Shearing Box Simulations Research Articles

Magnetically Driven Turbulence in the Inner Regions of Protoplanetary Disks

Given the important role turbulence plays in the settling and growth of dust grains in protoplanetary disks, it is crucial that we determine whether these disks are turbulent and to what extent. Protoplanetary disks are weakly ionized near the midplane, which has led to a paradigm in which largely laminar magnetic field structures prevail deeper in the disk, with angular momentum being transported via magnetically launched winds. Yet, there has been little exploration of the precise behavior of the gas within the bulk of the disk. We carry out 3D, local shearing box simulations that include all three low-ionization effects (ohmic diffusion, ambipolar diffusion, and the Hall effect) to probe the nature of magnetically driven gas dynamics 1–30 au from the central star. We find that gas turbulence can persist with a generous yet physically motivated ionization prescription (order unity Elsässer numbers). The gas velocity fluctuations range from 0.03 to 0.09 of the sound speed c s at the disk midplane to ∼c s near the disk surface, and are dependent on the initial magnetic field strength. However, the turbulent velocities do not appear to be strongly dependent on the field polarity, and thus appear to be insensitive to the Hall effect. The midplane turbulence has the potential to drive dust grains to collision velocities exceeding their fragmentation limit, and likely reduces the efficacy of particle clumping in the midplane, though it remains to be seen if this level of turbulence persists in disks with lower ionization levels.

Collisionless Magnetorotational Turbulence in Pair Plasmas: Steady-State Dynamics, Particle Acceleration, and Radiative Cooling.

We present 3D fully kinetic shearing-box simulations of pair-plasma magnetorotational turbulence with unprecedented macro-to-microscopic scale separation. While retrieving the expected fluid behavior of the plasma at large scales, we observe a steepening of turbulent spectra at kinetic scales and substantial angular-momentum transport linked with kinetic processes. For the first time, we provide a definitive demonstration of nonthermal particle acceleration in kinetic magnetorotational turbulence agnostically of shearing-box initial conditions by means of a novel strategy exploiting synchrotron cooling.

Runaway Eccentricity Growth: A Pathway for Binary Black Hole Mergers in AGN Disks

Binary black holes (BBHs) embedded within the accretion disks that fuel active galactic nuclei (AGN) are promising progenitors for the source of gravitational wave (GW) events detected by LIGO/VIRGO. Several recent studies have shown that when these binaries form, they are likely to be highly eccentric and retrograde. However, many uncertainties remain concerning the orbital evolution of these binaries as they either inspiral toward merger or disassociate. Previous hydrodynamical simulations exploring their orbital evolution have been predominantly two-dimensional or have been restricted to binaries on nearly circular orbits. We present the first high-resolution, three-dimensional local shearing-box simulations of both prograde and retrograde eccentric BBHs embedded in AGN disks. We find that retrograde binaries shrink several times faster than their prograde counterparts and exhibit significant orbital eccentricity growth, the rate of which monotonically increases with binary eccentricity. Our results suggest that retrograde binaries may experience runaway orbital eccentricity growth, which may bring them close enough together at pericenter for GW emission to drive them to coalescence. Although their eccentricity is damped, prograde binaries shrink much faster than their orbital eccentricity decays, suggesting they should remain modestly eccentric as they contract toward merger. Finally, binary precession driven by the AGN disk may dominate over precession induced by the supermassive black hole depending on the binary accretion rate and its location in the AGN disk, which can subdue the evection resonance and von Ziepel–Lidov–Kozai cycles.

