Turbulent Dynamo Research Articles

Efficient Highly Subsonic Turbulent Dynamo and Growth of Primordial Magnetic Fields.

We present the first study on the amplification of magnetic fields by the turbulent dynamo in the highly subsonic regime, with Mach numbers ranging from 10^{-3} to 0.4. We find that for the lower Mach numbers the saturation efficiency of the dynamo (E_{mag}/E_{kin})_{sat} increases as the Mach number decreases. Even in the case when injection of energy is purely through longitudinal forcing modes, (E_{mag}/E_{kin})_{sat}≳10^{-2} at a Mach number of 10^{-3}. We apply our results to magnetic field amplification in the early Universe and predict that a turbulent dynamo can amplify primordial magnetic fields to ≳10^{-16} G on scales up to 0.1pc and ≳10^{-13} G on scales up to 100pc. This produces fields compatible with lower limits of the intergalactic magnetic field inferred from blazar γ-ray observations.

Magnetic field amplification in accretion discs around the first stars: implications for the primordial IMF.

Magnetic fields play an important role in the dynamics of present-day molecular clouds. Recent work has shown that magnetic fields are equally important for primordial clouds, which form the first stars in the Universe. While the primordial magnetic field strength on cosmic scales is largely unconstrained, theoretical models strongly suggest that a weak seed field existed in the early Universe. We study how the amplification of such a weak field can influence the evolution of accretion discs around first stars, and thus affect the primordial initial mass function (IMF). We perform a suite of 3D ideal magneto-hydrodynamic simulations with different initial field strengths and numerical resolutions. We find that, in simulations with sufficient spatial resolution to resolve the Jeans scale during the collapse, even initially weak magnetic fields grow exponentially to become dynamically important due to both the so-called small-scale turbulent dynamo and the large-scale mean-field dynamo. Capturing the small-scale dynamo action depends primarily on how well we resolve the Jeans length, while capturing the large-scale dynamo depends on the Jeans resolution as well as the maximum absolute resolution. Provided enough resolution, we find that fragmentation does not depend strongly on the initial field strength, because even weak fields grow to become strong. However, fragmentation in runs with magnetic fields differs significantly from those without magnetic fields. We conclude that the development of dynamically strong magnetic fields during the formation of the first stars is likely inevitable, and that these fields had a significant impact on the primordial IMF.

Interaction of Large- and Small-scale Dynamos in Isotropic Turbulent Flows from GPU-accelerated Simulations

Abstract Magnetohydrodynamical (MHD) dynamos emerge in many different astrophysical situations where turbulence is present, but the interaction between large-scale dynamos (LSDs) and small-scale dynamos (SSDs) is not fully understood. We performed a systematic study of turbulent dynamos driven by isotropic forcing in isothermal MHD with magnetic Prandtl number of unity, focusing on the exponential growth stage. Both helical and nonhelical forcing was employed to separate the effects of LSD and SSD in a periodic domain. Reynolds numbers ( ) up to ≈250 were examined and multiple resolutions used for convergence checks. We ran our simulations with the Astaroth code, designed to accelerate 3D stencil computations on graphics processing units (GPUs) and to employ multiple GPUs with peer-to-peer communication. We observed a speedup of ≈35 in single-node performance compared to the widely used multi-CPU MHD solver Pencil Code. We estimated the growth rates from both the averaged magnetic fields and their power spectra. At low LSD growth dominates, but at high SSD appears to dominate in both helically and nonhelically forced cases. Pure SSD growth rates follow a logarithmic scaling as a function of . Probability density functions of the magnetic field from the growth stage exhibit SSD behavior in helically forced cases even at intermediate . We estimated mean field turbulence transport coefficients using closures like the second-order correlation approximation (SOCA). They yield growth rates similar to the directly measured ones and provide evidence of α quenching. Our results are consistent with the SSD inhibiting the growth of the LSD at moderate , while the dynamo growth is enhanced at higher .

