Accretion disks in which angular momentum transport is dominated by the magnetorotational instability (MRI) can also possess additional, purely hydrodynamic, drivers of turbulence. Even when the hydrodynamic processes, on their own, generate negligible levels of transport, they may still affect the evolution of the disk via their influence on the MRI. Here, we study the interaction between the MRI and hydrodynamic turbulence using local MRI simulations that include hydrodynamic forcing. As expected, we find that hydrodynamic forcing is generally negligible if it yields a saturated kinetic energy density that is small compared to the value generated by the MRI. For stronger hydrodynamic forcing levels, we find that hydrodynamic turbulence modifies transport, with the effect varying depending upon the spatial scale of hydrodynamic driving. Large scale forcing boosts transport by an amount that is approximately linear in the forcing strength, and leaves the character of the MRI (for example the ratio between Maxwell and Reynolds stresses) unchanged, up to the point at which the forced turbulence is an order of magnitude stronger than that generated by the MRI. Low amplitude small scale forcing may modestly suppress the MRI. We conclude that the impact of hydrodynamic turbulence on the MRI is generically ignorable in cases, such as convection, where the additional turbulence arises due to the accretion energy liberated by the MRI itself. Hydrodynamic turbulence may affect (and either enhance or suppress) the MRI if it is both strong, and driven by independent mechanisms such as self-gravity, supernovae, or solid-gas interactions in multiphase protoplanetary disks.
Quasi-periodic oscillations (QPOs) in the X-ray lightcurves of accreting neutron star and black hole binaries have been widely interpreted as being due to standing wave modes in accretion disks. These disks are thought to be highly turbulent due to the magnetorotational instability (MRI). We study wave excitation by MRI turbulence in the shearing box geometry. We demonstrate that axisymmetric sound waves and radial epicyclic motions driven by MRI turbulence give rise to narrow, distinct peaks in the temporal power spectrum. Inertial waves, on the other hand, do not give rise to distinct peaks …
Chaotic flow is studied in a series of numerical magnetohydrodynamical simulations that use the shearing box formalism. This mimics important features of local accretion disk dynamics. The magnetorotational instability gives rise to flow turbulence, and quantitative chaos parameters, such as the largest Lyapunov exponent, can be measured. Linear growth rates appear in these exponents even when the flow is fully turbulent. The extreme sensitivity to initial conditions associated with chaotic flows has practical implications, the most important of which is that hundreds of orbital times are need…
We study the behavior of magnetorotational turbulence in shearing box simulations with a radial and azimuthal extent up to ten scale heights. Maxwell and Reynolds stresses are found to increase by more than a factor two when increasing the box size beyond two scale heights in the radial direction. Further increase of the box size has little or no effect on the statistical properties of the turbulence. An inverse cascade excites magnetic field structures at the largest scales of the box. The corresponding 10% variation in the Maxwell stress launches a zonal flow of alternating sub- and super-Ke…
This paper reports on 3D numerical simulations of dynamics and thermodynamics in the diffuse ISM. Our models are local, account for sheared galactic rotation, magnetic fields, and realistic cooling, and resolve scales ~1-200 pc. This combination permits the study of quasi-steady turbulence in a cloudy medium representing the warm/cold atomic ISM. Turbulence is driven by the magnetorotational instability (MRI); our models are the first to study the saturated state of MRI under strongly inhomogeneous conditions, with cloud/intercloud contrasts of about 100. For volume-averaged densities n=0.25 -…
Using local 3D MHD simulations, we investigate ways in which galactic turbulence associated with the magnetorotational instability (MRI) may influence the formation and properties of GMCs. Our disk models are vertically stratified and subject to uniform shear. Initial magnetic fields are weak and purely vertical. For simplicity, we adopt an isothermal equation of state. We find that MRI-driven turbulence develops rapidly, with the saturated-state Shakura & Sunyaev parameter alpha~(0.15-0.3) dominated by Maxwell stresses. Many of the dimensionless characteristics of the turbulence (e.g. the rat…
We measure the turbulent diffusion coefficient of dust grains embedded in magnetorotational turbulence in a protoplanetary disc directly from numerical simulations and compare it to the turbulent viscosity of the flow. The simulations are done in a local coordinate frame comoving with the gas in Keplerian rotation. Periodic boundary conditions are used in all directions, and vertical gravity is not applied to the gas. Using a two-fluid approach, small dust grains of various sizes (with friction times up to $\varOmega_0 τ_{\rm f}=0.02$) are allowed to move under the influence of friction with t…
