Relativistic Magnetohydrodynamic Simulations Research Articles

Mildly Super-Eddington Accretion onto Slowly Spinning Black Holes Explains the X-Ray Weakness of the Little Red Dots

JWST has revealed a population of low-luminosity active galactic nuclei at z > 4 in compact, red hosts (the “Little Red Dots,” or LRDs), which are largely undetected in X-rays. We investigate this phenomenon using General Relativistic Radiation Magnetohydrodynamics simulations of super-Eddington accretion onto a supermassive black hole (SMBH) with M • = 107 M ⊙ at z ∼ 6, representing the median population; the spectral energy distributions (SEDs) that we obtain are intrinsically X-ray weak. The highest levels of X-ray weakness occur in SMBHs accreting at mildly super-Eddington rates (1.4 < f Edd < 4) with zero spin, viewed at angles >30° from the pole. X-ray bolometric corrections in the observed 2–10 keV band reach ∼104 at z = 6, ∼5 times higher than the highest constraint from X-ray stacking. Most SEDs are extraordinarily steep and soft in the X-rays (median photon index Γ = 3.1, mode of Γ = 4.4). SEDs strong in the X-rays have harder spectra with a high-energy bump when viewed near the hot (>108 K) and highly relativistic jet, whereas X-ray weak SEDs lack this feature. Viewing an SMBH within 10° of its pole, where beaming enhances the X-ray emission, has a ∼1.5% probability, matching the LRD X-ray detection rate. Next-generation observatories like AXIS will detect X-ray-weak LRDs at z ∼ 6 from any viewing angle. Although many SMBHs in the LRDs are already estimated to accrete at super-Eddington rates, our model explains 50% of their population by requiring that their masses are overestimated by a mere factor of ∼3. In summary, we suggest that LRDs host slowly spinning SMBHs accreting at mildly super-Eddington rates, with large covering factors and broad emission lines enhanced by strong winds, providing a self-consistent explanation for their X-ray weakness and complementing other models.

Measuring Black Hole Light Echoes with Very Long Baseline Interferometry

Light passing near a black hole can follow multiple paths from an emission source to an observer due to strong gravitational lensing. Photons following different paths take different amounts of time to reach the observer, which produces an echo signature in the image. The characteristic echo delay is determined primarily by the mass of the black hole, but it is also influenced by the black hole spin and inclination to the observer. In the Kerr geometry, echo images are demagnified, rotated, and sheared copies of the direct image and lie within a restricted region of the image. Echo images have exponentially suppressed flux, and temporal correlations within the flow make it challenging to directly detect light echoes from the total light curve. In this Letter, we propose a novel method to search for light echoes by correlating the total light curve with the interferometric signal at high spatial frequencies, which is a proxy for indirect emission. We explore the viability of our method using numerical general relativistic magnetohydrodynamic simulations of a near-face-on accretion system scaled to M87-like parameters. We demonstrate that our method can be used to directly infer the echo delay period in simulated data. An echo detection would be clear evidence that we have captured photons that have circled the black hole, and a high-fidelity echo measurement would provide an independent measure of fundamental black hole parameters. Our results suggest that detecting echoes may be achievable through interferometric observations with a modest space-based very long baseline interferometry mission.

Magnetic dissipation in short gamma-ray-burst jets

Aims. Short gamma-ray bursts originate when relativistic jets emerge from the remnants of binary neutron star (BNS) mergers, as observed in the first multi-messenger event GW170817–GRB 170817A, which coincided with a gravitational wave signal. Both the jet and the remnant are believed to be magnetized, and the presence of magnetic fields is known to influence the jet propagation across the surrounding post-merger environment. In the magnetic interplay between the jet and the environment itself, effects due to a finite plasma conductivity may be important, especially in the first phases of jet propagation. We aim to investigate these effects, from jet launching to its final breakout from the post-merger environment. Methods. We used the PLUTO numerical code to perform 2D axisymmetric and full 3D resistive relativistic magnetohydrodynamic (MHD) simulations, employing spherical coordinates with spatial radial stretching. We considered and compared different models for physical resistivity, which must be small but still dominating over the intrinsic numerical dissipation (which yields unwanted smearing of structures in any ideal MHD code). Stiff terms in the current density are treated with IMplicit-EXplicit Runge Kutta algorithms for time-stepping. A Synge-like gas (Taub equation of state) is also considered. All simulations were performed using an axisymmetric analytical model for both the jet propagation environment and the jet injection; we leave the case of jet propagation in a realistic environment (i.e., imported from an actual BNS merger simulation) to a future study. Results. As expected, no qualitative differences are detected due to the effect of a finite conductivity, but significant quantitative differences in the jet structure and induced turbulence are clearly seen in 2D axisymmetric simulations, and we also compare different resistivity models. We see the formation of regions with a resistive electric field parallel to the magnetic field, and nonthermal particle acceleration may be enhanced there. The level of dissipated Ohmic power is also dependent on the various recipes for resistivity. Most of the differences arise before the breakout from the inner environment, whereas once the jet enters the external weakly magnetized environment region, these differences are preserved during further propagation despite the lower grid refinement. Finally, we show and discuss the 3D evolution of the jet within the same environment in order to highlight the emergence of nonaxisymmetric features.