A global model of the magnetorotational instability in protoneutron stars

Context. Magnetars are isolated neutron stars characterized by their variable high-energy emission, which is powered by the dissipation of enormous internal magnetic fields. The measured spin-down of magnetars constrains the magnetic dipole to be in the range of 1014 − 1015 G. The magnetorotational instability (MRI) is considered to be a promising mechanism to amplify the magnetic field in fast-rotating protoneutron stars and form magnetars. This scenario is supported by many local studies that have shown that magnetic fields could be amplified by the MRI on small scales. However, the efficiency of the MRI at generating a dipole field is still unknown. Aims. To answer this question, we study the MRI dynamo in an idealized global model of a fast rotating protoneutron star with differential rotation. Methods. Using the pseudo-spectral code MagIC, we performed three-dimensional incompressible magnetohydrodynamics simulations in spherical geometry with explicit diffusivities where the differential rotation is forced at the outer boundary. We performed a parameter study in which we varied the initial magnetic field and investigated different magnetic boundary conditions. These simulations were compared to local shearing box simulations performed with the code Snoopy. Results. We obtain a self-sustained turbulent MRI-driven dynamo, whose saturated state is independent of the initial magnetic field. The MRI generates a strong turbulent magnetic field of B ≥ 2 × 1015 G and a nondominant magnetic dipole, which represents systematically about 5% of the averaged magnetic field strength. Interestingly, this dipole is tilted toward the equatorial plane. By comparing these results with shearing box simulations, we find that local models can reproduce fairly well several characteristics of global MRI turbulence such as the kinetic and magnetic spectra. The turbulence is nonetheless more vigorous in the local models than in the global ones. Moreover, overly large boxes allow for elongated structures to develop without any realistic curvature constraint, which may explain why these models tend to overestimate the field amplification. Conclusions. Overall, our results support the ability of the MRI to form magnetar-like large-scale magnetic fields. They furthermore predict the presence of a stronger small-scale magnetic field. The resulting magnetic field could be important to power outstanding stellar explosions, such as superluminous supernovae and gamma-ray bursts.

ДОВГОТРИВАЛІ ВАРІАЦІЇ МАГНІТНОЇ АКТИВНОСТІ СОНЦЯ

The main law of the evolution of the Earth’s climate is the cyclical nature of global changes in the latter. One of the possible explanations for the cyclical nature of global climate changes is provided by the astrophysical model of fluctuations in the insolation of the Earth’s surface by solar radiation. Modern climate change is mainly associated with variations in the magnetic activity of the Sun, one of the main proxies of which are sunspots. The decrease in the number of sunspots coincides with the epochs of cooling on the Earth, while during the maximum number of sunspots warming is observed. The paper reviews cosmogenic reconstructions of long-term variations in the Sun’s magnetic activity (large minima and large maxima) during the Holocene (last 12,000 years). The accidental appearance of large minima and maxima can to some extent be reproduced by modern models of a turbulent dynamo with a stochastic drive. An important key to studying the impact of solar activity variations on the Earth’s climate is the Maunder minimum (late 17th century), during which extremely little sunspots were observed. Applying the method of analysis of rare events to these observations led researchers to conclude that the appearance of sunspots at the Maunder minimum was characterized by a weak amplitude of 22 years. The concept of continuity of magnetic cycles at this time is also confirmed by measurements of cosmogenic radionuclides in natural terrestrial archives. Therefore, today it is believed that during the Maunder minimum, the cyclic magnetic activity of the Sun did not stop, although the amplitude of the cycles was quite low. In the αΩ-dynamo model, this may be due to the fact that the magnitude of the magnetic induction of the toroidal field excited by radial differential rotation in the solar convection zone at this time did not reach the threshold value required for lifting magnetic power tubes on the solar surface (nonlinear dynamo mode). Possible physical mechanisms describing the suppression of the dynamo process at intervals when no sunspots were observed are analyzed. A scenario for explaining the north-south asymmetry of magnetic activity during the Maunder minimum is proposed. A key role in the proposed scenario is played by the special nature of the internal rotation of the Sun, revealed as a result of helioseismological experiments. The modern grand maximum of solar activity, which began in the 1940s, has ceased after solar cycle 23, and activity of the Sun seems to be returning to its normal moderate level.