The saturation level of the magnetorotational instability (MRI) is investigated using three-dimensional MHD simulations. The shearing box approximation is adopted and the vertical component of gravity is ignored, so that the evolution of the MRI is followed in a small local part of the disk. We focus on the dependence of the saturation level of the stress on the gas pressure, which is a key assumption in the standard alpha disk model. From our numerical experiments it is found that there is a weak power-law relation between the saturation level of the Maxwell stress and the gas pressure in the…
(abridged) We apply a second-order Godunov code, Athena, to studies of the magnetorotational instability using unstratified shearing box simulations with a uniform net vertical field and a sinusoidally varying zero net vertical field. The Athena results agree well with similar studies that used different numerical algorithms. We conduct analyses to study the flow of energy from differential rotation to turbulent fluctuations to thermalization. A study of the temporal correlation between the time derivatives of volume-averaged energy components shows that energy injected into turbulent fluctuat…
The magnetorotational instability (MRI) plays an essential role in the formation of stars and black holes. By destabilizing hydrodynamically stable Keplerian flows, the MRI triggers turbulence and enables outward transport of angular momentum in accretion discs. We present the results of a liquid metal Taylor-Couette experiment under the influence of helical magnetic fields that show typical features of MRI at Reynolds numbers of the order 1000 and Hartmann numbers of the order 10. Particular focus is laid on an improved experiment in which split end caps are used to minimize the Ekman pumping…
We study the influence of the choice of transport coefficients (viscosity and resistivity) on MHD turbulence driven by the magnetorotational instability (MRI) in accretion disks. We follow the methodology described in paper I: we adopt an unstratified shearing box model and focus on the case where the net vertical magnetic flux threading the box vanishes. For the most part we use the finite difference code ZEUS, including explicit transport coefficients in the calculations. However, we also compare our results with those obtained using other algorithms (NIRVANA, the PENCIL code and a spectral …
We develop a local model for the exponential growth and saturation of the Reynolds and Maxwell stresses in turbulent flows driven by the magnetorotational instability. We first derive equations that describe the effects of the instability on the growth and pumping of the stresses. We highlight the relevance of a new type of correlations that couples the dynamical evolution of the Reynolds and Maxwell stresses and plays a key role in developing and sustaining the magnetorotational turbulence. We then supplement these equations with a phenomenological description of the triple correlations that …
We analyze the ionization state of the magnetohydrodynamically turbulent protoplanetary disks and propose a new mechanism of sustaining ionization. First, we show that in the quasi-steady state of turbulence driven by magnetorotational instability in a typical protoplanetary disk with dust grains, the amount of energy dissipation should be sufficient for providing the ionization energy that is required for activating magnetorotational instability. Second, we show that in the disk with dust grains the energetic electrons that compose electric currents in weakly ionized gas can provide collision…
In this letter, we present numerical calculations made to investigate the possible resolution dependence of the Shakura & Sunyaev (1973) viscosity parameter alpha from local magnetohydrodynamic simulations of the magnetorotational instability (MRI). We find that the values of alpha do indeed depend significantly on the numerical resolution but also that when the highest resolutions attainable by the computational resources available are used, the growth of the alpha-parameter seems to saturate. The values of alpha are at most of the order of 10^(-3), which indicates that the sole presence of t…
We carry out three-dimensional magnetohydrodynamical simulations of the magnetorotational (Balbus-Hawley) instability in weakly-ionized plasmas. We adopt a formulation in which the ions and neutrals each are treated as separate fluids coupled only through a collisional drag term. Ionization and recombination processes are not considered. The linear stability of the ion-neutral system has been previously considered by Blaes and Balbus (1994). Here we extend their results into the nonlinear regime by computing the evolution of Keplerian angular momentum distribution in the local shearing box app…