Convective Overstability in Radially Global Protoplanetary Disks. I. Pure Gas Dynamics

Protoplanetary disks are prone to several hydrodynamic instabilities. One candidate, convective overstability (COS), can drive radial semiconvection that may influence dust dynamics and planetesimal formation. However, the COS has primarily been studied in local models. This paper investigates the COS near the midplane of radially global disk models. We first conduct a global linear stability analysis, which shows that linear COS modes exist only radially inward of their Lindblad resonance (LR). The fastest-growing modes have LRs near the inner radial domain boundary with effective radial wavelengths that can be a substantial fraction of the disk radius. We then perform axisymmetric global simulations and find that the COS’s nonlinear saturation is similar to previous incompressible shearing box simulations. In particular, we observe the onset of persistent zonal and elevator flows for sufficiently steep radial entropy gradients. In full 3D, nonaxisymmetric global simulations, we find the COS produces large-scale, long-lived vortices, which induce the outward radial transport of angular momentum via the excitation of spiral density waves. The corresponding α-viscosity values of order 10−3 agree well with those found in previous 3D compressible shearing box simulations. However, in global disks, significant modifications to their radial structure are found, including the formation of pressure bumps. Interestingly, the COS typically generates an outward radial mass transport, i.e., decretion. We briefly discuss the possible implications of our results for planetesimal formation and for interpreting dust rings and asymmetries observed in protoplanetary disks.

Streaming Instability and Turbulence: Conditions for Planetesimal Formation

The streaming instability (SI) is a leading candidate for planetesimal formation, which can concentrate solids through two-way aerodynamic interactions with the gas. The resulting concentrations can become sufficiently dense to collapse under particle self-gravity, forming planetesimals. Previous studies have carried out large parameter surveys to establish the critical particle to gas surface density ratio (Z), above which SI-induced concentration triggers planetesimal formation. The threshold Z depends on the dimensionless stopping time (τ s , a proxy for dust size). However, these studies neglected both particle self-gravity and external turbulence. Here, we perform 3D stratified shearing box simulations with both particle self-gravity and turbulent forcing, which we characterize via a turbulent diffusion parameter, α D. We find that forced turbulence, at amplitudes plausibly present in some protoplanetary disks, can increase the threshold Z by up to an order of magnitude. For example, for τ s = 0.01, planetesimal formation occurs when Z ≳ 0.06, ≳0.1, and ≳0.2 at α D = 10−4, 10−3.5, and 10−3, respectively. We provide a single fit to the critical Z required for the SI to work as a function of α D and τ s (although limited to the range τ s = 0.01–0.1). Our simulations also show that planetesimal formation requires a mid-plane particle-to-gas density ratio that exceeds unity, with the critical value being largely insensitive to α D. Finally, we provide an estimation of particle scale height that accounts for both particle feedback and external turbulence.

The Origin of the Slow-to-Alfvén Wave Cascade Power Ratio and Its Implications for Particle Heating in Accretion Flows

The partition of turbulent heating between ions and electrons in radiatively inefficient accretion flows plays a crucial role in determining the observational appearance of accreting black holes. Modeling this partition is, however, a challenging problem because of the large scale-separation between the macroscopic scales at which energy is injected by turbulence and the microscopic ones at which it is dissipated into heat. Recent studies of particle heating from collisionless damping of turbulent energy have shown that the partition of energy between ions and electrons is dictated by the ratio of the energy injected into the slow and Alfvén wave cascades as well as the plasma β parameter. In this paper, we study the mechanism of the injection of turbulent energy into slow- and Alfvén-wave cascades in magnetized shear flows. We show that this ratio depends on the particular (r ϕ) components of the Maxwell and Reynolds stress tensors that cause the transport of angular momentum, the shearing rate, and the orientation of the mean magnetic field with respect to the shear. We then use numerical magnetohydrodynamic shearing-box simulations with background conditions relevant to black hole accretion disks to compute the magnitudes of the stress tensors for turbulence driven by the magneto-rotational instability and derive the injection power ratio between slow and Alfvén wave cascades. We use these results to formulate a local subgrid model for the ion-to-electron heating ratio that depends on the macroscopic characteristics of the 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.

