Axisymmetric Magnetorotational Instability Research Articles

Fully Kinetic Shearing-box Simulations of Magnetorotational Turbulence in 2D and 3D. I. Pair Plasmas

The magnetorotational instability (MRI) is a fundamental mechanism determining the macroscopic dynamics of astrophysical accretion disks. In collisionless accretion flows around supermassive black holes, MRI-driven plasma turbulence cascading to microscopic (i.e., kinetic) scales can result in enhanced angular-momentum transport and redistribution, nonthermal particle acceleration, and a two-temperature state where electrons and ions are heated unequally. However, this microscopic physics cannot be captured with standard magnetohydrodynamic (MHD) approaches typically employed to study the MRI. In this work, we explore the nonlinear development of MRI turbulence in a pair plasma, employing fully kinetic particle-in-cell (PIC) simulations in two and three dimensions. First, we thoroughly study the axisymmetric MRI with 2D simulations, explaining how and why the 2D geometry produces results that differ substantially from 3D MHD expectations. We then perform the largest (to date) 3D simulations, for which we employ a novel shearing-box approach, demonstrating that 3D PIC models can reproduce the mesoscale (i.e., MHD) MRI dynamics in sufficiently large runs. With our fully kinetic simulations, we are able to describe the nonthermal particle acceleration and angular-momentum transport driven by the collisionless MRI. Since these microscopic processes ultimately lead to the emission of potentially measurable radiation in accreting plasmas, our work is of prime importance to understand current and future observations from first principles, beyond the limitations imposed by fluid (MHD) models. While in this first study we focus on pair plasmas for simplicity, our results represent an essential step toward designing more realistic electron–ion simulations, on which we will focus in future work.

Angular momentum transport in thin magnetically arrested discs

In accretion disks with large-scale ordered magnetic fields, the magnetorotational instability (MRI) is marginally suppressed, so other processes may drive angular momentum transport leading to accretion. Accretion could then be driven by large-scale magnetic fields via magnetic braking, but large-scale magnetic flux can build-up onto the black hole and within the disk leading to a magnetically-arrested disk (MAD). Such a MAD state is unstable to the magnetic Rayleigh-Taylor (RT) instability, which itself leads to vigorous turbulence and the emergence of low-density highly-magnetized bubbles. This instability was studied in a thin (ratio of half-height H to radius R, $H/R \approx 0.1$) MAD simulation, where it has a more dramatic effect on the dynamics of the disk than for thicker disks. We find that the low-density bubbles created by the magnetic RT instability decrease the stress (leading to angular momentum transport) in the disk rather than increasing magnetic torques. Indeed, we find that the dominant component of the stress is due to turbulent magnetic fields, despite the suppression of the axisymmetric MRI and the dominant presence of large-scale magnetic fields. This suggests that the magnetic RT instability plays a significant role in driving angular momentum transport in MADs.

The Weakly Nonlinear Magnetorotational Instability in a Global, Cylindrical Taylor–Couette Flow

Abstract We conduct a global, weakly nonlinear analysis of the magnetorotational instability (MRI) in a Taylor–Couette flow. This is a multiscale, perturbative treatment of the nonideal, axisymmetric MRI near threshold, subject to realistic radial boundary conditions and cylindrical geometry. We analyze both the standard MRI, initialized by a constant vertical background magnetic field, and the helical MRI, with an azimuthal background field component. This is the first weakly nonlinear analysis of the MRI in a global Taylor–Couette geometry, as well as the first weakly nonlinear analysis of the helical MRI. We find that the evolution of the amplitude of the standard MRI is described by a real Ginzburg–Landau equation (GLE), whereas the amplitude of the helical MRI takes the form of a complex GLE. This suggests that the saturated state of the helical MRI may itself be unstable on long spatial and temporal scales.