We present global 3D MHD simulations of disks of gas and solids, aiming at developing models that can be used to study various scenarios of planet formation and planet-disk interaction in turbulent accretion disks. A second goal is to show that Cartesian codes are comparable to cylindrical and spherical ones in handling the magnetohydrodynamics of the disk simulations, as the disk-in-a-box models presented here develop and sustain MHD turbulence. We investigate the dependence of the magnetorotational instability on disk scale height, finding evidence that the turbulence generated by the magnet…
(abridged) MHD turbulence is known to exist in shearing boxes with either zero or nonzero net magnetic flux. However, the way turbulence survives in the zero-net-flux case is not explained by linear theory and appears as a purely numerical result. Aims: We look for a nonlinear mechanism able to explain the persistence of MHD turbulence in shearing boxes with zero net magnetic flux, and potentially leading to large-scale dynamo action. Method: Spectral nonlinear simulations of the magnetorotational instability are shown to exhibit a large-scale axisymmetric magnetic field, maintained for a few …
Aims: We study the changes in the properties of turbulence driven by the magnetorotational instability in a shearing box, as the computational domain size in the radial direction is varied relative to the height Methods: We perform 3D simulations in the shearing box approximation, with a net magnetic flux, and we consider computational domains with different aspect ratios Results: We find that in boxes of aspect ratio unity the transport of angular momentum is strongly intermittent and dominated by channel solutions in agreement with previous work. In contrast, in boxes with larger aspect rati…
By performing local three-dimensional MHD simulations of stratified accretion disks, we investigate disk winds driven by MHD turbulence. Initially given weak vertical magnetic fields are effectively amplified by magnetorotational instability and winding due to differential rotation. Large scale channel flows develop most effectively at 1.5 - 2 times the scale heights where the magnetic pressure is comparable to but slightly smaller than the gas pressure. The breakup of these channel flows drives structured disk winds by transporting the Poynting flux to the gas. These features are universally …
Magnetorotational Instability (MRI), the instability causing turbulent transport in accretion disks, is studied in the kinetic regime. Radiatively Inefficient Accretion Flows (RIAFs), like the one around the supermassive black hole in the center of our Galaxy, are believed to be collisionless. Kinetic MHD formalism, based on the moments of the Vlasov equation, is used for linear analysis and nonlinear simulations. ZEUS MHD code is modified to include key kinetic MHD terms: anisotropic pressure tensor and anisotropic thermal conduction. Simulations use the local shearing box approximation. Pres…
Interstellar turbulence is expected to dissipate quickly in the absence of continuous energy input. I examine the energy available for driving the turbulence from six likely mechanisms: magnetorotational instability, gravitational instability, protostellar outflows, H II region expansion, stellar winds, and supernovae. I conclude that supernovae contribute far more energy than the other mechanisms, and so form the most likely drving mechanism.
We examine the linear stability of a flow threaded by a weak, vertical magnetic field in a disk with a keplerian rotation profile and a vertical stratification of the ionization degree as that predicted for vast portions of protoplanetary disks. A quasi-global analysis is carried out, where the form of the perturbations in the vertical direction is determined. Considering the ohmic magnetic diffusivity of the gas, the conditions leading to the magnetorotational instability are analyzed as a function of the diffusivity at the disk surfaces, its vertical profile and the strength of the unperturb…
We present three-dimensional magnetohydrodynamic simulations of the nonlinear evolution of the magnetorotational instability (MRI) with a non-zero Ohmic resistivity. The properties of the saturated state depend on the initial magnetic field configuration. In simulations with an initial uniform vertical field, the MRI is able to support angular momentum transport even for large resistivities through the quasi-periodic generation of axisymmetric radial channel solutions rather than through the maintenance of anisotropic turbulence. Simulations with zero net flux show that the angular momentum tr…
We study the properties of MHD turbulence driven by the magnetorotational instability (MRI) in accretion disks. We adopt the local shearing box model and focus on the special case for which the initial magnetic flux threading the disk vanishes. We employ the finite difference code ZEUS to evolve the ideal MHD equations. Performing a set of numerical simulations in a fixed computational domain with increasing resolution, we demonstrate that turbulent activity decreases as resolution increases. We quantify the turbulent activity by measuring the rate of angular momentum transport through evaluat…