Parametric Survey of Nonaxisymmetric Accretion Disk Instabilities: Magnetorotational Instability to Super-Alfvénic Rotational Instability

Accretion disks are highly unstable to magnetic instabilities driven by shear flow, where classically, the axisymmetric, weak-field magnetorotational instability (MRI) has received much attention through local WKB approximations. In contrast, discrete nonaxisymmetric counterparts require a more involved analysis through a full global approach to deal with the influence of the nearby magnetohydrodynamic (MHD) continua. Recently, rigorous MHD spectroscopy identified a new type of ultralocalized, nonaxisymmetric instability in global disks with super-Alfvénic flow. These super-Alfvénic rotational instabilities (SARIs) fill vast unstable regions in the complex eigenfrequency plane with (near eigen)modes that corotate at the local Doppler velocity and are radially localized between Alfvénic resonances. Unlike discrete modes, they are utterly insensitive to the radial disk boundaries. In this work, we independently confirm the existence of these unprecedented modes using our novel spectral MHD code Legolas, reproducing and extending our earlier study with detailed eigenspectra and eigenfunctions. We calculate the growth rates of SARIs and MRI in a variety of disk equilibria, highlighting the impact of field strength and orientation, and find correspondence with analytical predictions for thin, weakly magnetized disks. We show that nonaxisymmetric modes can significantly extend instability regimes at high mode numbers, with maximal growth rates comparable to those of the MRI. Furthermore, we explicitly show a region filled with quasi-modes whose eigenfunctions are extremely localized in all directions. These modes must be ubiquitous in accretion disks, and play a role in local shearing box simulations. Finally, we revisit recent dispersion relations in the appendix, highlighting their relation to our global framework.

Reduction of dust radial drift by turbulence in protoplanetary disks

Reduction of dust radial drift by turbulence in protoplanetary disks

Length and Velocity Scales in Protoplanetary Disk Turbulence

In the theory of protoplanetary disk turbulence, a widely adopted ansatz, or assumption, is that the turnover frequency of the largest turbulent eddy, Ω L , is the local Keplerian frequency Ω K . In terms of the standard dimensionless Shakura–Sunyaev α parameter that quantifies turbulent viscosity or diffusivity, this assumption leads to characteristic length and velocity scales given respectively by αH and αc , in which H and c are the local gas scale height and sound speed. However, this assumption is not applicable in cases when turbulence is forced numerically or driven by some natural processes such as vertical shear instability. Here, we explore the more general case where Ω L ≥ Ω K and show that, under these conditions, the characteristic length and velocity scales are respectively α/R′H and αR′c , where R′≡ΩL/ΩK is twice the Rossby number. It follows that α=α˜/R′ , where α˜c is the root-mean-square average of the turbulent velocities. Properly allowing for this effect naturally explains the reduced particle scale heights produced in shearing box simulations of particles in forced turbulence, and it may help with interpreting recent edge-on disk observations; more general implications for observations are also presented. For R′>1 , the effective particle Stokes numbers are increased, which has implications for particle collision dynamics and growth, as well as for planetesimal formation.

Shedding light on the MRI-driven dynamo in a stratified shearing box

ABSTRACT We study the magneto-rotational instability (MRI) dynamo in a geometrically thin disc (H/R ≪ 1) using stratified zero net (vertical) flux shearing box simulations. We find that mean fields and electromotive forces (EMFs) oscillate with a primary frequency fdyn = 0.017 Ω (approximately nine orbital period), but also have higher harmonics at 3fdyn. Correspondingly, the current helicity has two frequencies 2fdyn and 4fdyn, which appear to be the beat frequencies of mean fields and EMFs, respectively, as expected from the magnetic helicity density evolution equation. Further, we adopt a novel inversion algorithm called the ‘Iterative Removal Of Sources’, to extract the turbulent dynamo coefficients in the mean-field closure using the mean magnetic fields and EMFs obtained from the shearing box simulation. We show that an α-effect (αyy) is predominantly responsible for the creation of the poloidal field from the toroidal field, while shear generates back a toroidal field from the poloidal field, indicating that an α–Ω-type dynamo is operative in MRI-driven accretion discs. We also find that both strong outflow ($\bar{v}_z$) and turbulent pumping (γz) transport mean fields away from the mid-plane. Instead of turbulent diffusivity, they are the principal sink terms in the mean magnetic energy evolution equation. We find encouraging evidence that a generative helicity flux is responsible for the effective α-effect. Finally, we point out potential limitations of horizontal (x − y) averaging in defining the ‘mean’ on the extraction of dynamo coefficients and their physical interpretations.

