Recent Developments in Magnetic Dynamo Theory

Karl-Heinz Rädler died on the 9th of February 2020 at his home near Potsdam at the age of 85 years. He is known for his path-breaking contributions to the development of cosmic dynamo theory. Together with Max Steenbeck and Fritz Krause, he presented a rigorous derivation of the α effect in 1966. A few years later, after their work became known in the West, many scientists around the globe applied their theory and started working on models of solar and Galactic dynamos. The field of mean-field dynamo theory was born, and it started a new industry in astrophysics. The roots of what is now sometimes referred to as the Potsdam Dynamo School go back to an earlier episode that started in Jena. After graduating from the University of Leipzig, Karl-Heinz Rädler developed his lifelong interest in the origin of cosmic magnetic fields when he took up a position as an assistant at the Institute for Magnetohydrodynamics at the Akademie der Wissenschaften in Jena. He joined the group of Max Steenbeck, where the theory of averaged magnetic fields was formulated and where he defended his PhD thesis “Zur Elektrodynamik turbulent bewegter leitender Medien.” It marked the beginning of a new discipline of mathematical physics, dealing with the equations of magnetohydrodynamics in turbulent media. In the 1970s, the research activities were moved to the Zentralinstitut für Astrophysik at Potsdam. This activity significantly shaped the profile of the Institute and contributed to its high international standing. Until today, many of the theoretical, numerical, and experimental studies of magnetohydrodynamics are based on the monograph “Mean-field Magnetohydrodynamics and Dynamo Theory” by Krause and Rädler of 1980. After the Fall of the Iron Curtain, owing to Rädler's scientific competence and personal integrity, as well as his democratic way of thinking, he was elected as the spokesperson of the scientific council of the Institute. Professor Rädler was chosen to become the founding director of what became the new Astrophysical Institute of Potsdam on the premises of the old observatory at Babelsberg. During that time, the Institute started collaboration with the Large Binocular Telescope in Arizona, as well as the telescope systems GREGOR and STELLA at Tenerife. It was also during this time when K.H. Rädler was active as the Editor-in-Chief of the Astronomische Nachrichten/Astronomical News. Another scientific highlight was his involvement in the theoretical underpinning of the Karlsruhe dynamo experiment in the late 1990s. Using the mean-field approach, he and his team in Potsdam prepared detailed predictions for the excitation conditions and saturation behavior for the dynamo experiment. Karl-Heinz Rädler remained an acting master of the field until his last years. Particularly noteworthy is his work of 2005, when he devised a numerical procedure for computing the full set of turbulent transport coefficients. With that, mean-field dynamo theory became an accurate and predictive tool beyond the realm of quasilinear theory. In celebration of Rädler's 75th birthday, friends and colleagues presented their latest research in Stockholm on the “α effect and beyond.” Some of the contributions in the special issue of Geophysical and Astrophysical Fluid Dynamics provide a lasting memory of this. His lifelong accomplishments have been honored by the Astronomische Gesellschaft through its award of the Karl Schwarzschild medal of 2013. His very last big contribution to the community was as one of the guest editors of the Journal of Plasma Physics of 2018 with the title “50 years of Mean Field Electrodynamics,” which fittingly characterizes a subject of which he was part of since the very first hour.

The nonlinear mean-field dynamo due to a shear-current effect in a nonhelical homogeneous turbulence with a mean velocity shear is discussed. The transport of magnetic helicity as a dynamical nonlinearity is taken into account. The shear-current effect is associated with the W × J term in the mean electromotive force, where W is the mean vorticity due to the large-scale shear motions and J is the mean electric current. This effect causes the generation of large-scale magnetic field in a turbulence with large hydrodynamic and magnetic Reynolds numbers. The dynamo action due to the shear-current effect depends on the spatial scaling of the correlation time τ( k ) of the background turbulence, where k is the wave number. For Kolmogorov scaling, τ(k)∝k −2/3, the dynamo instability occurs, while when τ(k) ∝k −2 (small hydrodynamic and magnetic Reynolds numbers) there is no the dynamo action in a sheared nonhelical turbulence. The magnetic helicity flux strongly affects the magnetic field dynamics in the nonlinear stage of the dynamo action. Numerical solutions of the nonlinear mean-field dynamo equations which take into account the shear-current effect, show that if the magnetic helicity flux is not small, the saturated level of the mean magnetic field is of the order of the equipartition field determined by the turbulent kinetic energy. Turbulence with a large-scale velocity shear is a universal feature in astrophysics, and the obtained results can be important for elucidation of origin of the large-scale magnetic fields in astrophysical sheared turbulence.