The magnetorotational instability (MRI) is the leading candidate for driving turbulence, angular momentum transport, and accretion in astrophysical disks. I consider the linear theory of the MRI in a thin, equatorial disk in the Kerr metric. I begin by analyzing a mechanical model for the MRI that consists of two point masses on nearly circular orbits connected by a spring. I then develop a local Cartesian coordinate system for thin, equatorial Kerr disks. In this local model general relativistic effects manifest themselves solely through changes in the Coriolis parameter and in the tidal expa…
We investigate the stability of differentially rotating proto-neutron stars (PNSs) with a toroidal magnetic field. Stability criteria for nonaxisymmetric MHD instabilities are derived using a local linear analysis. PNSs are expected to have much stronger radial shear in the rotation velocity compared to normal stars. We find that nonaxisymmetric magnetorotational instability (NMRI) with a large azimuthal wavenumber $m$ is dominant over the kink mode ($m=1$) in differentially rotating PNSs. The growth rate of the NMRI is of the order of the angular velocity $Ω$ which is faster than that of the …
The most efficient energy sources known in the Universe are accretion disks. Those around black holes convert 5 -- 40 per cent of rest-mass energy to radiation. Like water circling a drain, inflowing mass must lose angular momentum, presumably by vigorous turbulence in disks, which are essentially inviscid. The origin of the turbulence is unclear. Hot disks of electrically conducting plasma can become turbulent by way of the linear magnetorotational instability. Cool disks, such as the planet-forming disks of protostars, may be too poorly ionized for the magnetorotational instability to occur,…
Three completely different lines of work have recently led to the conclusion that the magnetorotational instability (MRI) may create a hoop-stress that collimates jets. One argument is based upon consideration that magnetohydrodynamic turbulence, in general, and turbulence driven by the MRI, in particular, is more nearly viscoelastic than it is viscous. Another argument is based upon the dispersion relation for the MRI in the context of 1D simulations of core collapse. Yet a third argument rests in the results of direct numerical magnetohydrodynamic simulations of collapsars and thick accretio…
Despite the importance of the magnetorotational instability (MRI) as a fundamental mechanism for angular momentum transport in magnetized accretion disks, it has yet to be demonstrated in the laboratory. A liquid sodium alpha-omega dynamo experiment at the New Mexico Institute of Mining and Technology provides an ideal environment to study the MRI in a rotating metal annulus (Couette flow). A local stability analysis is performed as a function of shear, magnetic field strength, magnetic Reynolds number, and turbulent Prandtl number. The later takes into account the minimum turbulence induced b…
We present 3D magnetohydrodynamic (MHD) numerical simulations of the evolution of self--gravitating and weakly magnetized disks with an adiabatic equation of state. Such disks are subject to the development of both the magnetorotational and gravitational instabilities, which transport angular momentum outward. As in previous studies, our hydrodynamical simulations show the growth of strong m=2 spiral structure. This spiral disturbance drives matter toward the central object and disappears when the Toomre parameter Q has increased well above unity. When a weak magnetic field is present as well,…
The nonlinear evolution of the magnetorotational instability (MRI) in weakly ionized accretion disks, including the effect of the Hall term and ohmic dissipation, is investigated using local three-dimensional MHD simulations and various initial magnetic field geometries. When the magnetic Reynolds number, Re_M \equiv v_A^2 / ηΩ(where v_A is the Alfven speed, ηthe magnetic diffusivity, and Ωthe angular frequency), is initially larger than a critical value Re_{M, crit}, the MRI evolves into MHD turbulence in which angular momentum is transported efficiently by the Maxwell stress. If Re_M < Re_{M…
The magnetorotational instability is thought to be responsible for the generation of magnetohydrodynamic turbulence that leads to enhanced outward angular momentum transport in accretion discs. Here, we present the first formal analytical proof showing that, during the exponential growth of the instability, the mean (averaged over the disc scale-height) Reynolds stress is always positive, the mean Maxwell stress is always negative, and hence the mean total stress is positive and leads to a net outward flux of angular momentum. More importantly, we show that the ratio of the Maxwell to the Reyn…