Local models of two-temperature accretion disc coronae – II. Ion thermal conduction and the absence of disc evaporation

ABSTRACT We use local stratified shearing-box simulations with magnetic field-aligned thermal conduction to study an idealized model of the coupling between a cold, radiatively efficient accretion disc, and an overlying, hot, two-temperature corona. Evaporation of a cold disc by conduction from the hot corona has been proposed as a means of mediating the soft-to-hard state transitions observed in X-ray binary systems. We model the coronal plasma in our local disc patch as an MHD fluid subject to both free-streaming ion conduction and a parametrized cooling function that captures the collisional transfer of energy from hot ions to colder, rapidly cooling leptons. In all of our models, independent of the initial net vertical magnetic flux (NF) threading the disc, we find no evidence of disc evaporation. The ion heat flux into the disc is radiated away before conduction can heat the disc’s surface layers. When an initial NF is present, steady-state temperature, density, and outflow velocities in our model coronae are unaffected by conduction. Instead of facilitating disc evaporation, thermal conduction is more likely to feed the disc with plasma condensing out of the corona, particularly in flows without NF. Our work indicates that uncertainties in the amount of NF threading the disc hold far greater influence over whether or not the disc will evaporate into a radiatively inefficient accretion flow compared to thermal conduction. We speculate that a change in net flux mediates disc truncation/evaporation.

Local models of two-temperature accretion disc coronae – I. Structure, outflows, and energetics

ABSTRACT We use local stratified shearing-box simulations to elucidate the impact of two-temperature thermodynamics on the thermal structure of coronae in radiatively efficient accretion flows. Rather than treating the coronal plasma as an isothermal fluid, we use a simple, parametrized cooling function that models the collisional transfer of energy from the ions to the rapidly cooling leptons. Two-temperature models naturally form temperature inversions, with a hot, magnetically dominated corona surrounding a cold disc. Simulations with net vertical flux (NF) magnetic fields launch powerful magnetocentrifugal winds that would enhance accretion in a global system. The outflow rates are much better converged with increasing box height than analogous isothermal simulations, suggesting that the winds into two-temperature coronae may be sufficiently strong to evaporate a thin disc and form a radiatively inefficient accretion flow under some conditions. We find evidence for multiphase structure in the corona, with broad density and temperature distributions, and we propose criteria for the formation of a multiphase corona. The fraction of cooling in the surface layers of the disc is substantially larger for NF fields compared to zero net-flux configurations, with moderate NF simulations radiating ≳30 per cent of the flow’s total luminosity above two mid-plane scale heights. Our work shows that NF fields may efficiently power the coronae of luminous Seyfert galaxies and quasars, providing compelling motivation for future studies of the heating mechanisms available to NF fields and the interplay of radiation with two-temperature thermodynamics.

Dust dynamics in current sheets within protoplanetary disks

Context.Chondrules originate from the reprocessing of dust grains. They are key building blocks of telluric planets, yet their formation, which must happen in strongly localized regions of high temperature, remains poorly understood.Aims.We examine the dust spatial distribution near regions of strong local heating produced by current sheets, as a step toward exploring a potential path for chondrule formation. We further aim to investigate current sheet formation under various conditions in protoplanetary disks in the presence of ambipolar diffusion and Ohmic resistivity and the effect of current sheet morphology on dust dynamics in their vicinity.Methods.We used the RAMSES code including modules for nonideal magnetohydrodynamics and the solution of the dynamics of multiple sizes of dust grains to compute unstratified shearing box simulations of current sheet formation. Through seven models, we investigated the effect of the ambipolar diffusion and Ohmic resistivity strength, the initial density, and magnetic field, as well as the resolution and box size.Results.We find that current sheets form in all our models, with typical widths of 10−3–10−2AU, and that strong dust fraction variations occur for millimeter-sized grains. These variations are typically of an order of magnitude and up to two orders of magnitude for the most favorable cases. We also show that the box size and resolution has a strong impact on the current sheet distribution and intensity.Conclusions.The formation of current sheets that can intensely heat their surroundings near strong dynamical dust fraction variations could have important implications for chondrule formation, as it appears likely to happen in regions with a large dust fraction.

