Mean-field Magnetohydrodynamics Research Articles

Karl‐Heinz Rädler (1935–2020)

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.

Flux Tubes Forming Instability Near the Base of the Rotating Convection Zone: A Possible Explanation for the Low Latitudes of Sunspots

Abstract The rise of flux tubes with intense magnetic fields from the base of the convection zone to the solar surface has been substantiated as a probable mechanism for sunspot formation. The origin of flux tubes of a sufficient strength (∼105 G) is, however, uncertain. This paper considers the instability of a large-scale toroidal magnetic field caused by the magnetic suppression of convective heat transport as a candidate for the flux-tube formation mechanism. The consideration employs the analytical dependence of eddy diffusion on the magnetic field supplied by mean-field magnetohydrodynamics. The instability tends to produce regions of increased field strength with spatial scales of the order of 100 Mm at the base of the convection zone. Characteristic growth times of the instability are short compared to the 11 yr cycle. The threshold field strength for the onset of the instability increases from several hundred Gauss in the vicinity of the equator to some kilo-Gauss at middle latitudes. Growth rates of unstable disturbances decrease with latitude. These latitudinal trends can be the reason for the observed confinement of sunspot activity to a near-equatorial belt.

Large-Scale Magnetic Field Fragmentation in Flux-Tubes Near the Base of the Solar Convection Zone

Magnetic quenching of turbulent thermal diffusivity leads to instability of the large-scale field with the production of spatially isolated regions of enhanced field. This conclusion follows from a linear stability analysis in the framework of mean-field magnetohydrodynamics that allows for thermal diffusivity dependence on the magnetic field. The characteristic growth time of the instability is short compared to the 11-year period of solar activity. The characteristic scale of the increased field regions measures in tens of mega-meters. The instability can produce magnetic inhomogeneities whose buoyant rise to the solar surface forms the solar active regions. The magnetic energy of the field concentrations coincides in order of magnitude with the energy of the active regions.

Effect of fluctuations on mean-field dynamos

We discuss the effect of different types of fluctuations on dynamos generated in the limit of scale separation. We first recall that the magnetic field observed in the VKS (von Karman flow of liquid sodium) experiment is not the one that would be generated by the mean flow alone and that smaller scale turbulent fluctuations therefore play an important role. We then consider how velocity fluctuations affect the dynamo threshold in the framework of mean-field magnetohydrodynamics. We show that the detrimental effect of turbulent fluctuations observed with many flows disappears in the case of helical flows with scale separation. We also find that fluctuations of the electrical conductivity of the fluid, for instance related to temperature fluctuations in convective flows, provide an efficient mechanism for dynamo action. Finally, we conclude by describing an experimental configuration that could be used to test the validity of mean-field magnetohydrodynamics in strongly turbulent flows.

The Tayler instability at low magnetic Prandtl numbers: between chiral symmetry breaking and helicity oscillations

The Tayler instability is a kink-type, current driven instability that plays an important role in plasma physics but might also be relevant in liquid metal applications with high electrical currents. In the framework of the Tayler–Spruit dynamo model of stellar magnetic field generation (Spruit 2002 Astron. Astrophys. 381 923–32), the question of spontaneous helical (chiral) symmetry breaking during the saturation of the Tayler instability has received considerable interest (Zahn et al 2007 Astron. Astrophys. 474 145–54; Gellert et al 2011 Mon. Not. R. Astron. Soc. 414 2696–701; Bonanno et al 2012 Phys. Rev. E 86 016313). Focusing on fluids with low magnetic Prandtl numbers, for which the quasistatic approximation can be applied, we utilize an integro-differential equation approach (Weber et al 2013 New J. Phys.15 043034) in order to investigate the saturation mechanism of the Tayler instability. Both the exponential growth phase and the saturated phase are analysed in terms of the action of the α and β effects of mean-field magnetohydrodynamics. In the exponential growth phase we always find a spontaneous chiral symmetry breaking which, however, disappears in the saturated phase. For higher degrees of supercriticality, we observe helicity oscillations in the saturated regime. For Lundquist numbers in the order of one we also obtain chiral symmetry breaking of the saturated magnetic field.