CONVECTION CAUSES ENHANCED MAGNETIC TURBULENCE IN ACCRETION DISKS IN OUTBURST

We present the results of local, vertically stratified, radiation MHD shearing box simulations of MRI turbulence appropriate for the hydrogen ionizing regime of dwarf nova and soft X-ray transient outbursts. We incorporate the frequency-integrated opacities and equation of state for this regime, but neglect non-ideal MHD effects and surface irradiation, and do not impose net vertical magnetic flux. We find two stable thermal equilibrium tracks in the effective temperature versus surface mass density plane, in qualitative agreement with the S-curve picture of the standard disk instability model. We find that the large opacity at temperatures near $10^4$K, a corollary of the hydrogen ionization transition, triggers strong, intermittent thermal convection on the upper stable branch. This convection strengthens the magnetic turbulent dynamo and greatly enhances the time-averaged value of the stress to thermal pressure ratio $\alpha$, possibly by generating vertical magnetic field that may seed the axisymmetric magnetorotational instability, and by increasing cooling so that the pressure does not rise in proportion to the turbulent dissipation. These enhanced stress to pressure ratios may alleviate the order of magnitude discrepancy between the $\alpha$-values observationally inferred in the outburst state and those that have been measured from previous local numerical simulations of magnetorotational turbulence that lack net vertical magnetic flux.

Revisiting linear dynamics of non-axisymmetric perturbations in weakly magnetized accretion discs

We investigate linear dynamics of non-axisymmetric perturbations in incompressible, vertically stratified Keplerian discs with a weak vertical magnetic field in the shearing box approximation. Perturbations are decomposed into shearing waves whose evolution is followed via numerical integration of the linearized ideal MHD equations. There are two basic modes in the system -- inertia-gravity waves and magnetic mode, which displays the magnetorotational instability (MRI). As distinct from previous studies, we introduce `eigenvariables' characterizing each (counter-propagating) component of the inertia-gravity and magnetic modes, which are governed by a set of four first order coupled ordinary differential equations. This allowed us to identify a new process of linear coupling of the two above non-axisymmetric modes due to the disc's differential rotation. We did a comparative analysis of the dynamics of non-axisymmetric and axisymmetric magnetic mode perturbations. It is shown that the growth of optimal and close-to-optimal non-axisymmetric harmonics of this mode, having transient nature, can prevail over the exponential growth of axisymmetric ones (i.e., over the axisymmetric MRI) during dynamical time. A possible implication of this result for axisymmetric channel solutions is discussed. Specifically, the formation of the channel may be affected/impeded by non-axisymmetric modes already at the early linear stage leading to its untimely disruption -- the outcome strongly depends on the amplitude and spectrum of initial perturbation. So, this competition may result in an uncertainty in the magnetic mode's non-linear dynamics. It is also shown that a maximum non-axisymmetric growth is at vertical wavelengths close to the scale-height for which compressibility effects are important. This indirectly suggests that compressibility plays a role in the dynamics of the non-axisymmetric MRI.

Quasi-global galactic magnetorotational instability with Braginskii viscosity

We present a global-in-radius linear analysis of the axisymmetric magnetorotational instability (MRI) in a collisional magnetized plasma with Braginskii viscosity. For a galactic angular velocity profile $\Omega$ we obtain analytic solutions for three magnetic field orientations: purely azimuthal, purely vertical and slightly pitched (almost azimuthal). In the first two cases the Braginskii viscosity damps otherwise neutrally stable modes, and reduces the growth rate of the MRI respectively. In the final case the Braginskii viscosity makes the MRI up to $2\sqrt{2}$ times faster than its inviscid counterpart, even for \emph{asymptotically small} pitch angles. We investigate the transition between the Lorentz-force-dominated and the Braginskii viscosity-dominated regimes in terms of a parameter $\sim \Omega \nub/B^2$ where $\nub$ is the viscous coefficient and $B$ the Alfv\'en speed. In the limit where the parameter is small and large respectively we recover the inviscid MRI and the magnetoviscous instability (MVI). We obtain asymptotic expressions for the approach to these limits, and find the Braginskii viscosity can magnify the effects of azimuthal hoop tension (the growth rate becomes complex) by over an order of magnitude. We discuss the relevance of our results to the local approximation, galaxies and other magnetized astrophysical plasmas. Our results should prove useful for benchmarking codes in global geometries.