3D Radiation Hydrodynamic Simulations of Gravitational Instability in AGN Accretion Disks: Effects of Radiation Pressure

We perform 3D radiation hydrodynamic local shearing-box simulations to study the outcome of gravitational instability (GI) in optically thick active galactic nuclei (AGNs) accretion disks. GI develops when the Toomre parameter Q T ≲ 1, and may lead to turbulent heating that balances radiative cooling. However, when radiative cooling is too efficient, the disk may undergo runaway gravitational fragmentation. In the fully gas-pressure-dominated case, we confirm the classical result that such a thermal balance holds when the Shakura–Sunyaev viscosity parameter (α) due to the gravitationally driven turbulence is ≲0.2, corresponding to dimensionless cooling times Ωt cool ≳ 5. As the fraction of support by radiation pressure increases, the disk becomes more prone to fragmentation, with a reduced (increased) critical value of α (Ωt cool). The effect is already significant when the radiation pressure exceeds 10% of the gas pressure, while fully radiation-pressure-dominated disks fragment at t cool ≲ 50 Ω−1. The latter translates to a maximum turbulence level α ≲ 0.02, comparable to that generated by magnetorotational instability. Our results suggest that gravitationally unstable (Q T ∼ 1) outer regions of AGN disks with significant radiation pressure (likely for high/near-Eddington accretion rates) should always fragment into stars, and perhaps black holes.

Contracting and Expanding Binary Black Holes in 3D Low-mass AGN Disks: The Importance of Separation

LIGO/Virgo has detected several binary black hole (BBH) merger events that may have originated in the accretion disks of active galactic nuclei (AGNs). These events require individual black hole masses that fall within the pair instability supernova mass gap, and therefore these black holes may have been grown from hierarchical mergers. AGN disks are a prime environment for hierarchical mergers and thus a potential location for the progenitors of BBH gravitational wave events. Understanding how a BBH embedded in an AGN disk interacts with the surrounding environment is thus crucial for determining whether this interaction can lead to its merger. However, there are few high-fidelity simulations of this process, and almost all are two-dimensional. We present the results from 3D, high-resolution, local shearing-box simulations of an embedded BBH interacting with an AGN disk. In these first simulations of their kind, we focus on determining the mass accretion rate and the orbital evolution rate at different BBH separations. We find that circular, equal-mass BBHs with separations greater than 10% of their Hill radius contract while accreting at a super-Eddington rate. At smaller separations, however, our 3D simulations find that BBHs expand their orbits. This result suggests that it may be difficult for an AGN disk to push a BBH to merger, but we discuss several mechanisms, including MHD turbulence and radiative and mechanical feedback, which could alleviate this difficulty.

Simulating the magnetorotational instability on a moving mesh with the shearing box approximation

ABSTRACT The magnetorotational instability (MRI) is an important process in sufficiently ionized accretion discs, as it can create turbulence that acts as an effective viscosity, mediating angular momentum transport. Due to its local nature, it is often analysed in the shearing box approximation with Eulerian methods, which otherwise would suffer from large advection errors in global disc simulations. In this work, we report on an extensive study that applies the quasi-Lagrangian, moving-mesh code arepo, combined with the Dedner cleaning scheme to control deviations from $\nabla \cdot \boldsymbol B=0$, to the problem of magnetized flows in shearing boxes. We find that we can resolve the analytical linear growth rate of the MRI with mean background magnetic field well. In the zero net flux case, there is a threshold value for the strength of the divergence cleaning above which the turbulence eventually dies out, and in contrast to previous Eulerian simulations, the strength of the MRI does not decrease with increasing resolution. In boxes with larger vertical aspect ratio we find a mean-field dynamo, as well as an active shear current effect that can sustain MRI turbulence for at least 200 orbits. In stratified simulations, we obtain an active αω dynamo and the characteristic butterfly diagram. Our results compare well with previous results obtained with static grid codes such as athena. We thus conclude that arepo represents an attractive approach for global disc simulations due to its quasi-Lagrangian nature, and for shearing box simulations with large density variations due to its continuously adaptive resolution.