Magnetic material in mean-field dynamos driven by small scale helical flows

We perform kinematic simulations of dynamo action driven by a helical small scale flow of a conducting fluid in order to deduce mean-field properties of the combined induction action of small scale eddies. We examine two different flow patterns in the style of the G O Roberts flow but with a mean vertical component and with internal fixtures that are modelled by regions with vanishing flow. These fixtures represent either rods that lie in the center of individual eddies, or internal dividing walls that provide a separation of the eddies from each other. The fixtures can be made of magnetic material with a relative permeability larger than one which can alter the dynamo behavior. The investigations are motivated by the widely unknown induction effects of the forced helical flow that is used in the core of liquid sodium cooled fast reactors, and from the key role of soft iron impellers in the von-Kármán-sodium dynamo. For both examined flow configurations the consideration of magnetic material within the fluid flow causes a reduction of the critical magnetic Reynolds number of up to 25%. The development of the growth-rate in the limit of the largest achievable permeabilities suggests no further significant reduction for even larger values of the permeability. In order to study the dynamo behavior of systems that consist of tens of thousands of helical cells we resort to the mean-field dynamo theory (Krause and Rädler 1980 Mean-field Magnetohydrodynamics and Dynamo Theory (Oxford: Pergamon)) in which the action of the small scale flow is parameterized in terms of an α- and β-effect. We compute the relevant elements of the α- and the β-tensor using the so called testfield method. We find a reasonable agreement between the fully resolved models and the corresponding mean-field models for wall or rod materials in the considered range . Our results may be used for the development of global large scale models with recirculation flow and realistic boundary conditions.

EFFECTS OF ANISOTROPIES IN TURBULENT MAGNETIC DIFFUSION IN MEAN-FIELD SOLAR DYNAMO MODELS

We study how anisotropy of turbulent convection affects diffusion of large-scale magnetic fields and the dynamo process on the Sun. The effect of anisotropy is calculated in a mean-field magneto-hydrodynamics framework using the minimal $\tau$-approximation. We examine two types of mean-field dynamo models: the well-known benchmark flux-transport model, and a distributed-dynamo model with the subsurface rotational shear layer. For both models we investigate effects of the double-cell meridional circulation, recently suggested by helioseismology. We introduce a parameter of anisotropy as a ratio of the radial and horizontal intensity of turbulent mixing, to characterize the anisotropy effects. It is found that the anisotropy of turbulent convection affects the distribution of magnetic fields inside the convection zone. The concentration of the magnetic flux near the bottom and top boundaries of the convection zone is greater when the anisotropy is stronger. It is shown that the critical dynamo number and the dynamo period approach to constant values for the large anisotropy parameter. The anisotropy reduces the overlap of the toroidal fields of subsequent cycles in the time-latitude ``butterfly' diagram. If we assume that sunspots are formed in the vicinity of the subsurface shear layer, then the distributed dynamo model with anisotropic diffusivity satisfies the observational constraints from heloseismology results and is consistent with the value of effective turbulent diffusion, estimated from the dynamics of surface magnetic fields.

Towards a hybrid dynamo model for the Milky Way

(Abridged) Based on the rapidly increasing all-sky data of Faraday rotation measures and polarised synchrotron radiation, the Milky Way's magnetic field is now modelled with an unprecedented level of detail and complexity. We aim to complement this heuristic approach with a physically motivated, quantitative Galactic dynamo model -- a model that moreover allows for the evolution of the system as a whole, instead of just solving the induction equation for a fixed static disc. Building on the framework of mean-field magnetohydrodynamics and extending it to the realm of a hybrid evolution, we perform three-dimensional global simulations of the Galactic disc. Closure coefficients embodying the mean-field dynamo are calibrated against resolved box simulations of supernova-driven interstellar turbulence. The emerging dynamo solutions comprise a mixture of the dominant axisymmetric S0 mode, with even parity, and a subdominant A0 mode, with odd parity. Notably, such a superposition of modes creates a strong localised vertical field on one side of the Galactic disc. We moreover find significant radial pitch angles, which decay with radius -- explained by flaring of the disc. In accordance with previous work, magnetic instabilities appear to be restricted to the less-stirred outer Galactic disc. Their main effect is to create strong fields at large radii such that the radial scale length of the magnetic field increases from 4 kpc (for the case of a mean-field dynamo alone) to about 10 kpc in the hybrid models. There remain aspects (e.g., spiral arms, X-shaped halo fields, fluctuating fields) that are not captured by the current model and that will require further development towards a fully dynamical evolution. Nevertheless, the work presented demonstrates that a hybrid modelling of the Galactic dynamo is feasible and can serve as a foundation for future efforts.

