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In this paper, written as a general historical and technical introduction to the various review papers collected in the special issue ``Solar and Stellar Dynamo: A New Era'', we review the evolution and current state of dynamo theory and modelling, with emphasis on the solar dynamo. Starting with a historical survey, we then focus on a set of ``tension points'' that are still left unresolved despite the remarkable progress of the past century. In our discussion of these tension points we touch upon the physical well-posedness of mean-field electrodynamics; constraints imposed by magnetic helicity conservation; the troublesome role of differential rotation; meridional flows and flux transpost dynamos; competing inductive mechanisms and Babcock-Leighton dynamos; the ambiguous precursor properties of the solar dipole; cycle amplitude regulation and fluctuation through nonlinear backreaction and stochastic forcing, including Grand Minima; and the promises and puzzles offered by global magnethydrodynamical numerical simulations of convection and dynamo action. We close by considering the potential bridges to be constructed between solar dynamo theory and modelling, and observations of magnetic activity in late-type stars., Comment: 37 pages, 5 figures, in collection "Solar and Stellar Dynamos: A New Era"(ISSI, June 2022) ; submitted to Space Science Reviews

Magnetic fields in galaxies exist on various spatial scales. Large-scale magnetic fields are thought to be generated by the $\alpha-\Omega$ dynamo. Small-scale galactic magnetic fields (1 kpc and below) can be generated by tangling the large-scale field or by the small-scale turbulent dynamo. The analysis of field structures with the help of polarized radio continuum emission is hampered by the effect of Faraday dispersion (due to fluctuations in magnetic field and/or thermal electron density) that shifts signals from large to small scales. At long observation wavelengths large-scale magnetic fields may become invisible, as in the case of spectro-polarimetric data cube of the spiral galaxy NGC~6946 observed with the Westerbork Radio Synthesis Telescope in the wavelength range 17-23 cm. The application of RM Synthesis alone does not overcome this problem. We propose to decompose the Faraday data cube into data cubes at different spatial scales by a wavelet transform. Signatures of the `magnetic arms' observed in NGC~6946 at shorter wavelengths become visible. Our method allows us to search for large-scale field patterns in data cubes at long wavelengths, as provided by new-generation radio telescopes., Comment: 7 pages, 7 figures, 1 table

Mean-field dynamo equations are addressed with the aid of the path-integral method. The evolution of magnetic field is treated as a three-dimensional Wiener random process, and the mean magnetic-field equations are obtained with the Wiener integral over all the trajectories of fluid particle. The form of the equations is just the same as the conventional mean-field equations, but the present equations are derived with the velocity-field realization affected by the magnetic-field force. In this sense, the present ones are nonlinear dynamo equations., Comment: 6 pages, 0 figures; submitted to Journal of Plasma Physics

Aims: Although the time of the Maunder minimum (1645--1715) is widely known as a period of extremely low solar activity, claims are still debated that solar activity during that period might still have been moderate, even higher than the current solar cycle #24. We have revisited all the existing pieces of evidence and datasets, both direct and indirect, to assess the level of solar activity during the Maunder minimum. Methods: We discuss the East Asian naked-eye sunspot observations, the telescopic solar observations, the fraction of sunspot active days, the latitudinal extent of sunspot positions, auroral sightings at high latitudes, cosmogenic radionuclide data as well as solar eclipse observations for that period. We also consider peculiar features of the Sun (very strong hemispheric asymmetry of sunspot location, unusual differential rotation and the lack of the K-corona) that imply a special mode of solar activity during the Maunder minimum. Results: The level of solar activity during the Maunder minimum is reassessed on the basis of all available data sets. Conclusions: We conclude that solar activity was indeed at an exceptionally low level during the Maunder minimum. Although the exact level is still unclear, it was definitely below that during the Dalton minimum around 1800 and significantly below that of the current solar cycle #24. Claims of a moderate-to-high level of solar activity during the Maunder minimum are rejected at a high confidence level., Comment: Accepted to Astron. Astrophys