Global axisymmetric stability of viscous, resistive, magnetized Couette flow is re-examined, with the emphasis on flows that would be hydrodynamically stable according to Rayleigh's criterion: opposing gradients of angular velocity and specific angular momentum. In this regime, magnetorotational instabilities [MRI] may occur. Previous work has focused on the Rayleigh-unstable regime. To prepare for an experimental study of MRI, which are of intense astrophysical interest, we solve for global linear modes in a wide gap with realistic dissipation coefficients. Exchange of stability appears to oc…
[Abridged] The turbulent stresses that lead to angular momentum transport in accretion discs have often been treated as resulting from an isotropic effective viscosity, related to the pressure through the alpha parametrization of Shakura & Sunyaev. This simple approach may be adequate for the simplest aspects of accretion disc theory, and was necessitated historically by an incomplete understanding of the origin of the turbulence. More recently, Balbus & Hawley have shown that the magnetorotational instability provides a robust mechanism of generating turbulent Reynolds and Maxwell stresses in…
In preparation for an experimental study of magnetorotational instability (MRI) in liquid metal, we present non-ideal two-dimensional magnetohydrodynamic simulations of the nonlinear evolution of MRI in the experimental geometry. The simulations adopt initially uniform vertical magnetic fields, conducting radial boundaries, and periodic vertical boundary conditions. No-slip conditions are imposed at the cylinders. Our linear growth rates compare well with existing local and global linear analyses. The MRI saturates nonlinearly with horizontal magnetic fields comparable to the initial axial fie…
Differential rotation occurs in conducting flows in accretion disks and planetary cores. In such systems, the magnetorotational instability can arise from coupling Lorentz and centrifugal forces to cause large radial angular momentum fluxes. We present the first experimental observation of the magnetorotational instability. Our system consists of liquid sodium between differentially rotating spheres, with an imposed coaxial magnetic field. We characterize the observed patterns, dynamics and torque increases, and establish that this instability can occur from a hydrodynamic turbulent background…
Migration of protoplanets in their gaseous host disks may be largely responsible for the observed orbital distribution of extrasolar planets. Recent simulations have shown that the magnetorotational turbulence thought to drive accretion in protoplanetary disks can affect migration by turning it into an orbital random walk. However, these simulations neglected the disk's ionization structure. Low ionization fraction near the midplane of the disk can decouple the magnetic field from the gas, forming a dead zone with reduced or no turbulence. Here, to understand the effect of dead zones on protop…
The effect of the Hall term on the evolution of the magnetorotational instability (MRI) in weakly ionized accretion disks is investigated using local axisymmetric simulations. First, we show that the Hall term has important effects on the MRI when the temperature and density in the disk is below a few thousand K and between 10^13 and 10^18 cm^{-3} respectively. Such conditions can occur in the quiescent phase of dwarf nova disks, or in the inner part (inside 10 - 100 AU) of protoplanetary disks. When the Hall term is important, the properties of the MRI are dependent on the direction of the ma…
Hydromagnetic stresses in accretion discs have been the subject of intense theoretical research over the past one and a half decades. Most of the disc simulations have assumed a small initial magnetic field and studied the turbulence that arises from the magnetorotational instability. However, gaseous discs in galactic nuclei and in some binary systems are likely to have significant initial magnetisation. Motivated by this, we performed ideal magnetohydrodynamic simulations of strongly magnetised, vertically stratified discs in a Keplerian potential. Our initial equilibrium configuration, whic…
Axisymmetric magnetorotational instability (MRI) in viscous accretion disks is investigated by linear analysis and two-dimensional nonlinear simulations. The linear growth of the viscous MRI is characterized by the Reynolds number defined as $R_{\rm MRI} \equiv v_A^2/νΩ$, where $v_A$ is the Alfv{é}n velocity, $ν$ is the kinematic viscosity, and $Ω$ is the angular velocity of the disk. Although the linear growth rate is suppressed considerably as the Reynolds number decreases, the nonlinear behavior is found to be almost independent of $R_{\rm MRI}$. At the nonlinear evolutionary stage, a two-c…
The question of a purely hydrodynamic origin of turbulence in accretion disks is reexamined, on the basis of a large body of experimental and numerical evidence on various subcritical (i.e., linearly stable) hydrodynamic flows. One of the main points of this paper is that the length scale and velocity fluctuation amplitude which are characteristic of turbulent transport in these flows scale like $Re_m^{-1/2}$, where $Re_m$ is the minimal Reynolds number for the onset of fully developed turbulence. From this scaling, a simple explanation of the dependence of $Re_m$ with relative gap width in su…
