A mean-field theory of the electrodynamics of a turbulent fluid is formulated under the assumption that the molecular electric conductivity is correlated with the turbulent velocity fluctuation in the (radial) direction, g. It is shown that for such homogeneous fluids a strong turbulence-induced field advection anti-parallel to g arises almost independently of rotation. For rotating fluids, an extra alpha effect appears with the known symmetries and with the expected maximum at the poles. Fast rotation, however, with Coriolis number exceeding unity suppresses this term. Numerical simulations of forced turbulence using the nirvana code demonstrate that the radial advection velocity, gamma, always dominates the alpha term. We show finally with simplified models that alpha(2) dynamos are strongly influenced by the radial pumping: for gamma < alpha the solutions become oscillatory, while for gamma > alpha they become highly exotic if they exist at all. In conclusion, dynamo models for slow and fast solid-body rotation on the basis of finite conductivity-velocity correlations are unlikely to work, at least for alpha(2)omega dynamos without strong shear.

Laminar electrically conducting Couette flows with the hydrodynamically stable quasi-Keplerian rotation profile and non-uniform conductivity are probed for dynamo instability. In spherical geometry the equations for the poloidal and the toroidal field components completely decouple, resulting in free decay, regardless of the spatial distribution of the electric conductivity. In cylindrical geometry the poloidal and toroidal components do not decouple, but here also we do not find dynamo excitations for the cases that the electric conductivity only depends on the radius or -- much more complex -- that it only depends on the azimuthal or the axial coordinate. The transformation of the plane-flow dynamo model of Busse \& Wicht (1992) to cylindrical or spherical geometry therefore fails. It is also shown that even the inclusion of axial flows of both directions does {\em not} support the dynamo mechanism. The Elsasser toroidal-velocity antidynamo theorem, according to which dynamos without any radial velocity component cannot work, is thus not softened by non-uniform conductivity distributions.

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.

We reveal and investigate a type of linear axisymmetric helical magnetorotational instability which is capable of destabilizing viscous and resistive rotational flows with radially increasing angular velocity, or positive shear. This instability is double-diffusive by nature and is different from the more familiar helical magnetorotational instability, operating at positive shear above the Liu limit, in that it works instead for a wide range of the positive shear when (i) a combination of axial and azimuthal magnetic fields is applied and (ii) the magnetic Prandtl number is not too close to unity. We study this instability first with radially local Wentzel-Kramers-Brillouin (WKB) analysis, deriving the scaling properties of its growth rate with respect to Hartmann, Reynolds, and magnetic Prandtl numbers. Then we confirm its existence using a global stability analysis of the magnetized flow confined between two rotating coaxial cylinders with purely conducting or insulating boundaries and compare the results with those of the local analysis. From an experimental point of view, we also demonstrate the presence of this instability in a magnetized viscous and resistive Taylor-Couette flow with positive shear for such values of the flow parameters, which can be realized in upcoming experiments at the DRESDYN facility. Finally, this instability might have implications for the dynamics of the equatorial parts of the solar tachocline and dynamo action there, since the above two necessary conditions for the instability to take place are satisfied in this region. Our global stability calculations for the tachocline-like configuration, representing a thin rotating cylindrical layer with the appropriate boundary conditions-conducting inner and insulating outer cylinders-and the values of the flow parameters, indicate that it can indeed arise in this case with a characteristic growth time comparable to the solar cycle period.

Magnetic fields of planets, stars, and galaxies are generated by self-excitation in moving electrically conducting fluids. Once produced, magnetic fields can play an active role in cosmic structure formation by destabilizing rotational flows that would be otherwise hydrodynamically stable. For a long time, both hydromagneticdynamo action and magnetically triggered flow instabilities had been the subject of purely theoretical research. Meanwhile, however, the dynamo effect has been observed in large-scale liquid sodium experiments in Riga, Karlsruhe, and Cadarache. In this chapter, we summarize the results of some smaller liquid metal experiments devoted to various magnetic instabilities, such as the helical and the azimuthal magnetorotational instability, the Tayler instability, and the different instabilities that appear in a magnetized spherical Couette flow. We conclude with an outlook on a large scale Tayler-Couette experiment using liquid sodium, and on the prospects to observe magnetically triggered instabilities of flows with positive shear.