Magnetic fields are an important ingredient of the interstellar medium (ISM). Besides their importance for star formation, they govern the transport of cosmic rays, relevant to the launch and regulation of galactic outflows and winds, which in turn are pivotal in shaping the structure of halo magnetic fields. Mapping the small-scale structure of interstellar magnetic fields in many nearby galaxies is crucial to understand the interaction between gas and magnetic fields, in particular how gas flows are affected. Elucidation of the magnetic role in, e.g., triggering star formation, forming and stabilising spiral arms, driving outflows, gas heating by reconnection and magnetising the intergalactic medium has the potential to revolutionise our physical picture of the ISM and galaxy evolution in general. Radio polarisation observations in the very nearest galaxies at high frequencies (>= 3 GHz) and with high spatial resolution (<= 5") hold the key here. The galaxy survey with SKA1 that we propose will also be a major step to understand the galactic dynamo, which is important for models of galaxy evolution and for astrophysical magnetohydrodynamics in general. Field amplification by turbulent gas motions, which is crucial for efficient dynamo action, has been investigated so far only in simulations, while compelling evidence of turbulent fields from observations is still lacking., Comment: 18 pages, 3 figures; to appear as part of 'Cosmic Magnetism' in Proceedings 'Advancing Astrophysics with the SKA' (AASKA14) as article PoS(AASKA14)094

Magnetic fields play an important role in shaping the structure and evolution of the interstellar medium (ISM) of galaxies, but the details of this relationship remain unclear. With SKA1, the 3D structure of galactic magnetic fields and its connection to star formation will be revealed. A highly sensitive probe of the internal structure of the magnetoionized ISM is the partial depolarization of synchrotron radiation from inside the volume. Different configurations of magnetic field and ionized gas within the resolution element of the telescope lead to frequency-dependent changes in the observed degree of polarization. The results of spectro-polarimetric observations are tied to physical structure in the ISM through comparison with detailed modeling, supplemented with the use of new analysis techniques that are being actively developed and studied within the community such as Rotation Measure Synthesis. The SKA will enable this field to come into its own and begin the study of the detailed structure of the magnetized ISM in a sample of nearby galaxies, thanks to its extraordinary wideband capabilities coupled with the combination of excellent surface brightness sensitivity and angular resolution., Comment: 11 pages, 1 figure; to appear as part of 'Cosmic Magnetism' in Proceedings 'Advancing Astrophysics with the SKA (AASKA14)', PoS(AASKA14)106

A VLA Sky Survey of the extragalactic sky at S band (2-4 GHz) with polarization information can uniquely probe the magneto-ionic medium in a wide range of astrophysical environments over cosmic time. For a shallow all-sky survey, we expect to detect over 4 million sources in total intensity $>$ 0.45 mJy beam$^{-1}$ and over 2.2$\times$10$^5$ sources in polarized intensity. With these new observations, we expect to discover new classes of polarized radio sources in very turbulent astrophysical environments and those with extreme values of Faraday depth. Moreover, by determining reliable Faraday depths and by modeling depolarization effects, we can derive properties of the magneto-ionic medium associated with AGNs, absorption line systems and galaxies, addressing the following unresolved questions: (1) What is the covering fraction, the degree of turbulence and the origin of absorption line systems? (2) What is the thermal content in AGNs and radio galaxies? (3) How do AGNs and galaxies evolve over cosmic time? (4) What causes the increase in percentage polarization with decreasing flux densities at the low flux density end of the polarized source count? (5) What is the growth rate of large-scale magnetic fields in galaxies?, Comment: submitted in response to NRAO's call for Community White Papers for a potential VLA Sky Survey

Several recent advances in turbulent dynamo theory are reviewed. High resolution simulations of small-scale and large-scale dynamo action in periodic domains are compared with each other and contrasted with similar results at low magnetic Prandtl numbers. It is argued that all the different cases show similarities at intermediate length scales. On the other hand, in the presence of helicity of the turbulence, power develops on large scales, which is not present in non-helical small-scale turbulent dynamos. At small length scales, differences occur in connection with the dissipation cutoff scales associated with the respective value of the magnetic Prandtl number. These differences are found to be independent of whether or not there is large-scale dynamo action. However, large-scale dynamos in homogeneous systems are shown to suffer from resistive slow-down even at intermediate length scales. The results from simulations are connected to mean field theory and its applications. Recent work on helicity fluxes to alleviate large-scale dynamo quenching, shear dynamos, nonlocal effects and magnetic structures from strong density stratification are highlighted. Several insights which arise from analytic considerations of small-scale dynamos are discussed., Comment: 36 pages, 11 figures, Spa. Sci. Rev., submitted to the special issue "Magnetism in the Universe" (ed. A. Balogh)