Chaotic magnetic disconnections trigger flux eruptions in accretion flows channeled onto magnetically saturated Kerr black holes

Magnetized accretion flow onto a black hole (BH) may lead to the accumulation of poloidal magnetic flux across its horizon, which for high BH spin can power far-reaching relativistic jets. The BH magnetic flux is subject to a saturation mechanism by means of magnetic flux eruptions involving relativistic magnetic reconnection. Such accretion flows have been described as magnetically arrested disks (MAD) or magnetically choked accretion flows (MCAF). The main goal of this work is to describe the onset of relativistic reconnection and initial development of magnetic flux eruption in accretion flow onto magnetically saturated BHs. We analyzed the results of 3D general relativistic ideal magnetohydrodynamic (GRMHD) numerical simulations in the Kerr metric, starting from weakly magnetized geometrically thick tori rotating either prograde or retrograde. We integrate d large samples of magnetic field lines in order to probe magnetic connectivity with the BH horizon. The boundary between magnetically connected and disconnected domains coincides roughly with enthalpy equipartition. The geometrically constricted innermost part of the disconnected domain develops a rigid structure of magnetic field lines -- rotating slowly and insensitive to the BH spin orientation. The typical shape of innermost disconnected lines is a double spiral converging to a sharp inner tip anchored at the single equatorial current layer. The foot points of magnetic flux eruptions are found to zip around the BH along with other azimuthal patterns. Magnetic flux eruptions from magnetically saturated accreting BHs can be triggered by minor density gaps in the disconnected domain, resulting from the chaotic disconnection of plasma-depleted magnetospheric lines. Accretion flow is effectively channeled along the disconnected lines toward the current layer, and further toward the BH by turbulent cross-field diffusion. Rotation of flux eruption foot points may contribute to the variability of BH crescent images.

Constraints on the accretion properties of quasi-periodic erupters from GRMHD simulations

Some apparently quiescent supermassive black holes (BHs) at centers of galaxies show quasi-periodic eruptions (QPEs) in the X-ray band, the nature of which is still unknown. A possible origin for the eruptions is an accretion disk. However, the properties of such disks are restricted by the timescales of recurrence and the duration of the flares. In this work, we test the possibility that the temporal properties of known QPEs can be explained by accretion from a compact accretion disk with an outer radius out g and we focus on a particular object, GSN 069. We ran several 3D general relativistic magnetohydrodynamic (GRMHD) simulations with the H-AMR code of thin and thick disks and studied how the initial disk parameters such as thickness, magnetic field configuration, magnetization, and Kerr parameter affect the observational properties of QPEs. We show that accretion onto a slowly rotating BH through a small, moderately thin accretion disk with an initially low plasma beta can explain the observed time between outbursts and the lack of evidence for a variable jet emission. In order to form such a disk, the accreting matter should have a low net angular momentum. A potential source for such low angular momentum matter with a quasi-periodic feeding mechanism might be a tight binary of wind-launching stars. Apart from their primary application, our results can also be useful for general studies of systems with small accretion disks, in which evolution occurs very rapidly so that the disks cannot be considered stationary. For such systems, it is important to understand how the initial conditions affect the results.