The role of boundaries in the magnetorotational instability

In this paper, we investigate numerically the flow of an electrically conducting fluid in a cylindrical Taylor-Couette flow when an axial magnetic field is applied. To minimize Ekman recirculation due to vertical no-slip boundaries, two independently rotating rings are used at the vertical endcaps. This configuration reproduces setup used in laboratory experiments aiming to observe the magnetorotational instability (MRI). Our 3D global simulations show that the nature of the bifurcation, the non-linear saturation, and the structure of axisymmetric MRI modes are significantly affected by the presence of boundaries. In addition, large scale non-axisymmetric modes are obtained when the applied field is sufficiently strong. We show that these modes are related to Kelvin-Helmholtz destabilization of a free Shercliff shear layer created by the combined action of the applied field and the rotating rings at the endcaps. Finally, we compare our numerical simulations to recent experimental results obtained in the Princeton MRI experiment.

Global Hall-MHD simulations of magnetorotational instability in a plasma Couette flow experiment

Global MHD and Hall-MHD numerical simulations relevant to the Madison plasma Couette flow experiment (MPCX) have been performed using the extended MHD code NIMROD. The MPCX has been constructed to study the magnetorotational instability (MRI) in a plasma. The two-fluid Hall effect, which is relevant to some astrophysical situations such as protostellar disks, is also expected to be important in the MPCX. Here, we first derive the local Hall dispersion relation including viscosity, extending earlier work by Balbus and Terquem [Astrophys. J. 552, 235 (2001)]. The predictions of the local analysis are then compared with nonlocal calculations of linear stability of the MRI for a parameter range relevant to the MPCX. It is found that the MHD stability limit and mode structure are altered by the Hall term, and nonlocal analysis is necessary to obtain quantitatively reliable predictions for MPCX. Two-fluid physics also significantly changes the nonlinear evolution and saturation of the axisymmetric MRI. Both the Reynolds and Maxwell stresses contribute significantly to momentum transport. In the Hall regime, when the magnetic field is parallel to the rotation axis, the Maxwell stress is larger than the Reynolds stress (similar to the MHD regime). However, when the magnetic field is antiparallel to the rotation axis in the Hall regime, the Reynolds stress is much larger than the Maxwell stress. To further study the role of non-axisymmetric modes, we have also carried out fully nonlinear MHD computations. Non-axisymmetric modes play an increasingly important role as the magnetic Reynolds number increases and grow to large amplitudes in a saturated turbulent state.

MRI channel flows in vertically stratified models of accretion discs

Simulations of the magnetorotational instability (MRI) in 'unstratified' shearing boxes exhibit powerful coherent flows, whereby the fluid vertically splits into countermoving planar jets or `channels'. Channel flows correspond to certain axisymmetric linear MRI modes, and their preponderance follows from the remarkable fact that they are approximate nonlinear solutions of the MHD equations in the limit of weak magnetic fields. We show in this paper, analytically and with one-dimensional numerical simulations, that this property is also shared by certain axisymmetric MRI modes in vertically-stratified shearing boxes. These channel flows rapidly capture significant amounts of magnetic and kinetic energy, and thus are vulnerable to secondary shear instabilities. We examine these parasites in the vertically stratified context, and estimate the maximum amplitudes that channels attain before they are destroyed. These estimates suggest that a dominant channel flow will usually drive the disk's magnetic field to thermal strengths. The prominence of these flows and their destruction place enormous demands on simulations, but channels in their initial stages also offer a useful check on numerical codes. These benchmarks are especially valuable given the increasing interest in the saturation of the stratified MRI. Lastly we speculate on the potential connection between 'run-away' channel flows and outburst behaviour in protostellar and dwarf nova disks.

The Magneto-Rotational Instability Near Threshold: Spatio-Temporal Amplitude Equation and Saturation

We show, by means of a perturbative weakly nonlinear analysis, that the axisymmetric magneto-rotational instability (MRI) in a magnetic Taylor-Couette (mTC) flow in a thin-gap gives rise, for very small magnetic Prandtl numbers (P_m), to a real Ginzburg-Landau equation for the disturbance amplitude. The saturation amplitude A_s is found to scale in this regime as P^\delta, with 1/2<\delta<2/3 (depending on the boundary conditions adopted). The asymptotic results are shown to comply with numerical calculations performed by using a spectral code. They suggest that the transport due to the nonlinearly developed MRI may be vanishingly small for P_m << 1.

