Local Shearing Box Approximation Research Articles

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.

Local Simulations of MRI turbulence with Meshless Methods

Abstract The magneto-rotational instability (MRI) is one of the most important processes in sufficiently ionized astrophysical disks. Grid-based simulations, especially those using the local shearing box approximation, provide a powerful tool to study the nonlinear turbulence the MRI produces. On the other hand, meshless methods have been widely used in cosmology, galactic dynamics, and planet formation, but have not been fully deployed on the MRI problem. We present local unstratified and vertically stratified MRI simulations with two meshless MHD schemes: a recent implementation of smoothed-particle magnetohydrodynamics (SPH MHD), and a meshless finite-mass (MFM) MHD scheme with constrained gradient divergence cleaning, as implemented in the GIZMO code. Concerning variants of the SPH hydro force formulation, we consider both the “vanilla” SPH and the PSPH variant included in GIZMO. We find, as expected, that the numerical noise inherent in these schemes significantly affects turbulence. Furthermore, a high-order kernel, free of the pairing instability, is necessary. Both schemes adequately simulate MRI turbulence in unstratified shearing boxes with net vertical flux. The turbulence, however, dies out in zero-net-flux unstratified boxes, probably due to excessive numerical dissipation. In zero-net-flux vertically stratified simulations, MFM can reproduce the MRI dynamo and its characteristic butterfly diagram for several tens of orbits before ultimately decaying. In contrast, extremely strong toroidal fields, as opposed to sustained turbulence, develop in equivalent simulations using SPH MHD. The latter unphysical state is likely caused by a combination of excessive artificial viscosity, numerical resistivity, and the relatively large residual errors in the divergence of the magnetic field.

IRON OPACITY BUMP CHANGES THE STABILITY AND STRUCTURE OF ACCRETION DISKS IN ACTIVE GALACTIC NUCLEI

ABSTRACT Accretion disks around supermassive black holes have regions where the Rosseland mean opacity can be larger than the electron scattering opacity due to the large number of bound–bound transitions in iron. We study the effects of this iron opacity “bump” on the thermal stability and vertical structure of radiation-pressure-dominated accretion disks, utilizing three-dimensional radiation magnetohydrodynamic (MHD) simulations in the local shearing box approximation. The simulations self-consistently calculate the heating due to MHD turbulence caused by magneto-rotational instability and radiative cooling by using the radiative transfer module based on a variable Eddington tensor in Athena. For a 5 × 108 solar mass black hole with ∼3% of the Eddington luminosity, a model including the iron opacity bump maintains its structure for more than 10 thermal times without showing significant signs of thermal runaway. In contrast, if only electron scattering and free–free opacity are included as in the standard thin disk model, the disk collapses on the thermal timescale. The difference is caused by a combination of (1) an anti-correlation between the total optical depth and the midplane pressure, and (2) enhanced vertical advective energy transport. These results suggest that the iron opacity bump may have a strong impact on the stability and structure of active galactic nucleus (AGN) accretion disks, and may contribute to a dependence of AGN properties on metallicity. Since this opacity is relevant primarily in UV emitting regions of the flow, it may help to explain discrepancies between observation and theory that are unique to AGNs.

MAGNETOROTATIONAL INSTABILITY: NONMODAL GROWTH AND THE RELATIONSHIP OF GLOBAL MODES TO THE SHEARING BOX

We study the magnetorotational instability (MRI) using nonmodal stability techniques. Despite the spectral instability of many forms of the MRI, this proves to be a natural method of analysis that is well-suited to deal with the non-self-adjoint nature of the linear MRI equations. We find that the fastest growing linear MRI structures on both local and global domains can look very different to the eigenmodes, invariably resembling waves shearing with the background flow (shear waves). In addition, such structures can grow many times faster than the least stable eigenmode over long time periods, and be localized in a completely different region of space. These ideas lead -- for both axisymmetric and non-axisymmetric modes -- to a natural connection between the global MRI and the local shearing box approximation. By illustrating that the fastest growing global structure is well described by the ordinary differential equations (ODEs) governing a single shear wave, we find that the shearing box is a very sensible approximation for the linear MRI, contrary to many previous claims. Since the shear wave ODEs are most naturally understood using nonmodal analysis techniques, we conclude by analyzing local MRI growth over finite time-scales using these methods. The strong growth over a wide range of wave-numbers suggests that nonmodal linear physics could be of fundamental importance in MRI turbulence.