General relativistic magnetohydrodynamic simulations with bam: Implementation and code comparison

General relativistic magnetohydrodynamic simulations with bam: Implementation and code comparison

Bayesian Black Hole Photogrammetry

We propose an analytic dual-cone accretion model for horizon-scale images of the cores of low-luminosity active galactic nuclei, including those observed by the Event Horizon Telescope (EHT). Our model is of synchrotron emission from an axisymmetric, magnetized plasma, constrained to flow within two oppositely oriented cones that are aligned with the black hole’s spin axis. We show this model can accurately reproduce images of a variety of time-averaged general relativistic magnetohydrodynamic simulations and that it accurately recovers the black hole spin, orientation, emission scale height, peak emission radius, and fluid flow direction from these simulations within a Bayesian inference framework using radio interferometric data. We show that nontrivial topologies in the images of relativistic accretion flows around black holes can result in nontrivial multimodal solutions when applied to observations with a sparse array, such as the EHT 2017 observations of M87*. The presence of these degeneracies underscores the importance of employing Bayesian techniques to adequately sample the posterior space for the interpretation of EHT measurements. We fit our model to the EHT observations of M87* and find a 95% highest posterior density interval for the mass-to-distance ratio of θ g ∈ (2.84, 3.75) μas, and give an inclination of θ o ∈ (11°, 24°). These new measurements are consistent with mass measurements from the EHT and stellar dynamical estimates and with the spin axis inclination inferred from properties of the M87* jet.

Bridging simulations of kink instability in relativistic magnetized jets with radio emission and polarisation

Relativistic outflows emanating from active galactic nuclei can extend up to kiloparsec scales in length, displaying a variety of complex morphologies. This study explores the intricate morphologies of such relativistic jets, mainly focusing on creating a bridge between magnetic instabilities in jets with observational signatures from complex radio galaxies. In particular, we aim to study the role of dynamical instabilities in forming distinctive morphological features by employing 3D relativistic magnetohydrodynamic (RMHD) simulations of rotating jets. Our simulations have further used the hybrid Eulerian-Lagrangian framework of the PLUTO code and generated the synthetic synchrotron emission and polarisation maps to compare with the observed signatures. Our analysis based on simulations of a continuously injected jet suggests that current-driven instabilities, notably the |m|=1 mode, generate ribs-like structures that are seen in some of the recent radio galaxies using MeerKat, e.g. MysTail. In our contrasting simulations of the restarted jet, the kink-instability driven ribs-like structures were formed relatively near the nozzle. In both cases, the jet dissipates its pre-existing magnetic energy through these instabilities, transitioning to a more kinetic energy dominant state. The turbulent structures resulting from this dissipation phase are filamentary and resemble the tethers as observed for the case of MysTail. This pilot study essentially provides a plausible qualitative explanation by bridging simulations of kink instability to produce synthetic radio features resembling the observed complex radio morphology of MysTail.

3D hybrid fluid-particle jet simulations and the importance of synchrotron radiative losses

Relativistic jets in active galactic nuclei are known for their exceptional energy output, and imaging the synthetic synchrotron emission of numerical jet simulations is essential for a comparison with observed jet polarization emission. Through the use of 3D hybrid fluid-particle jet simulations (with the PLUTO code), we overcome some of the commonly made assumptions in relativistic magnetohydrodynamic (RMHD) simulations by using non-thermal particle attributes to account for the resulting synchrotron radiation. Polarized radiative transfer and ray-tracing (via the RADMC-3D code) highlight the differences in total intensity maps when (i) the jet is simulated purely with the RMHD approach, (ii) a jet tracer is considered in the RMHD approach, and (iii) a hybrid fluid-particle approach is used. The resulting emission maps were compared to the example of the radio galaxy Centaurus A. We applied the Lagrangian particle module implemented in the latest version of the PLUTO code. This new module contains a state-of-the-art algorithm for modeling diffusive shock acceleration and for accounting for radiative losses in RMHD jet simulations. The module implements the physical postulates missing in RMHD jet simulations by accounting for a cooled ambient medium and strengthening the central jet emission. We find a distinction between the innermost structure of the jet and the back-flowing material by mimicking the radio emission of the Seyfert II radio galaxy Centaurus\,A when considering an edge-brightened jet with an underlying purely toroidal magnetic field. We demonstrate the necessity of synchrotron cooling as well as the improvements gained when directly accounting for non-thermal synchrotron radiation via non-thermal particles.