Axisymmetric magneto-rotational instability in viscous accretion disks

AbstractAxisymmetric MRI in viscous accretion disks is investigated. The linear growth of the viscous MRI is characterized by the Reynolds number RMRI ≡ v2A/νΩ, where vA is the Alfvén velocity, ν is the kinematic viscosity, and Ω is the angular velocity of the disk. Although the linear growth of the MRI is suppressed as the Reynolds number decreases, its nonlinear behavior is found to be almost independent of RMRI. At the nonlinear stage, the channel flow grows and the Maxwell stress increases even though RMRI is much smaller than unity. Nonlinear behavior of the MRI in the viscous regime can be explained by the characteristics of the linear dispersion relation. Applying our results to the case with both viscosity and resistivity, it is anticipated that the critical value of the Lundquist number SMRI ≡ v2A/ηΩ for active turbulence would depend on the magnetic Prandtl number Pm ≡ ν/η, where η is the magnetic diffusivity.

Axisymmetric magnetorotational instability in ideal and viscous laboratory plasmas

The original analysis of the axisymmetric magnetorotational instability (MRI) by Velikhov (Sov. Phys. JETP 9, 995 (1959)) and Chandrasekhar (Proc. Nat. Acad. Sci. 46, 253 (1960)), applied to the ideally conducting magnetized medium in the laboratory conditions and restricted to the incompressible approximation, is extended by allowing for the compressibility. Thereby, two additional driving mechanisms of MRI are revealed in addition to the standard drive due to the negative medium rotation frequency gradient (the Velikhov effect). One is due to the squared medium pressure gradient and another is a combined effect of the pressure and density gradients. For laboratory applications, the expression for the MRI boundary with all the above driving mechanisms and the stabilizing magnetoacoustic effect is derived. The effects of parallel and perpendicular viscosities on the MRI in the laboratory plasma are investigated. It is shown that, for strong viscosity, there is a family of MRI driven for the same condition as the ideal one. It is also revealed that the presence of strong viscosity leads to additional family of instabilities called the viscosity-driven MRI. Then the parallel-viscositydriven MRI looks as an overstability (oscillatory instability) possessing both the growth rate and the real part of oscillation frequency, while the perpendicular-viscosity MRI is the aperiodical instability.

Axisymmetric Magnetorotational Instability in Viscous Accretion Disks

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 -->RMRI ≡ v2A/ν Ω , where -->vA is the Alfven 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 -->RMRI. At the nonlinear evolutionary stage, a two-channel flow continues growing and the Maxwell stress increases until the end of calculations even though the Reynolds number is much smaller than unity. A large portion of the injected energy to the system is converted to the magnetic energy. The gain rate of the thermal energy, on the other hand, is found to be much larger than the viscous heating rate. Nonlinear behavior of the MRI in the viscous regime and its difference from that in the highly resistive regime can be explained schematically by using the characteristics of the linear dispersion relation. Applying our results to the case with both the viscosity and resistivity, it is anticipated that the critical value of the Lundquist number -->SMRI ≡ v2A/η Ω for active turbulence depends on the magnetic Prandtl number -->SMRI ,c Pm1/2 in the regime of -->Pm 1 and remains constant when -->Pm 1, where -->Pm ≡ SMRI/RMRI = ν/η and η is the magnetic diffusivity.

The azimuthal magnetorotational instability (AMRI)

AbstractWe consider the flow of an electrically conducting fluid between differentially rotating cylinders, in the presence of an externally imposed current‐free toroidal field B0(Rin/R) êϕ . It is known that the classical, axisymmetric magnetorotational instability does not exist for such a purely toroidal imposed field.We show here that a nonaxisymmetric magnetorotational instability does exist, having properties very similar to the axisymmetric magnetorotational instability in the presence of an axial field. In the nonlinear regime the magnetic energy of the perturbances is shifted (in the sense of an inverse cascade) to the axisymmetric mode rather than to the modes with m &gt; 1. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Nonlinear saturation of the magnetorotational instability near threshold in a thin-gap Taylor-Couette setup