ON THE THERMAL STABILITY OF RADIATION-DOMINATED ACCRETION DISKS

We study the long-term thermal stability of radiation dominated disks in which the vertical structure is determined self-consistently by the balance of heating due to dissipation of MHD turbulence driven by the magneto-rotational instability (MRI), and cooling due to radiation emitted at the photosphere. The calculations adopt the local shearing box approximation, and utilize the recently developed radiation transfer module in the Athena MHD code based on a variable Eddington tensor rather than an assumed local closure. After saturation of the MRI, in many cases the disk maintains a steady vertical structure for many thermal times. However, in every case in which the box size in the horizontal directions is at least one pressure scale height, fluctuations associated with MRI turbulence and dynamo action in the disk eventually trigger a thermal runaway which causes the disk to either expand or contract until the calculation must be terminated. During runaway, the dependence of the heating and cooling rates on total pressure satisfy the simplest criterion for classical thermal instability. We identify several physical reasons why the thermal runaway observed in our simulations differ from the standard alpha disk model, for example the advection of radiation contributes a non-negligible fraction to the vertical energy flux at the largest radiation pressure, most of the dissipation does not happen in the disk mid-plane, and the change of dissipation scale height with mid-plane pressure is slower than the change of density scale height. We discuss how and why our results differ from those published previously. Such thermal runaway behavior might have important implications for interpreting temporal variability in observed systems, but fully global simulations are required to study the saturated state before detailed predictions can be made.

LOW-MASS PLANETS IN PROTOPLANETARY DISKS WITH NET VERTICAL MAGNETIC FIELDS: THE PLANETARY WAKE AND GAP OPENING

We study wakes and gap opening by low mass planets in gaseous protoplanetary disks threaded by net vertical magnetic fields which drive magnetohydrodynamical (MHD) turbulence through the magnetorotational instabilty (MRI), using three dimensional simulations in the unstratified local shearing box approximation. The wakes, which are excited by the planets, are damped by shocks similar to the wake damping in inviscid hydrodynamic (HD) disks. Angular momentum deposition by shock damping opens gaps in both MHD turbulent disks and inviscid HD disks even for low mass planets, in contradiction to the "thermal criterion" for gap opening. To test the "viscous criterion", we compared gap properties in MRI-turbulent disks to those in viscous HD disks having the same stress, and found that the same mass planet opens a significantly deeper and wider gap in net vertical flux MHD disks than in viscous HD disks. This difference arises due to the efficient magnetic field transport into the gap region in MRI disks, leading to a larger effective \alpha within the gap. Thus, across the gap, the Maxwell stress profile is smoother than the gap density profile, and a deeper gap is needed for the Maxwell stress gradient to balance the planetary torque density. We also confirmed the large excess torque close to the planet in MHD disks, and found that long-lived density features (termed zonal flows) produced by the MRI can affect planet migration. The comparison with previous results from net toroidal flux/zero flux MHD simulations indicates that the magnetic field geometry plays an important role in the gap opening process. Overall, our results suggest that gaps can be commonly produced by low mass planets in realistic protoplanetary disks, and caution the use of a constant \alpha-viscosity to model gaps in protoplanetary disks.