General Relativistic Radiation Magnetohydrodynamics Simulations of Precessing Tilted Super-Eddington Disks

We perform a three-dimensional general relativistic radiation magnetohydrodynamics simulation of a tilted super-Eddington accretion disk around a spinning black hole (BH). The disk, which tilts and twists as it approaches the BH, precesses while maintaining its shape. The gas is mainly ejected around the rotation axis of the outer part of the disk rather than around the spin axis of the BH. The disk precession changes the ejection direction of the gas with time. The radiation energy is also released in approximately the same direction as the outflow, so the precession is expected to cause a quasiperiodic time variation of the observed luminosity. The timescale of the precession is about 10 s for a 10 M ⊙ BH and for the radial extent of a disk of several tens of gravitational radii, where M ⊙ is the solar mass. This timescale is consistent with the frequency of the low-frequency quasiperiodic oscillation (0.01–1 Hz) observed in some ultraluminous X-ray sources.

GRMHD Simulations of Accretion Structures with Different Angular Momentum Profiles

In this article, we explore the dynamics of accretion structures encircling spherically symmetric black holes, comparing three accretion disk models with distinct angular momentum profiles: (i) the geometrically thin Keplerian disk, (ii) the Fishbone–Moncrief torus; and (iii) the Polish Doughnut. Employing general relativistic magnetohydrodynamics simulations with the High Accuracy Relativistic Magnetohydrodynamics code, we investigate these three models, considering the magnetic field’s influence on the accretion disk angular momentum redistribution. We show that the magnetic field is a key factor in accretion disk structures, especially in regions with lower mass density. Our investigation verifies the well-established fact that the presence of a magnetic field significantly influences the accretion rate and its temporal variability.

In LIGO’s Sight? Vigorous Coherent Gravitational Waves from Cooled Collapsar Disks

We present the first numerical study of gravitational waves (GWs) from collapsar disks, using state-of-the-art 3D general relativistic magnetohydrodynamic simulations of collapsing stars. These simulations incorporate a fixed Kerr metric for the central black hole (BH) and employ simplified prescriptions for disk cooling. We find that cooled disks with an expected scale height ratio of H/R ≳ 0.1 at ∼10 gravitational radii induce Rossby instability in compact, high-density rings. The trapped Rossby vortices generate vigorous coherent emission regardless of disk magnetization and BH spin. For BH mass of ∼10 M ⊙, the GW spectrum peaks at ∼100 Hz, with some breadth due to various nonaxisymmetric modes. The spectrum shifts toward lower frequencies as the disk viscously spreads and the circularization radius of the infalling gas increases. Weaker-cooled disks with H/R ≳ 0.3 form a low-density extended structure of spiral arms, resulting in a broader, lower-amplitude spectrum. Assuming an optimistic detection threshold with a matched-filter signal-to-noise ratio of 20 and a rate similar to Type Ib/c supernovae, LIGO–Virgo–KAGRA (LVK) could detect ≲1 event annually, suggesting that GW events may already be hidden in observed data. Third-generation GW detectors could detect dozens to hundreds of collapsar disks annually, depending on the cooling strength and the disk formation rate. The GW amplitudes from collapsar disks are ≳100 times higher with a substantially greater event rate than those from core-collapse supernovae, making them potentially the most promising burst-type GW class for LVK and Cosmic Explorer. This highlights the importance of further exploration and modeling of disk-powered GWs, promising insights into collapsing star physics.

Dynamics and emission properties of flux ropes from two-temperature GRMHD simulations with multiple magnetic loops

Context. Flux ropes erupting in the vicinity of a black hole are thought to be a potential model for the flares observed in Sagittarius A*. Aims. In this study, we examine the radiative properties of flux ropes that emerged in the vicinity of a black hole. Methods. We performed three-dimensional two-temperature general relativistic magnetohydrodynamic (GRMHD) simulations of magnetized accretion flows with alternating multiple magnetic loops and general relativistic radiation transfer (GRRT) calculations. In the GRMHD simulations, we implemented two different sizes of initial magnetic loops. Results. In the small loop case, magnetic dissipation leads to a weaker excitement of magneto-rotational instability inside the torus, which generates a lower accretion rate compared to the large loop case. However, the small loop case generates more flux ropes due to frequent reconnection by magnetic loops with different polarities. By calculating the thermal synchrotron emission, we found that the variability of light curves and the emitting region are tightly related. At 230 GHz and higher, the emission from the flux ropes is relatively stronger compared to the background, which is responsible for the filamentary structure in the images. At lower frequencies (e.g., 43 GHz), emission comes from more extended regions, which have a less filamentary structure in the image. Conclusions. Our study shows that self-consistent electron temperature models are essential for the calculation of thermal synchrotron radiation and the morphology of the GRRT images. Flux ropes contribute considerable emission at frequencies ≳230 GHz.