Influence of ambipolar and Hall effects on vorticity in three-dimensional simulations of magneto-convection.

This paper presents the results of the analysis of three-dimensional simulations of solar magneto-convection that include the joint action of the ambipolar diffusion and the Hall effect. Three simulation runs are compared: one including both ambipolar diffusion and the Hall effect; one including only ambipolar diffusion and one without any of these two effects. The magnetic field is amplified from initial field to saturation level by the action of turbulent local dynamo. In each of these cases, we study 2 h of simulated solar time after the local dynamo reaches the saturation regime. We analyse the power spectra of vorticity, of magnetic field fluctuations and of the different components of the magnetic Poynting flux responsible for the transport of vertical or horizontal perturbations. Our preliminary results show that the ambipolar diffusion produces a strong reduction of vorticity in the upper chromospheric layers and that it dissipates the vortical perturbations converting them into thermal energy. The Hall effect acts in the opposite way, strongly enhancing the vorticity. When the Hall effect is included, the magnetic field in the simulations becomes, on average, more vertical and long-lived flux tube-like structures are produced. We trace a single magnetic structure to study its evolution pattern and the magnetic field intensification, and their possible relation to the Hall effect. This article is part of the Theo Murphy meeting issue 'High-resolution wave dynamics in the lower solar atmosphere'.

Renormalization Group Analysis of the Turbulent Hydromagnetic Dynamo: The Effect of Nonstationarity

Abstract The turbulent hydromagnetic dynamo is a process of magnetic field generation by chaotic flow of an electrically conducting fluid (plasma, liquid iron, etc.). It is responsible for generation of large-scale magnetic fields of astrophysical objects such as planets, stars, accretion disks, galaxies, galaxy clusters, etc. In particular, the dynamical process of induction of large-scale fields by highly conducting plasma has been very difficult to understand, as very low resistivity is not capable of creating a phase shift between magnetic and kinetic components of waves, making their interaction ineffective for generation of a large-scale electromotive force (EMF). The aim of the analysis presented here is to demonstrate that when the typically invoked statistical stationarity of turbulence is relaxed, large-scale magnetic fields can be very effectively generated by low-resistivity plasma. The renormalization group technique is applied to extract the final expression for the mean EMF from the fully nonlinear dynamical equations (Navier–Stokes, induction equation), and the mean field equations are solved for a force-free mode. Nonstationarity is shown to strongly enhance the process of large-scale EMF generation via wave interactions, and the dynamo effect induced by nonstationarity is proved to be effective. The results are also used to demonstrate the influence of magnetic fields and nonstationarity on energy and helicity spectra of turbulent flows.

Seed magnetic fields in turbulent small-scale dynamos

ABSTRACTMagnetic fields in galaxies and galaxy clusters are amplified from a very weak seed value to the observed $\mu$G strengths by the turbulent dynamo. The seed magnetic field can be of primordial or astrophysical origin. The strength and structure of the seed field, on the galaxy or galaxy cluster scale, can be very different, depending on the seed-field generation mechanism. The seed field first encounters the small-scale dynamo, thus we investigate the effects of the strength and structure of the seed field on the small-scale dynamo action. Using numerical simulations of driven turbulence and considering three different seed-field configurations: (1) uniform field, (2) random field with a power-law spectrum, and (3) random field with a parabolic spectrum, we show that the strength and statistical properties of the dynamo-generated magnetic fields are independent of the details of the seed field. We demonstrate that, even when the small-scale dynamo is not active, small-scale magnetic fields can be generated and amplified linearly due to the tangling of the large-scale field. In the presence of the small-scale dynamo action, we find that any memory of the seed field for the non-linear small-scale dynamo generated magnetic fields is lost and thus, it is not possible to trace back seed-field information from the evolved magnetic fields in a turbulent medium.