Establishing Dust Rings and Forming Planets within Them

Radio images of protoplanetary disks demonstrate that dust grains tend to organize themselves into rings. These rings may be a consequence of dust trapping within gas pressure maxima, wherein the local high dust-to-gas ratio is expected to trigger the formation of planetesimals and eventually planets. We revisit the behavior of dust near gas pressure perturbations enforced by a planet in two-dimensional, shearing-box simulations. While dust grains collect into generally long-lived rings, particles with a small Stokes parameter τ s < 0.1 tend to advect out of the ring within a few drift timescales. Scaled to the properties of ALMA disks, we find that rings composed of larger particles (τ s ≥ 0.1) can nucleate a dust clump massive enough to trigger pebble accretion, which proceeds to ingest the entire dust ring well within ∼1 Myr. To ensure the survival of the dust rings, we favor a nonplanetary origin and typical grain size τ s ≲ 0.05–0.1. Planet-driven rings may still be possible but if so we would expect the orbital distance of the dust rings to be larger for older systems.

MRI-driven dynamo at very high magnetic Prandtl numbers

ABSTRACT The dynamo driven by the magnetorotational instability (MRI) is believed to play an important role in the dynamics of accretion discs and may also explain the origin of the extreme magnetic fields present in magnetars. Its saturation level is an important open question known to be particularly sensitive to the diffusive processes through the magnetic Prandtl number Pm (the ratio of viscosity to resistivity). Despite its relevance to proto-neutron stars and neutron star merger remnants, the numerically challenging regime of high Pm is still largely unknown. Using zero-net flux shearing box simulations in the incompressible approximation, we studied MRI-driven dynamos at unprecedentedly high values of Pm reaching 256. The simulations show that the stress and turbulent energies are proportional to Pm up to moderately high values (Pm ∼ 50). At higher Pm, they transition to a new regime consistent with a plateau independent of Pm for $\rm Pm \gtrsim 100$. This trend is independent of the Reynolds number, which may suggest an asymptotic regime where the energy injection and dissipation are independent of the diffusive processes. Interestingly, large values of Pm not only lead to intense small-scale magnetic fields but also to a more efficient dynamo at the largest scales of the box.

Magnetorotational instability with smoothed particle hydrodynamics

The magnetorotational instability (MRI) is an important process in driving turbulence in sufficiently ionized accretion disks. It has been extensively studied using simulations with Eulerian grid codes, but remains fairly unexplored for meshless codes. Here, we present a thorough numerical study on the MRI using the smoothed particle magnetohydrodynamics method with the geometric density average force expression. We performed 37 shearing box simulations with different initial setups and a wide range of resolution and dissipation parameters. We show, for the first time, that MRI with sustained turbulence can be simulated successfully with smoothed-particle hydrodynamics (SPH), with results consistent with prior work with grid-based codes, including saturation properties such as magnetic and kinetic energies and their respective stresses. In particular, for the stratified boxes, our simulations reproduce the characteristic “butterfly” diagram of the MRI dynamo with saturated turbulence for at least 100 orbits. On the contrary, traditional SPH simulations suffer from runaway growth and develop unphysically large azimuthal fields, similar to the results from a recent study with meshless methods. We investigated the dependency of MRI turbulence on the numerical Prandtl number (Pm) in SPH, focusing on the unstratified, zero net-flux case. We found that turbulence can only be sustained with a Prandtl number larger than ∼2.5, similar to the critical values for the physical Prandtl number found in grid-code simulations. However, unlike grid-based codes, the numerical Prandtl number in SPH increases with resolution, and for a fixed Prandtl number, the resulting magnetic energy and stresses are independent of resolution. Mean-field analyses were performed on all simulations, and the resulting transport coefficients indicate no α-effect in the unstratified cases, but an active αω dynamo and a diamagnetic pumping effect in the stratified medium, which are generally in agreement with previous studies. There is no clear indication of a shear-current dynamo in our simulation, which is likely to be responsible for a weaker mean-field growth in the tall, unstratified, zero net-flux simulation.