Bayesian Constraints on the Ring Ellipticity of M87* 2017 Using Themis

Measuring the properties of black hole images has the potential to constrain deviations from general relativity on horizon scales. Of particular interest is the ellipticity of the ring that is sensitive to the underlying spacetime. In 2019, the Event Horizon Telescope (EHT) produced the first-ever image of a black hole on horizon scales. Here, we reanalyze the M87* EHT 2017 data using Bayesian imaging (BI) techniques, constructing a posterior of the ring shape. We find that BI recovers the true on-sky ring shape more reliably than the original imaging methods used in 2019. As a result, we find that M87*'s ring ellipticity is 0.09−0.06+0.07 and is consistent with the measured ellipticity from general relativistic magnetohydrodynamic simulations.

A Revisited Equilibrium Solution of the Fishbone and Moncrief Torus for Extended General Relativistic Magnetohydrodynamic Simulations

Accretion physics has become more important recently due to the detection of the first horizon-scale images of the supermassive black holes of M 87* and Sgr A* by the Event Horizon Telescope. General relativistic magnetohydrodynamic (GRMHD) simulations of magnetized accretion flows onto a Kerr black hole have been used to interpret them. However, further testing the theory of gravity by using horizon-scale images requires performing consistent GRMHD simulations in non-Kerr spacetime. In this paper, we revisited the hydrodynamical equilibrium solution of the Fishbone and Moncrief (FM) torus that can be used to study any stationary, axisymmetric, vacuum, or nonvacuum spacetime. Further, we check the stability of the FM torus in non-Kerr spacetime by general relativistic hydrodynamic simulations. We find that FM torus in non-Kerr spacetime is indeed stable under long-term evolution. We conclude that the generalized FM torus solution would be very useful for creating new GRMHD libraries in extended Kerr black holes.

Resolution analysis of magnetically arrested disc simulations

ABSTRACT Polarization measurements by the Event Horizon Telescope from M87* and Sgr A* suggest that there is a dynamically strong, ordered magnetic field, typical of what is expected of a magnetically arrested accretion disc (MAD). In such discs, the strong poloidal magnetic field can suppress the accretion flow and cause episodic flux eruptions. Recent work shows that general relativistic magnetohydrodynamic (GRMHD) MAD simulations feature dynamics of turbulence and mixing instabilities that are becoming resolved at higher resolutions. We perform a convergence study of MAD states exceeding the status quo by an order of magnitude in resolution. We use existing 3D simulations performed with the H-AMR code, up to a resolution of 5376 × 2304 × 2304 in a logarithmic spherical-polar grid. We find consistent time-averaged disc properties across all resolutions. However, higher resolutions reveal signs of inward angular momentum transport attributed to turbulent convection, particularly evident when mixing instabilities occur at the surfaces of flux tubes during flux eruptions. Additionally, we see wave-like features in the jet sheath, which become more prominent at higher resolutions, that may induce mixing between the jet and disc. At higher resolutions, we observe the sheath to be thinner, resulting in increased temperature, reduced magnetization, and greater variability. Those differences could affect the dissipation of energy, which would eventually result in distinct observable radiative emission from high-resolution simulations. With higher resolutions, we can delve into crucial questions about horizon-scale physics and its impact on the dynamics and emission properties of larger-scale jets.