Magnetizing the circumgalactic medium of disc galaxies

ABSTRACT The circumgalactic medium (CGM) is one of the frontiers of galaxy formation and intimately connected to the galaxy via accretion of gas on to the galaxy and gaseous outflows from the galaxy. Here, we analyse the magnetic field in the CGM of the Milky Way-like galaxies simulated as part of the auriga project that constitutes a set of high-resolution cosmological magnetohydrodynamical zoom simulations. We show that before z = 1 the CGM becomes magnetized via galactic outflows that transport magnetized gas from the disc into the halo. At this time, the magnetization of the CGM closely follows its metal enrichment. We then show that at low redshift an in situ turbulent dynamo that operates on a time-scale of Gigayears further amplifies the magnetic field in the CGM and saturates before z = 0. The magnetic field strength reaches a typical value of $0.1\, \mu \mathrm{ G}$ at the virial radius at z = 0 and becomes mostly uniform within the virial radius. Its Faraday rotation signal is in excellent agreement with recent observations. For most of its evolution, the magnetic field in the CGM is an unordered small-scale field. Only strong coherent outflows at low redshift are able to order the magnetic field in parts of the CGM that are directly displaced by these outflows.

Transition from laminar to turbulent dynamo: the effect of varying Prandtl number

We carry out numerical dynamo simulations in a spherical shell to assess the influence of varying Prandtl number on the morphology of the induced magnetic field in both cases when the convection is laminar and turbulent. We work in a regime of large magnetic Prandtl number () due to computational constraints of the full three-dimensional numerical model. We consider dynamos driven by thermal convection in the regimes of small and unit Prandtl numbers. Unlike the laminar convection, in the turbulent regime the morphology of the induced magnetic fields depends only weakly on the Prandtl number. For both values of considered the magnetic field is non-dipolar and small-scale. Together with the previous results of other authors, who found that the morphology of the magnetic field does not change substantially with when is small, the results of our study suggest that the induced magnetic field does not vary substantially with regardless of the strength of the magnetic diffusion.

Analysis of quiet-sun turbulence on the basis of SDO/HMI and goode solar telescope data

ABSTRACT We analysed line-of-sight magnetic fields and magnetic power spectra of an undisturbed photosphere using magnetograms acquired by the Helioseismic and Magnetic Imager (HMI) on-board the Solar Dynamic Observatory and the Near InfraRed Imaging Spectrapolarimeter (NIRIS) operating at the Goode Solar Telescope of the Big Bear Solar Observatory. In the NIRIS data, we revealed thin flux tubes of 200–400 km in diameter and of 1000–2000 G field strength. The HMI power spectra determined for a coronal hole, a quiet sun, and a plage areas exhibit the same spectral index of −1 on a broad range of spatial scales from 10–20 Mm down to 2.4 Mm. This implies that the same mechanism(s) of magnetic field generation operate everywhere in the undisturbed photosphere. The most plausible one is the local turbulent dynamo. When compared to the HMI spectra, the −1.2 slope of the NIRIS spectrum appears to be more extended into the short spatial range until the cut-off at 0.8–0.9 Mm, after which it continues with a steeper slope of −2.2. Comparison of the observed and Kolmogorov-type spectra allowed us to infer that the Kolmogorov turbulent cascade cannot account for more than 35 per cent of the total magnetic energy observed in the scale range of 3.5–0.3 Mm. The energy excess can be attributed to other mechanisms of field generation such as the local turbulent dynamo and magnetic superdiffusivity observed in an undisturbed photosphere that can slow down the rate of the Kolmogorov cascade leading to a shallower resulting spectrum.

