare with the results obtained when all the three independent components of the viscous stress are kept. We show that the centrifugal pressure supported shocks became weaker with the inclusion of viscosity. The shock is formed farther out when the viscosity is increased. When the viscosity is above a critical value, the shock disappears altogether and the flow becomes subsonic and Keplerian everywhere except in a region close to the horizon, where it remains supersonic. We also find that as the viscosity is increased, the amount of outflowing matter in the wind is decreased to less than a percentage of the inflow matter. Since the post-shock region could act as a reservoir of hot electrons or the so-called 'Compton cloud', the size of which changes with viscosity, the spectral properties are expected to depend on viscosity strongly: the harder states are dominated by low-angular momentum and the low viscosity flow with a significant outflows while the softer states are dominated by the high viscosity Keplerian flow having very little outflows.

The relationship between accretion disc age and stellar age and its consequences for protostellar discs

We show that for young stars which are still accreting and for which measurements of stellar age, disc mass and accretion rate are available, nominal disc age (Disc Age = Disc Mass / Accretion Rate) is approximately equal to the stellar age, at least within the considerable observational scatter. We then consider theoretical models of proto-stellar discs through analytic and numerical models. A variety of viscosity prescriptions including empirical power laws, magnetohydrodynamic turbulence and gravitational instability were considered within models describing the disc phenomena of dead zones, photoevaporation and planet formation. These models are generally poor fits to the observational data, showing values of 'Disc Age' which are too high by factors of 3 - 10. We then ask whether a systematic error in the measurement of one of the observational quantities might provide a reasonable explanation for this discrepancy. We show that for the observed systems only disc mass shows a systematic dependence on the value of 'Disc Age / Stellar Age' and we note that a systematic underestimate of the value of disc mass by a factor of around 3 - 5, would account for the discrepancy between theory and observations.

Structure and stability of a thin protostar disk with anomalous viscosity

The structure of a thin protostar disk has been investigated in the frame of anomalous magnetic viscosity prescription. The numerical results reveal that, the dynamical behavior of the disk with anomalous magnetic viscosity model is different from one with the well-known α-prescription, and the former can reveal the real behavior of the disk.

STUDIES OF THERMALLY UNSTABLE ACCRETION DISKS AROUND BLACK HOLES WITH ADAPTIVE PSEUDOSPECTRAL DOMAIN DECOMPOSITION METHOD. II. LIMIT-CYCLE BEHAVIOR IN ACCRETION DISKS AROUND KERR BLACK HOLES

For the first time ever, we derive equations governing the time-evolution of fully relativistic slim accretion disks in the Kerr metric, and numerically construct their detailed non-stationary models. We discuss applications of these general results to a possible limit-cycle behavior of thermally unstable disks. Our equations and numerical method are applicable in a wide class of possible viscosity prescriptions, but in this paper we use a diffusive form of the "standard alpha prescription" that assumes the viscous torque is proportional to the total pressure. In this particular case, we find that the parameters which dominate the limit-cycle properties are the mass-supply rate and the value of the alpha-viscosity parameter. Although the duration of the cycle (or the outburst) does not exhibit any clear dependence on the black hole spin, the maximal outburst luminosity (in the Eddington units) is positively correlated with the spin value. We suggest a simple method for a rough estimate of the black hole spin based on the maximal luminosity and the ratio of outburst to cycle durations. We also discuss a temperature-luminosity relation for the Kerr black hole accretion discs limit-cycle. Based on these results we discuss the limit-cycle behavior observed in microquasar GRS 1915+105. We also extend this study to several non-standard viscosity prescriptions, including a "delayed heating" prescription recently stimulated by the recent MHD simulations of accretion disks.

ON THE STRUCTURE OF ACCRETION DISKS WITH OUTFLOWS

In order to study the outflows from accretion disks, we solve the set of hydrodynamic equations for accretion disks in the spherical coordinates ($r\theta\phi$) to obtain the explicit structure along the $\theta$ direction. Using self-similar assumptions in the radial direction, we change the equations to a set of ordinary differential equations (ODEs) about the $\theta$-coordinate, which are then solved with symmetrical boundary conditions in the equatorial plane, and the velocity field is obtained. The $\alpha$ viscosity prescription is applied and an advective factor $f$ is used to simplify the energy equation.The results display thinner, quasi-Keplerian disks for Shakura-Sunyaev Disks (SSDs) and thicker, sub-Keplerian disks for Advection Dominated Accretion Flows (ADAFs) and slim disks, which are consistent with previous popular analytical models. However, an inflow region and an outflow region always exist, except when the viscosity parameter $\alpha$ is too large, which supports the results of some recent numerical simulation works. Our results indicate that the outflows should be common in various accretion disks and may be stronger in slim disks, where both advection and radiation pressure are dominant. We also present the structure dependence on the input parameters and discuss their physical meanings. The caveats of this work and possible improvements in the future are discussed.