EFFECT OF BACKGROUND MAGNETIC FIELD ON TURBULENCE DRIVEN BY MAGNETOROTATIONAL INSTABILITY IN ACCRETION DISKS

We investigate the background magnetic field dependence of the saturated state of a magnetorotational instability (MRI) in an accretion disk by performing three-dimensional magnetohydrodynamic simulations. We assume an unstratified disk by employing the local shearing box approximation. Three different uniform background magnetic field configurations are treated for a wide range of field intensities. These simulations indicate that the time variations of the turbulent stress and the magnetic energy are altered by the presence of a poloidal component of the background field. We find that the saturation amplitude of the turbulent stress and the magnetic energy are determined by both the poloidal and azimuthal components of the field. In particular, when the poloidal component has the same intensity, the obtained turbulent stress for β y0 ≈ 200 becomes smaller than those for a purely poloidal field case. Despite the fact that the background field affects the MRI turbulence, the correlation between the obtained turbulent stress and the magnetic energy in the nonlinear stage is independent of the field topology. Our results indicate that the saturated turbulent stress has a stronger correlation with the power of the perturbed component of the magnetic field than with the power of the total magnetic field. These results suggest that both the intensity and the direction of the background magnetic field significantly affect the turbulent motion of the MRI in accretion disks.

LOCAL SIMULATIONS OF THE MAGNETOROTATIONAL INSTABILITY IN CORE-COLLAPSE SUPERNOVAE

Bearing in mind the application to core-collapse supernovae, we study nonlinear properties of the magneto-rotational instability (MRI) by means of three- dimensional simulations in the framework of a local shearing box approximation. By changing systematically the shear rates that symbolize the degree of differential rotation in nascent proto-neutron stars (PNSs), we derive a scaling relation between the turbulent stress sustained by the MRI and the shear- vorticity ratio. Our parametric survey shows a power-law scaling between the turbulent stress ($<< w_{\rm tot}>>$) and the shear- vorticity ratio ($g_q$) as $<<w_{\rm tot}>> \propto g_q^{\delta}$ with its index $\delta \sim 0.5$. The MRI-amplified magnetic energy has a similar scaling relative to the turbulent stress, while the Maxwell stress has slightly smaller power-law index ($\sim 0.36$). By modeling the effect of viscous heating rates due to the MRI turbulence, we show that the stronger magnetic fields or the larger shear rates initially imposed lead to the higher dissipation rates. For a rapidly rotating PNS with the spin period in milliseconds and with strong magnetic fields of $10^{15}$ G, the energy dissipation rate is estimated to exceed $10^{51} {\rm erg\ sec^{-1}}$. Our results suggest that the conventional magnetohydrodynamic (MHD) mechanism of core-collapse supernovae is likely to be affected by the MRI-driven turbulence, which we speculate, on one hand, could harm the MHD-driven explosions due to the dissipation of the shear rotational energy at the PNS surface, on the other hand the energy deposition there might be potentially favorable for the working of the neutrino-heating mechanism.

Subcritical fluctuations and suppression of turbulence in differentially rotating gyrokinetic plasmas

Differential rotation is known to suppress linear instabilities in fusion plasmas. However, numerical experiments show that even in the absence of growing eigenmodes, subcritical fluctuations that grow transiently can lead to sustained turbulence, limiting the ability of the velocity shear to suppress anomalous transport. Here transient growth of electrostatic fluctuations driven by the parallel velocity gradient (PVG) and the ion temperature gradient (ITG) in the presence of a perpendicular (E × B) velocity shear is considered. The maximally simplified (but, as numerical simulations suggest, most promising for transport reduction) case of zero magnetic shear is treated in the framework of a local shearing box approximation. In this case there are no linearly growing eigenmodes, so all excitations are transient. In the PVG-dominated regime, the maximum amplification factor is found to be eN with N ∝ q/ϵ (safety factor/inverse aspect ratio), the maximally amplified wavenumbers perpendicular and parallel to the magnetic field are related by kyρi ≈ (ϵ/q)1/3k∥vthi/S, where ρi is the ion Larmor radius, vthi the ion thermal speed and S the E × B shear. In the ITG-dominated regime, N is independent of wavenumber and N ∝ vthi/(LTS), where LT is the ion-temperature scale length. Intermediate ITG–PVG regimes are also analysed and N is calculated as a function of q/ϵ, LT and S. Analytical results are corroborated and supplemented by linear gyrokinetic numerical tests. Regimes with N ≲ 1 for all wavenumbers are possible for sufficiently low values of q/ϵ (≲7 in our model); ion-scale turbulence is expected to be fully suppressed in such regimes. For cases when it is not suppressed, an elementary heuristic theory of subcritical PVG turbulence leading to a scaling of the associated ion heat flux with q, ϵ, S and LT is proposed; it is argued that the transport is much less ‘stiff’ than in the ITG regime.