Nonlinear Turbulent Dynamo during Gravitational Collapse

Abstract Via amplification by turbulent dynamo, magnetic fields can be potentially important for the formation of the first stars. To examine the dynamo behavior during the gravitational collapse of primordial gas, we extend the theory of the nonlinear turbulent dynamo to include the effect of gravitational compression. The relative importance between dynamo and compression varies during contraction, with the transition from dynamo- to compression-dominated amplification of magnetic fields with the increase of density. In the nonlinear stage of magnetic field amplification with the scale-by-scale energy equipartition between turbulence and magnetic fields, reconnection diffusion of magnetic fields in ideal magnetohydrodynamic turbulence becomes important. It causes the violation of the flux-freezing condition and accounts for (a) the small growth rate of the nonlinear dynamo, (b) the weak dependence of magnetic energy on density during contraction, (c) the saturated magnetic energy, and (d) the large correlation length of magnetic fields. The resulting magnetic field structure and the scaling of magnetic field strength with density are radically different from the expectations of flux freezing.

Magnetic power spectrum in the undisturbed solar photosphere

Using the magnetic field data obtained with the Helioseismic and Magnetic Imager (HMI) onboard the Solar Dynamics Observatory (SDO), an investigation of magnetic power spectra in the undisturbed solar photosphere was performed. The results are as follows. 1) To get a reliable estimate of a magnetic power spectrum from the uniformly distributed quiet-sun magnetic flux, a sample pattern of no less than 300 pixels length should be adopted. With smaller patterns, energy on all observable scales might be overestimated. 2) For patterns of different magnetic intensity (e.g., a coronal hole, a quiet-sun area, an area of supergranulation), the magnetic power spectra in a range of (2.5-10) Mm exhibit very close spectral indices of about -1. The observed spectrum is more shallow than the Kolmogorov-type spectrum (with a slope of -5/3) and it differs from steep spectra of active regions. Such a shallow spectrum cannot be explained by the only direct Kolmogorov’s cascade, but it can imply a small-scale turbulent dynamo action in a wide range of scales: from tens of megameters down to at least 2.5 Mm. On smaller scales, the HMI/SDO data do not allow us to reliably derive the shape of the spectrum. 3) Data make it possible to conclude that a uniform mechanism of the small-scale turbulent dynamo is at work all over the solar surface outside active regions.

Magnetic Helicity Dissipation and Production in an Ideal MHD Code

Abstract We study a turbulent helical dynamo in a periodic domain by solving the ideal magnetohydrodynamic (MHD) equations with the FLASH code using the divergence-cleaning eight-wave method and compare our results with direct numerical simulations (DNS) using the Pencil Code. At low resolution, FLASH reproduces the DNS results qualitatively by developing the large-scale magnetic field expected from DNS, but at higher resolution, no large-scale magnetic field is obtained. In all those cases in which a large-scale magnetic field is generated, the ideal MHD results yield too little power at small scales. As a consequence, the small-scale current helicity is too small compared with that of the DNS. The resulting net current helicity has then always the wrong sign, and its statistical average also does not approach zero at late times, as expected from the DNS. Our results have implications for astrophysical dynamo simulations of stellar and galactic magnetism using ideal MHD codes.