This paper deals with the problem of hydrodynamic shear turbulence in non-magnetized Keplerian disks. We wish to draw attention to a route to hydrodynamic turbulence which seems to be little known by the astrophysical community, but which has been intensively discussed among fluid dynamicists during the past decade. In this so-called `bypass' concept for the onset of turbulence, perturbations undergo a transient growth, and they may reach an amplitude that is sufficiently large to allow positive feedback through nonlinear interactions. This transient growth is linear in nature, and thus it dif…
We show by means of a perturbative weakly nonlinear analysis that the axisymmetric magnetorotational instability (MRI) of a viscous, resistive, incompressible rotating shear flow in a thin channel gives rise to a real Ginzburg-Landau equation for the disturbance amplitude. For small magnetic Prandtl number (${\cal P}_{\rm m}$), the saturation amplitude is $\propto \sqrt{{\cal P}_{\rm m}}$ and the resulting momentum transport scales as ${\cal R}^{-1}$, where $\cal R$ is the {\em hydrodynamic} Reynolds number. Simplifying assumptions, such as linear shear base flow, mathematically expedient boun…
(Abridged) In this paper we have used the RIEMANN code for computational astrophysics to study the interaction of a realistic distribution of dust grains with gas in a vertically stratified protostellar accretion disc. The disc was modeled to have the density and temperature of a minimum mass solar nebula, and was driven to a fully-developed turbulence via the magnetorotational instability (MRI). We find that the inclusion of standard dust to gas ratios does not have any significant effect on the MRI even when the dust sediments to the midplane of the accretion disc. The density distribution o…
We study the probability distribution function (PDF) of the mass density in simulations of supersonic turbulence with properties appropriate for molecular clouds. For this study we use Athena, a new higher-order Godunov code. We find there are surprisingly similar relationships between the mean of the time-averaged PDF and the turbulent Mach number for driven hydrodynamic and strong-field MHD turbulence. There is, however, a large scatter about these relations, indicating a high level of temporal and spatial variability in the PDF. Thus, the PDF of the mass density is unlikely to be a good mea…
We consider the magnetorotational instability (MRI) of a hydrodynamically stable Taylor-Couette flow with a helical external magnetic field in the inductionless approximation defined by a zero magnetic Prandtl number ($\Pm=0)$. This leads to a considerable simplification of the problem eventually containing only hydrodynamic variables. First, we point out that the energy of any perturbation growing in the presence of magnetic field has to grow faster without the field. This is a paradox because the base flow is stable without the magnetic while it is unstable in the presence of a helical magne…
We analyze numerically the magnetorotational instability of a Taylor-Couette flow in a helical magnetic field (HMRI) using the inductionless approximation defined by a zero magnetic Prandtl number (Pm=0). The Chebyshev collocation method is used to calculate the eigenvalue spectrum for small amplitude perturbations. First, we carry out a detailed conventional linear stability analysis with respect to perturbations in the form of Fourier modes that corresponds to the convective instability which is not in general self-sustained. The helical magnetic field is found to extend the instability to a…
Origin of hydrodynamic turbulence in rotating shear flow, e.g. plane Couette flow including the Coriolis force, is a big puzzle. While the flow often exhibits turbulence in laboratory experiments, according to the linear perturbation theory it should always be stable. We demonstrate that the secondary disturbance to the primarily perturbed flow triggers the elliptical instability in such a system and hence an exponential growing eigenmode. This result has an immediate application to astrophysics and geophysics which often exhibit a turbulent flow with the Keplerian angular momentum profile. We…
We study the possible origin of hydrodynamic turbulence in cold accretion disks such as those in star-forming systems and quiescent cataclysmic variables. As these systems are expected to have neutral gas, the turbulent viscosity is likely to be hydrodynamic in origin, not magnetohydrodynamic. Therefore MRI will be sluggish or even absent in such disks. Although there are no exponentially growing eigenmodes in a hydrodynamic disk, because of the non-normal nature of the eigenmodes, a large transient growth in the energy is still possible, which may enable the system to switch to a turbulent st…