Recent results from simulations of the magnetorotational instability

AbstractThe nonlinear saturation of the magnetorotational instability (MRI) is best studied through numerical MHD simulations. Recent results of simulations that adopt the local shearing box approximation, and fully global models that follow the entire disk, are described. Outstanding issues remain, such as a first-principles understanding of the dynamo processes that control saturation with no net magnetic flux. Important directions for future work include a better understanding of basic plasma processes, such as reconnection, dissipation, and particle acceleration, in the MHD turbulence driven by the MRI.

PLANETESIMAL AND PROTOPLANET DYNAMICS IN A TURBULENT PROTOPLANETARY DISK: IDEAL UNSTRATIFIED DISKS

The dynamics of planetesimals and planetary cores may be strongly influenced by density perturbations driven by magneto-rotational turbulence in their natal protoplanetary gas disks. Using the local shearing box approximation, we perform numerical simulations of planetesimals moving as massless particles in a turbulent, magnetized, unstratified gas disk. Our fiducial disk model shows turbulent accretion characterized by a Shakura-Sunyaev viscosity parameter of α ~ 10^-2, with rms density perturbations of ~10%. We measure the statistical evolution of particle orbital properties in our simulations including mean radius, eccentricity, and velocity dispersion. We confirm random walk growth in time of all three properties, the first time that this has been done with direct orbital integration in a local model. We find that the growth rate increases with the box size used at least up to boxes of eight scale heights in horizontal size. However, even our largest boxes show velocity dispersions sufficiently low that collisional destruction of planetesimals should be unimportant in the inner disk throughout its lifetime. Our direct integrations agree with earlier torque measurements showing that type I migration dominates over diffusive migration by stochastic torques for most objects in the planetary core and terrestrial planet mass range. Diffusive migration remains important for objects in the mass range of kilometer-sized planetesimals. Discrepancies in the derived magnitude of turbulence between local and global simulations of magneto-rotationally unstable disks remains an open issue, with important consequences for planet formation scenarios. (Less)

Nonlinear Evolution of the Magnetorotational Instability in Ion‐Neutral Disks

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 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 & Balbus. Here we extend their results to the nonlinear regime by computing the evolution of the Keplerian angular momentum distribution in the local shearing box approximation. We find significant turbulence and angular momentum transport when the collisional frequency is on the order of 100 times the orbital frequency Ω. At higher collision rates, the two-fluid system studied here behaves much like the fully ionized systems studied previously. At lower collision rates, the evolution of the instability is determined primarily by the properties of the ions, with the neutrals acting as a sink for the turbulence. Since in this regime saturation occurs when the magnetic field is superthermal with respect to the ion pressure, we find that the amplitude of the magnetic energy and the corresponding angular momentum transport rate is proportional to the ion density. Our calculations show that the ions and neutrals are essentially decoupled when the collision frequency is less than 0.01Ω; in this case, the ion fluid behaves as in the single-fluid simulations and the neutrals remain quiescent. We find that purely toroidal initial magnetic field configurations are unstable to the magnetorotational instability across the range of coupling frequencies.