Decades ago S. Lundquist, S. Chandrasekhar, P. H. Roberts and R. J. Tayler first posed questions about the stability of Taylor-Couette flows of conducting material under the influence of large-scale magnetic fields. These and many new questions can now be answered numerically where the nonlinear simulations even provide the instability-induced values of several transport coefficients. The cylindrical containers are axially unbounded and penetrated by magnetic background fields with axial and/or azimuthal components. The influence of the magnetic Prandtl number Pm on the onset of the instabilities is shown to be substantial. The potetial flow subject to axial fields becomes unstable against axisymmetric perturbations for a certain supercritical value of the averaged Reynolds number (Rm) over bar = root Re.Rm (with Re the Reynolds number of rotation, Rm its magnetic Reynolds number). Rotation profiles as flat as the quasi-Keplerian rotation law scale similarly but only for Pm >> 1 while for Pm << 1 the instability instead sets in for supercritical Rm at an optimal value of the magnetic field. Among the considered instabilities of azimuthal fields, those of the Chandrasekhar-type, where the background field and the background flow have identical radial profiles, are particularly interesting. They are unstable against nonaxisymmetric perturbations if at least one of the diffusivities is non-zero. For Pm << 1 the onset of the instability scales with Re while it scales with (Rm) over bar for Pm >> 1. Even superrotation can be destabilized by azimuthal and current-free magnetic fields; this recently discovered nonaxisymmetric instability is of a double-diffusive character, thus excluding Pm = 1. It scales with Re for Pm -> 0 and with Rm for Pm -> infinity. The presented results allow the construction of several new experiments with liquid metals as the conducting fluid. Some of them are described here and their results will be discussed together with relevant diversifications of the magnetic instability theory including nonlinear numerical studies of the kinetic and magnetic energies, the azimuthal spectra and the influence of the Hall effect. (C) 2018 The Authors. Published by Elsevier B.V.

The magnetohydrodynamic stability of axially unbounded cylindrical flows is considered which contain a toroidal magnetic background field with the same radial profile as the linear azimuthal velocity. Chandrasekhar (1956) has shown for ideal fluids the stability of this configuration if the Alfven velocity of the field equals the velocity of the background flow. It is demonstrated for magnetized Taylor-Couette flows at the Rayleigh line, however, that for finite diffusivity such flows become unstable against nonaxisymmetric perturbations where the critical magnetic Reynolds number of the rotation rate does not depend on the magnetic Prandtl number Pm if Pm much << 1. In order to study this new diffusive azimuthal magnetorotational instability, flows and fields with the same radial profile but with different amplitudes are considered. For Pm << 1 the instability domain with the weakest fields and the slowest rotation rates lies below the Chandrasekhar line of equal amplitudes for Alfven velocity and rotation velocity. We find that then the lines of marginal instability scale with the Reynolds number and the Hartmann number. The minimum values of the field strength and the rotation rate which are needed for the instability (slightly) grow for more and more flat rotation. Finally, the corresponding electric current of the background field becomes so strong that the Tayler instability (which even exists without rotation) also appears in the bifurcation map at small Hartmann numbers displacing after all the azimuthal magnetorotational instability.

With application to inner stellar radiative zones, a linear theory is used to analyze the instability of a dipole-parity toroidal background field, in the presence of density stratification, differential rotation, and realistically small Prandtl numbers. The physical parameters are the normalized latitudinal shear $a$ and the normalized field amplitude $b$. Only the solutions for the wavelengths with the maximal growth rates are considered. If these scales are combined to the radial values of velocity, one finds that the (very small) radial velocity only depends slightly on $a$ and $b$, so that it can be used as the free parameter of the eigenvalue system. The resulting instability-generated tensors of magnetic diffusivity and eddy viscosity are highly anisotropic. The eddy diffusivity in latitudinal direction exceeds the eddy diffusivity in radial direction by orders of magnitude. Its latitudinal profile shows a strong concentration toward the poles which is also true for the effective viscosity which has been calculated via the angular momentum transport of the instability pattern. The resulting effective magnetic Prandtl number reaches values of $O(10^2)$, so that the differential rotation decays much faster than the toroidal background field, which is {the} necessary condition to explain the observed slow rotation of the early red-giant and sub-giant cores by means of magnetic instabilities.