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., Comment: 2 pages, contribution to the proceedings of the IAU Symposium 259 "Cosmic Magnetic Fields: from Planets, to Stars and Galaxies", Puerto Santiago, Tenerife, Spain, November 3-7, 2008 (in press)

Aims. We investigate the cosmological evolution of large- and small-scale magnetic fields in galaxies in the light of present models of formation and evolution of galaxies. Methods. We use the dynamo theory to derive the timescales of amplification and ordering of magnetic fields in disk and puffy galaxies. Turbulence in protogalactic halos generated by thermal virialization can drive an efficient turbulent dynamo. Results from simulations of hierarchical structure formation cosmology provide a tool to develop an evolutionary model of regular magnetic fields coupled with galaxy formation and evolution. Results. The turbulent (small-scale) dynamo was able to amplify a weak seed magnetic field in halos of protogalaxies to a few muG strength within a few 10^8 yr. This turbulent field served as a seed to the mean-field (large-scale) dynamo. Galaxies similar to the Milky Way formed their disks at z~10 and regular fields of muG strength and a few kpc coherence length were generated within 2 Gyr (at z~3), but field-ordering on the coherence scale of the galaxy size required an additional 6 Gyr (at z~0.5). Giant galaxies formed their disks at z~10, allowing more efficient dynamo generation of strong regular fields (with kpc coherence length) already at z~4. However, the age of the Universe is short for fully coherent fields in giant galaxies larger than 15 kpc to have been achieved. Dwarf galaxies should have hosted fully coherent fields at z~1. After a major merger, the strength of the turbulent field is enhanced by a factor of a few. Conclusions. This evolutionary scenario can be tested by measurements of polarized synchrotron emission and Faraday rotation with the planned SKA. We predict an anticorrelation between galaxy size and ratio between ordering scale and galaxy size (abridged)., Comment: 13 pages, 3 figures, accepted for publication to A&A; new version which incorporates suggestions of a language editor

Nonlinear behaviour of galactic dynamos is studied, allowing for magnetic helicity removal by the galactic fountain flow. A suitable advection speed is estimated, and a one-dimensional mean-field dynamo model with dynamic alpha-effect is explored. It is shown that the galactic fountain flow is efficient in removing magnetic helicity from galactic discs. This alleviates the constraint on the galactic mean-field dynamo resulting from magnetic helicity conservation and thereby allows the mean magnetic field to saturate at a strength comparable to equipartition with the turbulent kinetic energy., Comment: 4 pages, 2 figures, accepted by A&A Letters

The thin-disc global asymptotics are discussed for axisymmetric mean-field dynamos with vacuum boundary conditions allowing for non-local terms arising from a finite radial component of the mean magnetic field at the disc surface. This leads to an integro-differential operator in the equation for the radial distribution of the mean magnetic field strength, $Q(r)$ in the disc plane at a distance $r$ from its centre; an asymptotic form of its solution at large distances from the dynamo active region is obtained. Numerical solutions of the integro-differential equation confirm that the non-local effects act similarly to an enhanced magnetic diffusion. This leads to a wider radial distribution of the eigensolution and faster propagation of magnetic fronts, compared to solutions with the radial surface field neglected. Another result of non-local effects is a slowly decaying algebraic tail of the eigenfunctions outside the dynamo active region, $Q(r)\sim r^{-4}$, which is shown to persist in nonlinear solutions where $\alpha$-quenching is included. The non-local nature of the solutions can affect the radial profile of the regular magnetic field in spiral galaxies and accretion discs at large distances from the centre., Comment: Revised version, as accepted; Geophys. Astrophys. Fluid Dynam

We reconsider thin-disc global asymptotics for kinematic, axisymmetric mean-field dynamos with vacuum boundary conditions. Non-local terms arising from a small but finite radial field component at the disc surface are consistently taken into account for quadrupole modes. As in earlier approaches, the solution splits into a local part describing the field distribution along the vertical direction and a radial part describing the radial (global) variation of the eigenfunction. However, the radial part of the eigenfunction is now governed by an integro-differential equation whose kernel has a weak (logarithmic) singularity. The integral term arises from non-local interactions of magnetic fields at different radii through vacuum outside the disc. The non-local interaction can have a stronger effect on the solution than the local radial diffusion in a thin disc, however the effect of the integral term is still qualitatively similar to magnetic diffusion.