ABSTRACTSolar, stellar and galactic large-scale magnetic fields are originated due to a combined action of non-uniform (differential) rotation and helical motions of plasma via mean-field dynamos. Usually, non-linear mean-field dynamo theories take into account algebraic and dynamic quenching of alpha effect and algebraic quenching of turbulent magnetic diffusivity. However, the theories of the algebraic quenching do not take into account the effect of modification of the source of turbulence by the growing large-scale magnetic field. This phenomenon is due to the dissipation of the strong large-scale magnetic field resulting in an increase of the total turbulent energy. This effect has been studied using the budget equation for the total turbulent energy (which takes into account the feedback of the generated large-scale magnetic field on the background turbulence) for (i) a forced turbulence, (ii) a shear-produced turbulence, and (iii) a convective turbulence. As the result of this effect, a non-linear dynamo number decreases with increase of the large-scale magnetic field, so that that the mean-field αΩ, α2, and α2Ω dynamo instabilities are always saturated by the strong large-scale magnetic field.

A nonlinear, time-dependent, magnetically driven dynamo theory that shows how magnetically dominated configurations can relax to become helical on the largest scale available is presented. Coupled time-dependent differential equations for large-scale magnetic helicity, small-scale magnetic helicity, velocity, and the electromotive force are solved. The magnetic helicity on small scales relaxes to drive significant large-scale helical field growth on dynamical (Alfvén crossing) time scales, independent of the magnitude of finite microphysical transport coefficients, after which the growing kinetic helicity slows the growth to a viscously limited pace. This magnetically driven dynamo complements the nonlinear kinetic helicity-driven dynamo; for the latter, the growing magnetic helicity fluctuations suppress, rather than drive, large-scale magnetic helicity growth. A unified set of equations accommodates both types of dynamos.

We present a numerical and analytical study of incompressible homogeneous conducting fluids using a helical Fourier representation. We analytically study both small- and large-scale dynamo properties, as well as the inverse cascade of magnetic helicity, in the most general minimal subset of interacting velocity and magnetic fields on a closed Fourier triad. We mainly focus on the dependency of magnetic field growth as a function of the distribution of kinetic and magnetic helicities among the three interacting wavenumbers. By combining direct numerical simulations of the full magnetohydrodynamics equations with the helical Fourier decomposition, we numerically confirm that in the kinematic dynamo regime the system develops a large-scale magnetic helicity with opposite sign compared to the small-scale kinetic helicity, a sort of triad-by-triad α-effect in Fourier space. Concerning the small-scale perturbations, we predict theoretically and confirm numerically that the largest instability is achived for the magnetic component with the same helicity of the flow, in agreement with the Stretch–Twist–Fold mechanism. Vice versa, in the presence of Lorentz feedback on the velocity, we find that the inverse cascade of magnetic helicity is mostly local if magnetic and kinetic helicities have opposite signs, while it is more nonlocal and more intense if they have the same sign, as predicted by the analytical approach. Our analytical and numerical results further demonstrate the potential of the helical Fourier decomposition to elucidate the entangled dynamics of magnetic and kinetic helicities both in fully developed turbulence and in laminar flows.

A shell model of magnetohydrodynamic turbulence, which allows one to conserve all the integrals of motion in both two and three dimensions, is proposed and studied. We demonstrate that this model reproduces basic facts known in the small-scale turbulent dynamo theory. In particular, we consider a process of redistribution of magnetic helicity generated by the mean-field dynamo, described in the model as magnetic forcing, into a small-scale magnetic field. We argue that the resulting equilibrium magnetic field spectrum strongly depends on the level of magnetic helicity and cross helicity, introduced by the large scales. The spectra with spectral index ``-5/3'' dominate if the cross helicity vanishes. If the level of cross helicity is high (correlated velocity and magnetic field) the spectra depend on the magnetic helicity: the strong magnetic helicity suppresses any cascade providing steep spectra, while the vanishing helicity of turbulent magnetic fields results in the occurrence of Kraichnan-Iroshnikov spectral index ``-3/2.''

The inter-relationships between magnetic helicity, magnetic reconnection, and dynamo effects are discussed. In laboratory experiments, where two plasmas are driven to merge, the helicity content of each plasma strongly affects the reconnection rate, as well as the shape of the diffusion region. Conversely, magnetic reconnection events also strongly affect the global helicity, resulting in efficient helicity cancellation (but not dissipation) during counter-helicity reconnection and a finite helicity increase or decrease (but less efficiently than dissipation of magnetic energy) during co-helicity reconnection. Close relationships also exist between magnetic helicity and dynamo effects. The turbulent electromotive force along the mean magnetic field (alpha-effect), due to either electrostatic turbulence or the electron diamagnetic effect, transports mean-field helicity across space without dissipation. This has been supported by direct measurements of helicity flux in a laboratory plasma. When the dynamo effect is driven by electromagnetic turbulence, helicity in the turbulent field is converted to mean-field helicity. In all cases, however, dynamo processes conserve total helicity except for a small battery effect, consistent with the observation that the helicity is approximately conserved during magnetic relaxation.