Rotational effects on the negative magnetic pressure instability

The surface layers of the Sun are strongly stratified. In the presence of turbulence with a weak mean magnetic field, a large-scale instability resulting in the formation of non-uniform magnetic structures, can be excited over the scale of many turbulent eddies or convection cells. This instability is caused by a negative contribution of turbulence to the effective (mean-field) magnetic pressure and has previously been discussed in connection with the formation of active regions and perhaps sunspots. We want to understand the effects of rotation on this instability in both two and three dimensions. We use mean-field magnetohydrodynamics in a parameter regime in which the properties of the negative effective magnetic pressure instability have previously been found to be in agreement with those of direct numerical simulations. We find that the instability is suppressed already for relatively slow rotation with Coriolis numbers (i.e. inverse Rossby numbers) around 0.2. The suppression is strongest at the equator. In the nonlinear regime, we find traveling wave solutions with propagation in the prograde direction at the equator with additional poleward migration away from the equator. The prograde rotation of the magnetic pattern near the equator is argued to be a possible explanation for the faster rotation speed of magnetic tracers found on the Sun. In the bulk of the domain, kinetic and current helicities are negative in the northern hemisphere and positive in the southern.

Helicity–vorticity turbulent pumping of magnetic fields in the solar convection zone

We study the effect of turbulent drift of a large-scale magnetic field that results from the interaction of helical convective motions and differential rotation in the solar convection zone. The principal direction of the drift corresponds to the direction of the large-scale vorticity vector. Thus, the effect produces a latitudinal transport of the large-scale magnetic field in the convective zone wherever the angular velocity has a strong radial gradient. The direction of the drift depends on the sign of helicity and it is defined by the Parker–Yoshimura rule. The analytic calculations are done within the framework of mean-field magnetohydrodynamics using the minimal τ-approximation. We estimate the magnitude of the drift velocity and find that it can be a few m/s near the base of the solar convection zone. The implications of this effect for the solar dynamo are illustrated on the basis of an axisymmetric mean-field dynamo model with a subsurface shear layer. The model shows that near the bottom of the convection zone the helicity–vorticity pumping results mostly from the kinetic helicity contributions. We find that the magnetic helicity contributions to the pumping effect are dominant at the subsurface shear layer. There the magnitude of the drift velocity is found to be a few cm/s. We find that the helicity–vorticity pumping effect can have an influence on the features of the sunspot time–latitude diagram, producing a fast drift of the sunspot activity maximum at the rise phase of the cycle and a slow drift at the decay phase of the cycle.

Yoshizawa's cross-helicity effect and its quenching

A central quantity in mean-field magnetohydrodynamics is the mean electromotive force , which in general depends on the mean magnetic field. It may however also have a part independent of the mean magnetic field. Here we study an example of a rotating conducting body of turbulent fluid with non-zero cross-helicity, in which a contribution to proportional to the angular velocity occurs (Yoshizawa, A., Self-consistent turbulent dynamo modeling of reversed field pinches and planetary magnetic fields. Phys. Fluids B 1990, 2, 1589–1600). If the forcing is helical, it also leads to an α effect, and large-scale magnetic fields can be generated. For not too rapid rotation, the field configuration is such that Yoshizawa's contribution to is considerably reduced compared to the case without α effect. In that case, large-scale flows are also found to be generated.

Mean electromotive force proportional to mean flow in MILD turbulence

AbstractIn mean‐field magnetohydrodynamics the mean electromotive force due to velocity and magnetic‐field fluctuations plays a crucial role. In general it consists of two parts, one independent of and another one proportional to the mean magnetic field. The first part may be nonzero only in the presence of mhd turbulence, maintained, e.g., by small‐scale dynamo action. It corresponds to a battery, which lets a mean magnetic field grow from zero to a finite value. The second part, which covers, e.g., the α effect, is important for large‐scale dynamos. Only a few examples of the aforementioned first part of the mean electromotive force have been discussed so far. It is shown that a mean electromotive force proportional to the mean fluid velocity, but independent of the mean magnetic field, may occur in an originally homogeneous isotropic mhd turbulence if there are nonzero correlations of velocity and electric current fluctuations or, what is equivalent, of vorticity and magnetic field fluctuations. This goes beyond the Yoshizawa effect, which consists in the occurrence of mean electromotive forces proportional to the mean vorticity or to the angular velocity defining the Coriolis force in a rotating frame and depends on the cross‐helicity defined by the velocity and magnetic field fluctuations. Contributions to the mean electromotive force due to inhomogeneity of the turbulence are also considered. Possible consequences of the above findings for the generation of magnetic fields in cosmic bodies are discussed (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Self-consistent mean-field magnetohydrodynamics