Faraday rotation of the polarization plane in magnetized thermal plasma provides one of the most efficient methods to deduce regular magnetic fields from radio astronomical observations. Since the Faraday rotation measure RM is proportional to an integral, along the line of sight, of magnetic field weighted with thermal electron density, RM is believed to yield the regular magnetic field averaged over large volume. Here we show that this is not the case in a turbulent medium where fluctuations in magnetic field and electron density are not statistically independent, and so contribute to RM. For example, in the case of pressure equilibrium, magnetic field can be anticorrelated with plasma density to produce a negative contribution. As a result, the strength of the regular magnetic field obtained from RM can be underestimated if the fluctuations in electron density and magnetic field are neglected. The anticorrelation also reduces the standard deviation of RM. We further discuss the effect of the positive correlations where the standard treatment of RM leads to an overestimated magnetic field. Because of the anisotropy of the turbulent magnetic field, the regular magnetic fields strength, obtained from synchrotron emission using standard formulae, can be overestimated. A positive correlation between cosmic-ray number density and magnetic field leads to an overestimate of the strengths of the regular and total fields. These effects can explain the difference between the strengths of the regular Galactic magnetic field as indicated by RM and synchrotron emissivity data and reconcile the magnetic field strength in the Solar vicinity with typical strength of regular magnetic fields in external galaxies., Comment: 10 pages with 2 figures. Accepted for publication in A&A

A theory of clustering of inertial particles advected by a turbulent velocity field caused by an instability of their spatial distribution is suggested. The reason for the clustering instability is a combined effect of the particles inertia and a finite correlation time of the velocity field. The crucial parameter for the clustering instability is a size of the particles. The critical size is estimated for a strong clustering (with a finite fraction of particles in clusters) associated with the growth of the mean absolute value of the particles number density and for a weak clustering associated with the growth of the second and higher moments. A new concept of compressibility of the turbulent diffusion tensor caused by a finite correlation time of an incompressible velocity field is introduced. In this model of the velocity field, the field of Lagrangian trajectories is not divergence-free. A mechanism of saturation of the clustering instability associated with the particles collisions in the clusters is suggested. Applications of the analyzed effects to the dynamics of droplets in the turbulent atmosphere are discussed. An estimated nonlinear level of the saturation of the droplets number density in clouds exceeds by the orders of magnitude their mean number density. The critical size of cloud droplets required for clusters formation is more than $20 \mu$m., Comment: REVTeX 4, 15 pages, 2 figures(included), PRE submitted

We suggested a theory of clustering of inertial particles advected by a turbulent velocity field caused by an instability of their spatial distribution. The reason of the {\em clustering instability} is a combined effect of the particle inertia and finite correlation time of the velocity field. The crucial parameter for the instability is a size of the particles. The critical size is estimated for a {\em strong clustering} (with a finite fraction of particles in clusters) associated with the growth of the mean absolute value of the particles number density and for a {\em weak clustering} associated with the growth of the second and higher moments. A nonlinear mechanism for a saturation of the clustering instability (particles collisions in the clusters) is suggested. Applications of the analyzed effects to the dynamics of aerosols and droplets in the turbulent atmosphere are discussed. The critical size of atmospheric aerosols and droplets in clustering is of the order of $(20 - 30)\mu$m, and a lower estimate of the number of particles in a cluster is about hundreds., Comment: Europhys. Letters, submitted, 5 pages, REVTeX 3

Various approaches to estimate turbulent transport coefficients from numerical simulations of hydromagnetic turbulence are discussed. A quantitative comparison between the averaged magnetic field obtained from a specific three-dimensional simulation of a rotating turbulent shear flow in a slab and a simple one-dimensional alpha-omega dynamo model is given. A direct determination of transport coefficients is attempted by calculating the correlation matrix of different components of the field and its derivatives. The alpha-effect operating on the toroidal field is found to be negative and of similar magnitude as the value obtained previously by correlating the electromotive force with the mean magnetic field. The turbulent diffusion of the toroidal field is comparable to the kinematic viscosity that was determined earlier by comparing the stress with the shear. However, the turbulent diffusion of the radial field component is smaller and can even be formally negative. The method is then modified to obtain the spectral dependence of the turbulent transport coefficients on the wavenumber. There is evidence for nonlocal behavior in that most of the response comes from the smallest wavenumbers corresponding to the largest scale possible in the simulation., Comment: 30 pages, 14 figures, to appear in Geophys. & Astrophys. Fluid Dynamics