Stability of a thin magnetized accretion disk surrounding a YSO with anomalous magnetic viscosity

The stability of a thin magnetized accretion disk surrounding a young stellar object (YSO) is numerically examined. Here we adopt a somewhat more physical viscosity prescription (anomalous magnetic viscosity) than previous authors did. Our results show that, under the short wavelength approximation (kr≫1), viscous and thermal modes are stable, while two magneto-acoustic modes are unstable. Particularly, for the unstable modes, the values of the growth rates are smaller in the outer region than ones in the inner. It means that the growth time scales for these instabilities in the inner region are shorter than in the outer, which maybe provide an explanation for fast burst.

Exact time-dependent solutions for the thin accretion disc equation: boundary conditions at finite radius

We discuss Green's-function solutions of the equation for a geometrically thin, axisymmetric Keplerian accretion disc with a viscosity prescription "\nu ~ R^n". The mathematical problem was solved by Lynden-Bell & Pringle (1974) for the special cases with boundary conditions of zero viscous torque and zero mass flow at the disc center. While it has been widely established that the observational appearance of astrophysical discs depend on the physical size of the central object(s), exact time-dependent solutions with boundary conditions imposed at finite radius have not been published for a general value of the power-law index "n". We derive exact Green's-function solutions that satisfy either a zero-torque or a zero-flux condition at a nonzero inner boundary R_{in}>0, for an arbitrary initial surface density profile. Whereas the viscously dissipated power diverges at the disc center for the previously known solutions with R_{in}=0, the new solutions with R_{in}>0 have finite expressions for the disc luminosity that agree, in the limit t=>infinity, with standard expressions for steady-state disc luminosities. The new solutions are applicable to the evolution of the innermost regions of thin accretion discs.

Conservative, special-relativistic smoothed particle hydrodynamics

We present and test a new, special-relativistic formulation of smoothed particle hydrodynamics (SPH). Our approach benefits from several improvements with respect to earlier relativistic SPH formulations. It is self-consistently derived from the Lagrangian of an ideal fluid and accounts for the terms that stem from non-constant smoothing lengths, usually called “grad-h terms”. In our approach, we evolve the canonical momentum and the canonical energy per baryon and thus circumvent some of the problems that have plagued earlier formulations of relativistic SPH. We further use a much improved artificial viscosity prescription which uses the extreme local eigenvalues of the Euler equations and triggers selectively on (a) shocks and (b) velocity noise. The shock trigger accurately monitors the relative density slope and uses it to fine-tune the amount of artificial viscosity that is applied. This procedure substantially sharpens shock fronts while still avoiding post-shock noise. If not triggered, the viscosity parameter of each particle decays to zero. None of these viscosity triggers is specific to special relativity, both could also be applied in Newtonian SPH. The performance of the new scheme is explored in a large variety of benchmark tests where it delivers excellent results. Generally, the grad-h terms deliver minor, though worthwhile, improvements. As expected for a Lagrangian method, it performs close to perfect in supersonic advection tests, but also in strong relativistic shocks, usually considered a particular challenge for SPH, the method yields convincing results. For example, due to its perfect conservation properties, it is able to handle Lorentz factors as large as γ = 50,000 in the so-called wall shock test. Moreover, we find convincing results in a rarely shown, but challenging test that involves so-called relativistic simple waves and also in multi-dimensional shock tube tests.

DISSIPATION EFFICIENCY IN TURBULENT CONVECTIVE ZONES IN LOW-MASS STARS

We extend the analysis of Penev et al. (2007) to calculate effective viscosities for the surface convective zones of three main sequence stars of 0.775Msun, 0.85Msun and the present day Sun. In addition we also pay careful attention to all normalization factors and assumptions in order to derive actual numerical prescriptions for the effective viscosity as a function of the period and direction of the external shear. Our results are applicable for periods that are too long to correspond to eddies that fall within the inertial subrange of Kolmogorov scaling, but no larger than the convective turnover time, when the assumptions of the calculation break down. We find linear scaling of effective viscosity with period and magnitudes at least three times larger than the Zahn (1966, 1989) prescription.