We consider the linear stability of two-dimensional nonlinear magnetohydrodynamic basic states to long-wavelength three-dimensional perturbations. Following Hughes & Proctor (Hughes & Proctor 2009Proc. R. Soc. A465, 1599–1616 (doi:10.1098/rspa.2008.0493)), the two-dimensional basic states are obtained from a specific forcing function in the presence of an initially uniform mean field of strength. By extending to the nonlinear regime the kinematic analysis of Roberts (Roberts 1970Phil. Trans. R. Soc. Lond. A266, 535–558 (doi:10.1098/rsta.1970.0011)), we show that it is possible to predict the growth rate of these perturbations by applying mean-field theory toboththe momentum and the induction equations. If, these equations decouple and large-scale magnetic and velocity perturbations may grow via the kinematicα-effect and the anisotropic kinetic alpha instability, respectively. However, if, the momentum and induction equations are coupled by the Lorentz force; in this case, we show that four transport tensors are now necessary to determine the growth rate of the perturbations. We illustrate these situations by numerical examples; in particular, we show that a mean-field description of the nonlinear regime based solely on a quenchedαcoefficient is incorrect.

MIGRATION OF EXTRASOLAR PLANETS: EFFECTS FROM X -WIND ACCRETION DISKS

Magnetic fields are dragged in from the interstellar medium during the gravitational collapse that forms star/disk systems. Consideration of mean field magnetohydrodynamics (MHD) in these disks shows that magnetic effects produce subkeplerian rotation curves and truncate the inner disk. This letter explores the ramifications of these predicted disk properties for the migration of extrasolar planets. Subkeplerian flow in gaseous disks drives a new migration mechanism for embedded planets and modifies the gap opening processes for larger planets. This subkeplerian migration mechanism dominates over Type I migration for sufficiently small planets (m_P < 1 M_\earth) and/or close orbits (r < 1 AU). Although the inclusion of subkeplerian torques shortens the total migration time by only a moderate amount, the mass accreted by migrating planetary cores is significantly reduced. Truncation of the inner disk edge (for typical system parameters) naturally explains final planetary orbits with periods P=4 days. Planets with shorter periods P=2 days can be explained by migration during FU-Ori outbursts, when the mass accretion rate is high and the disk edge moves inward. Finally, the midplane density is greatly increased at the inner truncation point of the disk (the X-point); this enhancement, in conjunction with continuing flow of gas and solids through the region, supports the in situ formation of giant planets.

Closure tests for mean field magnetohydrodynamics using a self-consistent reduced model

The mean electromotive force and alpha effect are computed for a forced turbulent flow using a simple nonlinear dynamical model. The results are used to check the applicability of two basic analytic ansatze of mean-field magnetohydrodynamics - the second order correlation approximation (SOCA) and the tau approximation. In the numerical simulations the effective Reynolds number Re is 2-20, while the magnetic Prandtl number varies from 0.1 to $10^{7}$. We present evidence that the $\tau$ approximation may be appropriate in dynamical regimes where there is a small-scale dynamo. Catastrophic quenching of the $\alpha$ effect is found for high $P_{m}$. Our results indicate that for high $P_{m}$ SOCA gives a very large value of the $\alpha$ coefficient compared with the ``exact'' solution. The discrepancy depends on the properties of the random force that drives the flow, with a larger difference occuring for $\delta$-correlated force compared with that for a steady random force.

