Karl‐Heinz Rädler (1935–2020)

Two spectral regimes of magnetic field amplification in magnetohydrodynamic (MHD) flows can be distinguished by the scale on which fields are amplified relative to the primary forcing scale of the turbulence. For field amplification at or below the forcing scale, the amplification can be called a “small-scale dynamo.” For amplification at and above the forcing scale the process can be called a “large-scale dynamo.” Non - local (in wave number) effects play a key role in both the growth of the small-scale field in non-helical turbulence and the growth of large and smallscale fields in helical turbulence. Mean field dynamo (MFD) theory represents a simple semi-analytic way to get a handle on large-scale field amplification in MHD turbulence. Helicity has long been known to be important for large scale, flux generating, externally forced MFDs. The extent to which such MFDs operate “slow” or “fast” (dependent or independent on magnetic Reynolds number) has been controversial, but there has been recent progress. Simulations of α2 dynamos in a periodic box dynamo and their quenching can now be largely understood within a simplified dynamical non-linear paradigm in which the MFD growth equation is supplemented by the total magnetic helicity evolution equation. For α2 dynamos, the large-scale field growth is directly related to the large-scale magnetic helicity growth. Magnetic helicity conservation then implies that growth of the large-scale magnetic helicity induces growth of small-scale magnetic (and current) helicity of the opposite sign, which eventually suppresses the α effect driving the MFD growth. Although the α2 MFD then becomes slow in the long time limit, substantial large-scale field growth proceeds in a kinematic, “fast” phase before non-linear asymptotic quenching of the “slow” phase applies. Ultimately, the MFD emerges as a process that transfers magnetic helicity between small and large scales. How these concepts apply to more general dynamos with shear, and open boundary dynamos is a topic of ongoing research. Some unresolved issues are identified. Overall, the following summarizes the most recent progress in mean-field dynamo theory: For a closed turbulent flow, the non-linear mean field dynamo, is first fast and kinematic, then slow and dynamic, and magnetic helicity transfer makes it so.KeywordsMagnetic EnergyMagnetic HelicityMagnetic Reynolds NumberDynamo TheoryForce ScaleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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 < 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 < 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.

A mechanism of the polarity reversal of the solar magnetic field is explored on the basis of the mean field or turbulent dynamo theory. In the low-latitude region of the convective zone, the toroidal magnetic field, which is the origin of sunspots, is generated by the rotational motion of fluids, with the turbulent cross helicity as the intermediary. This field generates the poloidal field of dipole type through the alpha or turbulent helicity effect. The latter, in turn, contributes to the annihilation of the turbulent cross helicity, resulting in the decay of the toroidal magnetic field. This process indicates less room for the occurrence of the fully developed poloidal field in the low-latitude region and paves the way for the polarity reversal through the change of the sign of the turbulent cross helicity. A simple model mimicking the periodic polarity reversal is presented, and the relationship of the reversal period to the ratio of the poloidal to toroidal fields is given. The meridional-flow velocity at the solar surface is estimated, giving a result consistent with observations.

Presented here is a novel formulation of the mean-field dynamo as a modulational instability of magnetohydrodynamic (MHD) turbulence. This formulation, termed mean-field wave kinetics (MFWK), is based on the Weyl symbol calculus and allows describing the interaction between the mean fields (magnetic field and fluid velocity) and turbulence without requiring scale separation that is commonly assumed in the literature. The turbulence is described by the Wigner–Moyal equation for the spectrum of the two-point correlation matrix (Wigner matrix) of magnetic-field and velocity fluctuations and depicts the turbulence as an effective plasma of quantum-like particles that interact via the mean fields. Eddy–eddy interactions, which serve as ‘collisions’ in this effective plasma, are modelled within the standard minimal tau approximation to aid comparison with existing theories. Using MFWK, the non-local electromotive force is calculated for generic turbulence from first principles, modulo the limitations of MFWK. This result is then used to study, both analytically and numerically, the modulational modes of MHD turbulence, which appear as linear instabilities of the said effective quantum-like plasma of fluctuations. The standard $\alpha ^2$ -dynamo and other known results are reproduced as special cases. A new dynamo effect is predicted that is driven by correlations between the turbulent flow velocity and the turbulent current.

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.

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 & 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.

Numerical aspects of dynamos in periodic domains are discussed. Modifications of the solutions by numerically motivated alterations of the equations are being reviewed using the examples of magnetic hyperdiffusion and artificial diffusion when advancing the magnetic field in its Euler potential representation. The importance of using integral kernel formulations in mean-field dynamo theory is emphasized in cases where the dynamo growth rate becomes comparable with the inverse turnover time. Finally, the significance of microscopic magnetic Prandtl number in controlling the conversion from kinetic to magnetic energy is highlighted.

We discuss the properties of the large-scale galactic magnetic fields which can help to discriminate between theories of their origin. We argue that the mean-field dynamo theory in its simplest form is unique in that it can explain most of the known features from a single conceptual viewpoint.

We investigate the cosmological evolution of large- and small-scale magnetic fields in galaxies at high redshifts. Results from simulations of hierarchical structure formation cosmology provide a tool to develop an evolutionary model of regular magnetic fields coupled to galaxy formation and evolution. Turbulence in protogalactic halos generated by thermal virialization can drive an efficient turbulent dynamo. The mean-field dynamo theory is used to derive the timescales of amplification and ordering of regular magnetic fields in disk and dwarf galaxies. For future observations with the SKA, we predict an anticorrelation at fixed redshift between galaxy size and the ratio between ordering scale and galaxy size. Undisturbed dwarf galaxies should host fully coherent fields at z < 1, spiral galaxies at z < 0.5.