3D turbulent reconnection: Theory, tests, and astrophysical implications

Magnetic reconnection, topological changes in magnetic fields, is a fundamental process in magnetized plasmas. It is associated with energy release in regions of magnetic field annihilation, but this is only one facet of this process. Astrophysical fluid flows normally have very large Reynolds numbers and are expected to be turbulent, in agreement with observations. In strong turbulence, magnetic field lines constantly reconnect everywhere and on all scales, thus making magnetic reconnection an intrinsic part of the turbulent cascade. We note in particular that this is inconsistent with the usual practice of magnetic field lines as persistent dynamical elements. A number of theoretical, numerical, and observational studies starting with the paper done by Lazarian and Vishniac [Astrophys. J. 517, 700–718 (1999)] proposed that 3D turbulence makes magnetic reconnection fast and that magnetic reconnection and turbulence are intrinsically connected. In particular, we discuss the dramatic violation of the textbook concept of magnetic flux-freezing in the presence of turbulence. We demonstrate that in the presence of turbulence, the plasma effects are subdominant to turbulence as far as the magnetic reconnection is concerned. The latter fact justifies a magnetohydrodynamiclike treatment of magnetic reconnection on all scales much larger than the relevant plasma scales. We discuss the numerical and observational evidence supporting the turbulent reconnection model. In particular, we demonstrate that the tearing reconnection is suppressed in 3D, and unlike the 2D settings, 3D reconnection induces turbulence that makes magnetic reconnection independent of resistivity. We show that turbulent reconnection dramatically affects key astrophysical processes, e.g., star formation, turbulent dynamo, and acceleration of cosmic rays. We provide criticism of the concept of “reconnection-mediated turbulence” and explain why turbulent reconnection is very different from enhanced turbulent resistivity and hyper-resistivity and why the latter have fatal conceptual flaws.

Спектр мощности магнитного поля в невозмущенной фотосфере Солнца

Проведено исследование спектров мощности магнитного поля в невозмущенной фотосфере Солнца по данным инструмента Helioseismic and Magnetic Imager (HMI) на борту станции Solar Dynamics Observatory (SDO). Результаты можно сформулировать следующим образом. 1) Для достоверной оценки спектра мощности в обширной зоне квазиравномерного распределения поля следует выбирать участок длиной не менее 300 пикселей. По оценкам по меньшему участку, мощность на всех доступных частотах оказывается завышенной. 2) Для магнитных зон разной интенсивности, а именно для корональной дыры, спокойного Солнца и супергрануляционной сетки, спектр мощности в диапазоне (2.5-10) Мм проявляет одинаковый спектральный индекс, близкий к -1. Наблюдаемый спектр более пологий, чем колмогоровский (с наклоном -5/3), и отличается от более крутых спектров активных областей. Такой пологий спектр нельзя объяснить только прямым колмогоровским каскадом. Его можно объяснить дополнительной накачкой магнитной энергии за счет мелкомасштабного турбулентного динамо, работающего в широком диапазоне масштабов: от десятков мегаметров до, по крайней мере, 2.5 Мм. На масштабах менее 2.5 Мм SDO/HMI-данные не позволяют адекватно оценить форму спектра. 3) Данные позволяют заключить, что однообразный механизм мелкомасштабного турбулентного динамо работает по всей поверхности Солнца вне активных областей.

Suppression of Small-Scale Magnetic Field Generation by Transformation of the Kinetic Energy Spectrum

The stabilization of a small-scale dynamo caused by a change in the energy spectrum of a conducting plasma flow is investigated. Such a change in kinetic properties can be due to the reverse conversion of generated magnetic energy into kinetic energy. We model the reverse conversion process using the classical Kazantsev model, which describes the operation of a small-scale dynamo in a mirror-symmetric turbulent flow. Our results suggest that the appearance of magnetic energy at small scales in the kinetic spectrum can stop the generation process. This stabilization turns out to be no less efficient than the braking of a turbulent dynamo through the conservation of total magnetic and kinetic energies. However, it looks more realistic, because for most dynamo systems the observed magnetic energy is lower than the kinetic one by several orders of magnitude. We show that in this method of nonlinear suppression just the appearance of energy at small scales is often not enough; in particular, a simple shift of the Gaussian spectrum toward small scales only amplifies the generation. In this paper, we describe a reasonable spectrum transformation method that allows the generation in both critical and supercritical regimes of small-scale dynamo operation to be stabilized. The fact that the described spectrum transformation can be directly recorded in an experimental test is equally important.