We study the saturation near threshold of the axisymmetric magnetorotational instability (MRI) of a viscous, resistive, incompressible fluid in a thin-gap Taylor-Couette configuration. A vertical magnetic field, Keplerian shear, and no-slip conducting radial boundary conditions are adopted. The weakly nonlinear theory leads to a real Ginzburg-Landau equation for the disturbance amplitude, as in our previous idealized analysis. For small magnetic Prandtl number (Pm<<1) , the saturation amplitude scales as Pm2/3 while the magnitude of angular momentum transport scales as Pm4/3. The difference from the previous scalings (proportional to Pm1/2 and Pm respectively) is attributed to the emergence of radial boundary layers. Away from those, steady-state nonlinear saturation is achieved through a modest reduction in the destabilizing shear. These results will be useful in understanding MRI laboratory experiments and associated numerical simulations.

Weakly Nonlinear Analysis of the Magnetorotational Instability in a Model Channel Flow

We show by means of a perturbative weakly nonlinear analysis that the axisymmetric magnetorotational instability (MRI) of a viscous, resistive, incompressible rotating shear flow subject to a background vertical magnetic field in a thin channel gives rise to a Ginzburg-Landau equation for the disturbance amplitude. For small magnetic Prandtl number (P(m)), the saturation amplitude is proportional square root P(m) and the resulting momentum transport scales as R(-1), where R is the hydrodynamic Reynolds number. Simplifying assumptions, such as linear shear base flow, mathematically expedient boundary conditions, and continuous spectrum of the vertical linear modes, are used to facilitate this analysis. The asymptotic results are shown to comply with numerical calculations using a spectral code. They suggest that the transport due to the nonlinearly developed MRI may be very small in experimental setups with P(m)<<1.

MHD Stability in Flowing Plasmas: Connection between Fusion Plasma and Astrophysics Research

Axisymmetric magneto-rotational instability (MRI) is studied in comparison with interchange instability (IntI) in a rotating cylindrical plasma. MRI is driven by the shear of plasma rotation, and the IntI by the density gradient with effective gravity due to the plasma rotation. The eigenmode equation for the MRI has the same form as that for the IntI. The local stability criterion is also summarized in a similar statement as “the spatial gradient of centrifugal force greater than the square of Alfven frequency causes instability.” However, the MRI is essentially different from the IntI because of the non-Hermitian property. The Keplerian rotation generates irregular singularity at the center of the disk, which yields a continuum of eigenvalues with non-orthogonal and square-integrable eigenfunctions.

Magnetorotational instability in electrically driven flow of liquid metal: Spectral analysis of global modes

The spectral magnetohydrodynamics stability of liquid metal differentially rotating in transverse magnetic field is studied numerically by solving the eigenvalue problem with rigid-wall boundary conditions. The equilibrium velocity profile used in calculations corresponds to the electrically driven flow in circular channel with the rotation law Ω(r)∝1∕r2. This type of flow profile is planned to be used in new experimental devices to test the magnetorotational instability (MRI) in the laboratory. Our analysis includes calculations of the eigenfrequency spectra for both axisymmetric (with azimuthal wavenumber m=0) and nonaxisymmetric (m≠0) modes. It is shown that for certain parameters the flow is unstable with respect to MRI with the fastest growth rate corresponding to the axisymmetric mode. For other parameters, the axisymmetric MRI modes can be suppressed and the instability develops only for modes with m≠0.

Nonaxisymmetric Magnetorotational Instability in Proto–Neutron Stars

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 $\Omega$ which is faster than that of the kink-type instability by several orders of magnitude. The stability criteria are analogous to those of the axisymmetric magnetorotational instability with a poloidal field, although the effects of leptonic gradients are considered in our analysis. The NMRI can grow even in convectively stable layers if the wavevectors of unstable modes are parallel to the restoring force by the Brunt-V\ais\al\a oscillation. The nonlinear evolution of NMRI could amplify the magnetic fields and drive MHD turbulence in PNSs, which may lead to enhancement of the neutrino luminosity.

WKB Analysis of Axisymmetric Magneto-Rotational Instability in a Thin Accretion Disk

The temporal behavior of axisymmetric magneto-rotational instability in a thin accretion disk is analyzed via the Wentzel-Kramars-Brillouin (WKB) method. The height of the thin disk is used as a small parameter. It is found that the oscillation of the envelope of the mode accelerates with time because of the density distribution in the direction of the disk height.