Hybrid GRMHD and force-free simulations of black hole accretion

ABSTRACT We present a new approach for stably evolving general relativistic magnetohydrodynamic (GRMHD) simulations in regions where the magnetization $\sigma =b^2/\rho c^2$ becomes large. GRMHD codes typically struggle to evolve plasma above $\sigma \approx 100$ in simulations of black hole accretion. To ensure stability, GRMHD codes will inject mass density artificially to the simulation as necessary to keep the magnetization below a ceiling value $\sigma _{\rm max}$. We propose an alternative approach where the simulation transitions to solving the equations of general relativistic force-free electrodynamics (GRFFE) above a magnetization $\sigma _{\rm trans}$. We augment the GRFFE equations in the highly magnetized region with approximate equations to evolve the decoupled field-parallel velocity and plasma energy density. Our hybrid scheme is explicit and easily added to the framework of standard-volume GRMHD codes. We present a variety of tests of our method, implemented in the GRMHD code koral, and we show results from a 3D hybrid GRMHD + GRFFE simulation of a magnetically arrested disc (MAD) around a spinning black hole. Our hybrid MAD simulation closely matches the average properties of a standard GRMHD MAD simulation with the same initial conditions in low magnetization regions, but it achieves a magnetization $\sigma \approx 10^6$ in the evacuated jet funnel. We present simulated horizon-scale images of both simulations at 230 GHz with the black hole mass and accretion rate matched to M87*. Images from the hybrid simulation are less affected by the choice of magnetization cut-off $\sigma _{\rm cut}$ imposed in radiative transfer than images from the standard GRMHD simulation.

The three-dimensional structure of black hole accretion flows within the plunging region

ABSTRACT We analyse, using new analytical models and numerical general relativistic magnetohydrodynamic simulations, the three-dimensional properties of accretion flows inside the plunging region of black hole spacetimes (i.e. at radii smaller than the innermost stable circular orbit). These simulations are of thick discs, with aspect ratios of order unity $h/r \sim 1$, and with a magnetic field geometry given by the standard low-magnetization ‘SANE’ configuration. This work represents the first step in a wider analysis of this highly relativistic region. We show that analytical expressions derived in the ‘thin disc’ limit describe the numerical results remarkably well, despite the large aspect ratio of the flow. We further demonstrate that accretion within this region is typically mediated via spiral arms, and that the geometric properties of these spiral structures can be understood with a simple analytical model. These results highlight how accretion within the plunging region is fundamentally two-dimensional in character, which may have a number of observational implications. We derive a modified theoretical description of the pressure within the plunging region which accounts for turbulent heating and may be of use to black hole image modelling.

What is the hard spectral state in X-ray binaries? Insights from GRRMHD accretion flows simulations and polarization of their X-ray emission

X-ray binaries are known to exhibit different spectral states which are often associated with different black hole accretion modes. The exact geometry and properties of these accretion modes is still uncertain. Recent IXPE measurements of linear polarization of X-ray emission in canonical X-ray binary system Cygnus X-1 allow us to test models for the hard spectral state of accretion in a unique way. We show that general relativistic radiative magnetohydrodynamic (GRRMHD) simulations of accreting stellar black hole in a hard X-ray state may be consistent with the new observational information. In the presented framework, where first-principle models have limited number of free parameters, the polarimetric X-ray observations put constraints on the viewing angle of the inner hot accretion flow.

Magnetorotational dynamo can generate large-scale vertical magnetic fields in 3D GRMHD simulations of accreting black holes

ABSTRACT Jetted astrophysical phenomena with black hole engines, including binary mergers, jetted tidal disruption events, and X-ray binaries, require a large-scale vertical magnetic field for efficient jet formation. However, a dynamo mechanism that could generate these crucial large-scale magnetic fields has not been identified and characterized. We have employed three-dimensional global general relativistic magnetohydrodynamical simulations of accretion discs to quantify, for the first time, a dynamo mechanism that generates large-scale magnetic fields. This dynamo mechanism primarily arises from the non-linear evolution of the magnetorotational instability (MRI). In this mechanism, large non-axisymmetric MRI-amplified shearing wave modes, mediated by the axisymmetric azimuthal magnetic field, generate and sustain the large-scale vertical magnetic field through their non-linear interactions. We identify the advection of magnetic loops as a crucial feature, transporting the large-scale vertical magnetic field from the outer regions to the inner regions of the accretion disc. This leads to a larger characteristic size of the, now advected, magnetic field when compared to the local disc height. We characterize the complete dynamo mechanism with two time-scales: one for the local magnetic field generation, $t_{\rm gen}$, and one for the large-scale scale advection, $t_{\rm adv}$. Whereas the dynamo we describe is non-linear, we explore the potential of linear mean field models to replicate its core features. Our findings indicate that traditional $\alpha$-dynamo models, often computed in stratified shearing box simulations, are inadequate and that the effective large-scale dynamics is better described by the shear current effects or stochastic $\alpha$-dynamos.