Primordial magnetogenesis

Earlier we have computed a Lorentz violation (LV) bound for torsion terms via galactic dynamos and found bounds similar to the one obtained by Kostelecky et al. (Phys Rev Lett 100:111102, 2008) which is of the order of 10^{-31} GeV. Their result was found making use of the axial torsion vector in terms of Dirac spinors and minimal torsion coupling in flat space-time of fermions. In this paper, a torsion dynamo equation obtained using the variation of the torsion trace and galaxy M51 data of 500 pc are used to place an upper bound of 10^{-26} GeV in LV, which agrees with the one by Kostelecky and his group using an astrophysical framework background. Their lowest bound was obtained in earth laboratory using dual masers. One of the purposes of this paper is to apply the Faraday self-induction magnetic equation, recently extended to torsioned space-time, by the author to show that it lends support to physics in Riemann–Cartan space-time, in several distinct physical backgrounds. Backreaction magnetic effects are used to obtain the LV bounds. Previously Bamba et al. (JCAP 10:058, 2012) have used the torsion trace in their teleparallel investigation of the IGMF, with the argument that the torsion trace leads to less weaker effects than the other irreducible components of the torsion tensor. LV is computed in terms of a chiral-torsion-like current in the new dynamo equation analogous to the Dvornikov and Semikoz dynamo equation with chiral magnetic currents. Making use of the chiral-torsion dynamo equation we estimate the LV bounds in the early universe to be of the order of 10^{-24} GeV, which was the order of the charged-lepton sector. Our main result is that it is possible to obtain more stringent bounds than the ones found in the fermion sector of astrophysics in the new revised 2017 data table for CPT and Lorentz violation by Kostelecky and Mewes. They found in several astrophysical backgrounds, orders of magnitude such as 10^{-24} and 10^{-23} GeV which are not so stringent as the one found here, of 10^{-26} GeV, in the torsion fermionic sector with the help of galaxy M51 data. It is also shown that non-gauge invariance of chiral dynamos and axial anomalies are obtained from the noninvariance of the electric fields under torsion spatial translations.

We show that the magnetic analogue of the Rdler effect of mean-field dynamo theory leads to a non-linear backreaction that quenches a large-scale galactic dynamo, and can result in saturation of the large-scale magnetic field at near-equipartition with turbulent kinetic energy density. In a rotating fluid containing small-scale magnetic fluctuations, anisotropic terms in the mean electromotive force are induced via the Coriolis effect and these terms lead to a reduction of the growth rate in a predominantly -type galactic dynamo. By including the generation of small-scale magnetic fluctuations by turbulent tangling of the large-scale magnetic field, one obtains a negative feedback effect that quenches the dynamo and leads to the saturation of the large-scale field. This saturation mechanism is found to be competitive with the dynamical -quenching mechanism for realistic galactic parameter values. Furthermore, in the context of the dynamical -quenching model, a separate non-linear term is obtained which has the same form as the helicity flux term of Vishniac & Cho, but which depends on the strength of small-scale magnetic fluctuations. We briefly discuss the observational implications of the magnetic Rdler effect for galaxies.

We introduce a class of stochastic population models based on "patch dynamics." The size of the patch may be varied, and this allows one to quantify the departures of these stochastic models from various mean-field theories, which are generally valid as the patch size becomes very large. These models may be used to formulate a broad range of biological processes in both spatial and nonspatial contexts. Here, we concentrate on two-species competition. We present both a mathematical analysis of the patch model, in which we derive the precise form of the competition mean-field equations (and their first-order corrections in the nonspatial case), and simulation results. These mean-field equations differ, in some important ways, from those which are normally written down on phenomenological grounds. Our general conclusion is that mean-field theory is more robust for spatial models than for a single isolated patch. This is due to the dilution of stochastic effects in a spatial setting resulting from repeated rescue events mediated by interpatch diffusion. However, discrete effects due to modest patch sizes lead to striking deviations from mean-field theory even in a spatial setting.

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.

We present a mean-field theory for the dynamics of driven flow with exclusion in graphenelike structures, and numerically check its predictions. We treat first a specific combination of bond transmissivity rates, where mean field predicts, and numerics to a large extent confirms, that the sublattice structure characteristic of honeycomb networks becomes irrelevant. Dynamics, in the various regions of the phase diagram set by open boundary injection and ejection rates, is then in general identical to that of one-dimensional systems, although some discrepancies remain between mean-field theory and numerical results, in similar ways for both geometries. However, at the critical point for which the characteristic exponent is z=3/2 in one dimension, the mean-field value z=2 is approached for very large systems with constant (finite) aspect ratio. We also treat a second combination of bond (and boundary) rates where, more typically, sublattice distinction persists. For the two rate combinations, in continuum or late-time limits, respectively, the coupled sets of mean-field dynamical equations become tractable with various techniques and give a two-band spectrum, gapless in the critical phase. While for the second rate combination quantitative discrepancies between mean-field theory and simulations increase for most properties and boundary rates investigated, theory still is qualitatively correct in general, and gives a fairly good quantitative account of features such as the late-time evolution of density profile differences from their steady-state values.