A realization of the turbulent vortex dynamo in the atmosphere: based on the 21st century knowledge

A brief summary of the results of numerical modeling undertaken to substantiate the action of the turbulent vortex dynamo (TVD) in the atmosphere is presented. Two series of numerical experiments are discussed. The first of them was aimed at examining a mathematical model of TVD formulated as the mean-field equations, which describe the dynamo-effect in a non-uniformly heated medium. To simulate and demonstrate the main peculiarities of the new large-scale instability in turbulence, an ingenious statement of the problem for numerical analysis was produced, namely, within a simpler framework – the Rayleigh-Bénard problem on laminar thermal convection in an extended horizontal layer. Within the second series, our developed numerical approach, used and tested in the initial series as well as obtained results therein allowed applying the TVD theory to diagnosis of tropical cyclogenesis. To this purpose, the Russian-American collaboration was initiated and the most advanced to that date tools of atmospheric analysis were utilized. Tropical cyclone formation was examined through the analysis of the American data of near-cloud-resolving (2-3 km horizontal grid spacing) atmospheric numerical simulation.

Turbulence Dynamo in the Stratified Medium of Galaxy Clusters

The existence of microgauss magnetic fields in galaxy clusters have been established through observations of synchrotron radiation and Faraday rotation. They are conjectured to be generated via small-scale dynamo by turbulent flow motions in the intracluster medium (ICM). Some of giant radio relics, on the other hand, show the structures of synchrotron polarization vectors, organized over the scales of $\sim$ Mpc, challenging the turbulence origin of cluster magnetic fields. Unlike turbulence in the interstellar medium, turbulence in the ICM is subsonic. And it is driven sporadically in highly stratified backgrounds, when major mergers occur during the hierarchical formation of clusters. To investigate quantitatively the characteristics of turbulence dynamo in such ICM environment, we performed a set of turbulence simulations using a high-order-accurate, magnetohydrodynamic (MHD) code. We find that turbulence dynamo could generate the cluster magnetic fields up to the observed level from the primordial seed fields of $10^{-15}$ G or so within the age of the universe, if the MHD description of the ICM could be extended down to $\sim$ kpc scales. However, highly organized structures of polarization vectors, such as those observed in the Sausage relic, are difficult to be reproduced by the shock compression of turbulence-generated magnetic fields. This implies that the modeling of giant radio relics may require the pre-existing magnetic fields organized over $\sim$ Mpc scales.

Grand Activity Minima and Maxima via Dual Dynamos

Reconstructions of past solar activity based on cosmogenic radioisotopes have reavealed that the Sun spends a significant fraction ( ${\approx}\, 20$ %) of its time in aperiodically recurring states of so-called Grand Minima or Grand Maxima, namely epochs of strongly supressed and markedly above-average levels of magnetic activity, respectively. The physical origin of these episodes is not yet understood. In this article we present a dual-dynamo model of the solar cycle, combining a dominant dynamo based on differential-rotation shear and surface decay of bipolar active regions, and a weak, deep-seated turbulent dynamo. The resulting dynamo simulations are found to exhibit the equivalent of observed Grand Minima and Maxima. By adjusting the magnitude and saturation level of the secondary turbulent dynamo, we can reproduce well the duration and waiting-time distributions of Grand Minima and Maxima inferred from the cosmogenic-isotope record. The exit from Grand Minima episodes is typically characterized by strong hemispheric asymmetries, in agreement with sunspot observations during the 1645 – 1715 Maunder Minimum. In these simulations, Grand Maxima can be unambiguously identified as a distinct dual-dynamo state resulting from constructive interference between the two dynamos mechanisms operating within the simulation. This interaction leads to the autonomous production of long quasi-periodicities in the millennial range, commensurate with the Halstatt cycle. Such a quasi-periodic modulation, readily produced through dynamical backreaction on large-scale flows in non-kinematic dynamo models, is quite uncommon in a purely kinematic solar-cycle model such as the one developed herein. We argue that these long periodicities are set by the long diffusion time of magnetic field accumulating in the stable layers underlying the turbulent convection zone.