The inter-relationships between magnetic helicity, magnetic reconnection, and dynamo effects are discussed. In laboratory experiments, where two plasmas are driven to merge, the helicity content of each plasma strongly affects the reconnection rate, as well as the shape of the diffusion region. Conversely, magnetic reconnection events also strongly affect the global helicity, resulting in efficient helicity cancellation (but not dissipation) during counter-helicity reconnection and a finite helicity increase or decrease (but less efficiently than dissipation of magnetic energy) during co-helicity reconnection. Close relationships also exist between magnetic helicity and dynamo effects. The turbulent electromotive force along the mean magnetic field (alpha-effect), due to either electrostatic turbulence or the electron diamagnetic effect, transports mean-field helicity across space without dissipation. This has been supported by direct measurements of helicity flux in a laboratory plasma. When the dynamo effect is driven by electromagnetic turbulence, helicity in the turbulent field is converted to mean-field helicity. In all cases, however, dynamo processes conserve total helicity except for a small battery effect, consistent with the observation that the helicity is approximately conserved during magnetic relaxation.

The magnetorotational instability (MRI) may dominate outward transport of angular momentum in accretion disks, allowing material to fall onto the central object. Previous work has established that the MRI can drive a mean-field dynamo, possibly leading to a self-sustaining accretion system. Recently, however, simulations of the scaling of the angular momentum transport parameter $\alphaSS$ with the magnetic Prandtl number $\Prandtl$ have cast doubt on the ability of the MRI to transport astrophysically relevant amounts of angular momentum in real disk systems. Here, we use simulations including explicit physical viscosity and resistivity to show that when vertical stratification is included, mean field dynamo action operates, driving the system to a configuration in which the magnetic field is not fully helical. This relaxes the constraints on the generated field provided by magnetic helicity conservation, allowing the generation of a mean field on timescales independent of the resistivity. Our models demonstrate the existence of a critical magnetic Reynolds number $\Rmagc$, below which transport becomes strongly $\Prandtl$-dependent and chaotic, but above which the transport is steady and $\Prandtl$-independent. Prior simulations showing $\Prandtl$-dependence had $\Rmag < \Rmagc$. We conjecture that this steady regime is possible because the mean field dynamo is not helicity-limited and thus does not depend on the details of the helicity ejection process. Scaling to realistic astrophysical parameters suggests that disks around both protostars and stellar mass black holes have $\Rmag >> \Rmagc$. Thus, we suggest that the strong $\Prandtl$ dependence seen in recent simulations does not occur in real systems.

Aims. Even though magnetic fields play an important role in galaxy evolution, the redshift evolution of galactic-scale magnetic fields is not well constrained observationally. In this paper we provide an observational constraint on the timescale of the mean-field dynamo, and derive the magnetic field in a distant galaxy at z = 0.414. Methods. We obtained broadband spectro-polarimetric 1−8 GHz Very Large Array observations of the lensing system B1600+434, which is a background quasar gravitationally lensed by a foreground spiral galaxy into two images. We applied rotation measure (RM) synthesis and Stokes QU fitting to derive the RM of the two lensed images, which we used to estimate the lensing galaxy’s magnetic field. Results. We measured the RM difference between the lensed images and detected Faraday dispersion caused by the magneto-ionic medium of the lensing galaxy at z = 0.414. Assuming that the RM difference is due to the large-scale regular field of the galaxy’s halo, we measured a coherent magnetic field with a strength of 0.2−3.0 μG at 0.7 kpc and 0.01−2.8 μG at 6.2 kpc vertical distance from the disk of the galaxy. We derive an upper limit on the dynamo e-folding time: τ dynamo &lt; 2.9 × 10 8 yr. We find turbulence on scales below 50 pc and a turbulent field strength of 0.2−12.1 μG. Conclusions. We measured the magnetic field in the halo of a spiral galaxy and find turbulence on scales of &lt; 50 pc. If the RM difference is due to large-scale fields, our result follows the expectation from mean-field dynamo theory and shows that galaxies at z ≃ 0.4 already have magnetic field strengths similar to present-day galaxies. There is one caveat, however: we note the possibility of the turbulent field of the lensing galaxy contributing to the observed RM difference.