Mean Field Magnetohydrodynamics of Accretion Disks

We consider the accretion process in a disk with magnetic fields that are dragged in from the interstellar medium by gravitational collapse. Two diffusive processes are at work in the system: (1) viscous torques exerted by turbulent and magnetic stresses, and (2) resistive redistribution of mass with respect to the magnetic flux arising from the imperfect conduction of current. In steady state, self-consistency between the two rates of drift requires that a relationship exists between the coefficients of turbulent viscosity and turbulent resistivity. Ignoring any interactions with a stellar magnetosphere, we solve the steady-state equations for a magnetized disk under the gravitational attraction of a mass point and threaded by an amount of magnetic flux consistent with calculations of magnetized gravitational collapse in star formation. Our model mean field equations have an exact analytical solution that corresponds to magnetically diluted Keplerian rotation about the central mass point. The solution yields the strength of the magnetic field and the surface density as functions of radial position in the disk and their connection with the departure from pure Keplerian rotation in representative cases. We compare the predictions of the theory with the available observations concerning T Tauri stars, FU Orionis stars, and low- and high-mass protostars. Finally, we speculate on the physical causes for high and low states of the accretion disks that surround young stellar objects. One of the more important results of this study is the physical derivation of analytic expressions for the turbulent viscosity and turbulent resistivity.

The mean electro-motive force and current helicity under the influence of rotation, magnetic field and shear

The mean electromotive force (MEMF) in a rotating stratified magnetohydrodynamical turbulence is studied. Our study is based on the mean-field magnetohydrodynamics framework and τ approximation. We compute the effects of the large-scale magnetic fields (LSMF), global rotation and large-scale shear flow on the different parts of the MEMF (such as α-effect, turbulent diffusion, turbulent transport, etc.) in an explicit form. The influence of the helical magnetic fluctuations which stem from the small-scale dynamo is taken into account, as well. In the article, we derive the equation governing the current helicity evolution. It is shown that the joint effect of the differential rotation and magnetic fluctuations in the stratified media can be responsible for the generation, maintenance and redistribution of the current helicity. The implication of the obtained results to astrophysical dynamos is considered.

Induction, helicity, and alpha effect in a toroidal screw flow of liquid gallium

We investigate experimentally induction mechanisms in a screw flow of gallium in a toroidal channel. The flow is nonstationary and operated in a spin-down regime: the channel (and fluid) are initially set into solid body rotation; as the channel is stopped the fluid is set into strong helical motion by diverters located inside the channel. In this study, we put a particular emphasis on the induction generated by these helical motions, which are expected to develop over the entire range of turbulent scales. We apply an external magnetic field either perpendicular to the channel axis parallel to it. At large scales the nonlinear induction mechanisms are associated with the Parker stretch and twist effect and with the expulsion due to overall rotation. Induction mechanisms can also originate in the small scale helicity as in the alpha induction effect of mean-field magnetohydrodynamics. Our measurements yield an upper bound for the alpha coefficient, significantly lower than estimates based on dimensional analysis. We discuss the consequences of our observations for the engineering of homogeneous dynamos in the laboratory.

The long-time evolution of mean field magnetohydrodynamics

The equations of mean field magnetohydrodynamics with constantmean velocity are proved to posses solutions bounded in the$H^{1}$-norm for all time, and a compact attractor whose dimensionis estimated. It is shown that depending on the functional form ofthe so-called alpha term the attractor may reduce to zero or be alarger set. If, as usual in physical situations, there exists aset of solutions with a minimum size $N$, the dimension of thisset decreases rapidly with increasing $N$. Finally, the dependenceof the dimension on the magnetic diffusivity is analyzed,suggesting that the evolution of a magnetic field under the meanfield equation is much more restricted than the one deduced fromthe full magnetohydrodynamic system.

The nonlinear properties of a large-scale dynamo driven by helical forcing

By considering an idealized model of helically forced flow in an extended domain that allows scale separation, we have investigated the interaction between dynamo action on different spatial scales. The evolution of the magnetic field is studied numerically, from an initial state of weak magnetization, through the kinematic and into the dynamic regime. We show how the choice of initial conditions is a crucial factor in determining the structure of the magnetic field at subsequent times. For a simulation with initial conditions chosen to favour the growth of the small-scale field, the evolution of the large-scale magnetic field can be described in terms of the α-effect of mean field magnetohydrodynamics. We have investigated this feature further by a series of related numerical simulations in smaller domains. Of particular significance is that the results are consistent with the existence of a nonlinearly driven α-effect that becomes saturated at very small amplitudes of the mean magnetic field.