The azimuthal version of the magnetorotational instability (MRI) is a nonaxisymmetric instability of a hydrodynamically stable differentially rotating flow under the influence of a purely or predominantly azimuthal magnetic field. It may be of considerable importance for destabilizing accretion disks, and plays a central role in the concept of the MRI dynamo. We report the results of a liquid metal Taylor-Couette experiment that shows the occurrence of an azimuthal MRI in the expected range of Hartmann numbers.

Differential rotation and meridional flow are key ingredients in flux transport dynamo models of the solar activity cycle. As the subsurface flow pattern is not sufficiently constrained by observations, it is a major source of uncertainty in solar and stellar dynamo models. We discuss the current mean field theory of stellar differential rotation and meridional flows and its predicitons for the Sun and stars on the lower main sequence.

Many astrophysical phenomena (such as the slow rotation of neutron stars or the rigid rotation of the solar core) can be explained by the action of the Tayler instability of toroidal magnetic fields in the radiative zones of stars. In order to place the theory of this instability on a safe fundament, it has been realized in a laboratory experiment measuring the critical field strength, the growth rates, as well as the shape of the supercritical modes. A strong electrical current flows through a liquid metal confined in a resting columnar container with an insulating outer cylinder. As the very small magnetic Prandtl number of the gallium–indium–tin alloy does not influence the critical Hartmann number of the field amplitudes, the electric currents for marginal instability can also be computed with direct numerical simulations. The results of this theoretical concept are confirmed by the experiment. Also the predicted growth rates on the order of minutes for the nonaxisymmetric perturbations are certified by the measurements. That they do not directly depend on the size of the experiment is shown as a consequence of the weakness of the applied fields and the absence of rotation.

Magnetic diffusion is a key ingredient in mean-field dynamo models but neither observations nor theory are able to produce reliable values. Numerical simulations provide an alternative way to determine the turbulent electromotive force. Cross helicity allows us to determine the turbulent magnetic diffusion coefficient in simulations of stellar magnetoconvection.

In the current-driven, kink-type Tayler instability (TI) a sufficiently strong azimuthal magnetic field becomes unstable against non-axisymmetric perturbations. The TI has been discussed as a possible ingredient of the solar dynamo mechanism and a source of the helical structures in cosmic jets. It is also considered as a size limiting factor for liquid metal batteries. We report on a liquid metal TI experiment using a cylindrical column of the eutectic alloy GaInSn to which electrical currents of up to 8 kA are applied. We present results of external magnetic field measurements that indicate the occurrence of the TI in good agreement with numerical predictions. The interference of TI with the competing large scale convection, resulting from Joule heating, is also discussed.

It is shown that the magnetic current-driven (`kink-type') instability produces flow and field patterns with helicity and even with \alpha-effect but only if the magnetic background field possesses non-vanishing current helicity \bar{\vec{B}}\cdot curl \bar{\vec{B}} by itself. Fields with positive large-scale current helicity lead to negative small-scale kinetic helicity. The resulting \alpha-effect is positive. These results are very strict for cylindric setups without z/I>-dependence of the background fields. The sign rules also hold for the more complicated cases in spheres where the toroidal fields are the result of the action of differential rotation (induced from fossil poloidal fields) at least for the case that the global rotation is switched off after the onset of the instability.