Three closely related stumbling blocks of solar mean field dynamo theory are discussed: how dominant are the small scale fields, how is the alpha effect quenched, and whether magnetic and current helicity fluxes alleviate the quenching? It is shown that even at the largest currently available resolution there is no clear evidence of power law scaling of the magnetic and kinetic energy spectra in turbulence. However, using subgrid scalemodeling, some indications of asymptotic equipartition can be found. The frequently used first order smoothing approach to calculate the alpha effect and other transport coefficients is contrasted with the superior minimal tau approximation. The possibility of catastrophic alpha quenching is discussed as a result of magnetic helicity conservation. Magnetic and current helicity fluxes are shown to alleviate catastrophic quenching in the presence of shear. Evidence for strong large scale dynamo action, even in the absence of helicity in the forcing, is presented. (© 2005 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Astrophysical jets are launched from strongly magnetized systems that host an accretion disk surrounding a central object. Here we address the question of how accretion-disk magnetization and the field structure required for jet launching are generated. We continue our work from Mattia &amp; Fendt (Paper I) by considering a nonscalar accretion-disk mean-field α 2Ω dynamo in the context of large-scale disk-jet simulations. We now investigate a disk dynamo that follows analytical solutions of the mean-field dynamo theory, essentially based only on a single parameter, the Coriolis number. We thereby confirm the anisotropy of the dynamo tensor acting in accretion disks, allowing both the resistivity and mean-field dynamo to be related to the disk turbulence. Our new model recovers previous simulations by applying a purely radial initial field while allowing for a more stable evolution for seed fields with a vertical component. We also present correlations between the strength of the disk dynamo coefficients and the dynamical parameters of the jet that is launched, and discuss their implications for observed jet quantities.

Context. There are indications that the magnetic field evolution in galaxies is influenced by tidal interactions and mergers between galaxies. Aims. We carried out a parameter study of interacting disk galaxies with impact parameters ranging from central collisions to weakly interacting scenarios. The orientations of the disks were also varied. In particular, we investigated how magnetic field amplification depends on these parameters. Methods. We used magnetohydrodynamics for gas disks in combination with live dark-matter halos in adaptive mesh refinement simulations. The disks were initialized using a setup for isolated disks in hydrostatic equilibrium. Since we focused on the impact of tidal forces on magnetic field evolution, adiabatic physics was applied. Small-scale filtering of the velocity and magnetic field allowed us to estimate the turbulent electromotive force (EMF) and kinetic helicity. Results. Time series of the average magnetic field in central and outer disk regions show pronounced peaks during close encounters and mergers. This agrees with observed magnetic fields at different interaction stages. The central field strength exceeds 10 μG (corresponding to an amplification factor of 2–3) for small impact parameters. As the disks are increasingly disrupted and turbulence is produced by tidal forces, the small-scale EMF reaches a significant fraction of the total EMF. The small-scale kinetic helicity is initially antisymmetric across the disk plane. Though its evolution is sensitive to both the impact parameter and inclinations of the rotation axes with respect to the relative motion of the disks, antisymmetry is generally broken through interactions and the merger remnant loses most of the initial helicity. The EMF and the magnetic field also decay rapidly after coalescence. Conclusions. The strong amplification during close encounters of the interacting galaxies is mostly driven by helical flows and a mean-field dynamo. The small-scale dynamo contributes significantly in post-interaction phases. However, the amplification of the magnetic field cannot be sustained.

Fluxes of the magnetic helicity density play an important role in large-scale turbulent dynamos, allowing the growth of large-scale magnetic fields while overcoming catastrophic quenching. We show here, analytically, how several important types of magnetic helicity fluxes can arise from terms involving triple correlators of fluctuating fields in the helicity density evolution equation. For this, we assume incompressibility and weak inhomogeneity, and use a quasi-normal closure approximation: fourth-order correlators are replaced by products of second-order ones, and the effect of the fourth-order cumulants on the evolution of the third moments is modeled by a strong damping term. First, we show how a diffusive helicity flux, until now only measured in simulations, arises from the triple correlation term. This is accompanied by what we refer to as a random advective flux, which predominantly transports magnetic helicity along the gradients of the random fields. We also find that a new helicity flux contribution, in some aspects similar to that first proposed by Vishniac, can arise from the triple correlator. This contribution depends on the gradients of the random magnetic and kinetic energies along the large-scale vorticity, and thus arises in any rotating, stratified system, even if the turbulence is predominantly non-helical. It can source a large-scale dynamo by itself while spatially transporting magnetic helicity within the system.

To investigate the characteristics of large-scale and long-term evolution of magnetic helicity with solar cycles, we use the method of Local Correlation Tracking to estimate the magnetic helicity evolution over solar cycle 23 from 1996 to 2009 using 795 MDI magnetic synoptic charts. The main results are as follows: the hemispheric helicity rule still holds in general, i.e., the large-scale negative (positive) magnetic helicity dominates the northern (southern) hemisphere. However, the large-scale magnetic helicity fluxes show the same sign in both hemispheres around 2001 and 2005. The global, large-scale magnetic helicity flux over the solar disk changes from a negative value at the beginning of solar cycle 23 to a positive value at the end of the cycle, while the net accumulated magnetic helicity is negative in the period between 1996 and 2009.