The fractal shape and multi-component nature of the interstellar medium together with its vast range of dynamical scales provides one of the great challenges in theoretical and numerical astrophysics. Here we will review recent progress in the direct modelling of interstellar hydromagnetic turbulence, focusing on the role of energy injection by supernova explosions. The implications for dynamo theory will be discussed in the context of the mean-field approach. Results obtained with the test field-method are confronted with analytical predictions and estimates from quasilinear theory. The simulation results enforce the classical understanding of a turbulent Galactic dynamo and, more importantly, yield new quantitative insights. The derived scaling relations enable confident global mean-field modelling.

AbstractCurrent-driven instabilities in stellar radiation zones, to which we refer as Tayler instabilities, can lead to complex nonlinear evolutions. It is of fundamental interest whether magnetically driven turbulence can lead to dynamo action in these radiative zones. We investigate initial-value simulations in a 3D spherical shell including differential rotation. The Tayler instability is connected with a very weak kinetic helicity, stronger current helicity, and a positive αφφ in the northern hemisphere. The amplitudes are small compared to the effect of the tangential cylinder producing an eddy with negative kinetic helicity and negative αφφ in the northern hemisphere. The αφφ from the Tayler instability reaches about 1% of the rms velocity.

The magnetorotational instability (MRI) is thought to play a key role in the formation of stars and black holes by sustaining the turbulence in hydrodynamically stable Keplerian accretion disks. In previous experiments the MRI was observed in a liquid metal Taylor-Couette flow at moderate Reynolds numbers by applying a helical magnetic field. The observation of this helical MRI (HMRI) was interfered with a significant Ekman pumping driven by solid end caps that confined the instability only to a part of the Taylor-Couette cell. This paper describes the observation of the HMRI in an improved Taylor-Couette setup with the Ekman pumping significantly reduced by using split end caps. The HMRI, which now spreads over the whole height of the cell, appears much sharper and in better agreement with numerical predictions. By analyzing various parameter dependencies we conclude that the observed HMRI represents a self-sustained global instability rather than a noise-sustained convective one.

It is well established that magnetic fields exist in astronomical objects of all scales, in planets, stars,galaxies and clusters of galaxies. Magnetic fields play a crucial role in star formation, solar and stel-lar activity, pulsars, magnetars, accretion disks, formation and stability jets, origin of cosmic rays,and stability of galactic disks. It is generally accepted that cosmic magnetic fields are producedby dynamo processes operating on various scales. In these processes the magnetic field is main-tained against Ohmic dissipation by turbulent motions, and despite the turbulent nature it showsremarkable self-organization properties, forming sunspots and starspots, magnetic loop structures,magnetic spiral arms etc. These phenomena show striking similarity suggesting that the basic phys-ical mechanisms are essentially the same.The Sun is our Rosetta stone when it comes to magnetic-field studies in the entire Universe. Thesolar magnetism is studied in great details, from global fields of the interior by helioseismology,to the smallest resolved and even unresolved scales by new large ground-based telescopes (SST,GREGOR, BBSO NST) and from space (SOHO, RHESSI, STEREO, Hinode, and SDO (scheduledfor launch in 2009). In addition, significant progress has been made in realistic numerical MHDsimulations. This progress in observations and modeling provides a good basis for solving theproblem of the solar dynamo and formation of self-organized magnetic structures during the nextdecade. This will have tremendous impact in many fields of astrophysics.However, this new science opportunity requires focused coordinated efforts in observations,modeling and theory. This opportunity can be realized with relatively modest investments, mostly,for supporting the projects that already exist or are under development. The Sun is the only objectthat can be observed to the level of details sufficient for investigation of the basic physical processesin a magnetized astrophysical plasma. These processes are a cornerstone of modern astrophysics.A key element of this opportunity is the understanding of interlinks between small-scale turbu-lent properties of magnetized plasma and large-scale dynamo processes. It has been long assumedthat the turbulent properties, such as turbulent diffusivity and helicity, define the large-scale behaviorof the magnetized plasma with some simple back-reaction, but recent plasma experiments and theo-retical studies showed that the large-scale flows and structures may significantly alter the turbulence,and that this may cause large-scale organization in plasma