Mean-field Dynamo Research Articles

Coherent structures and the saturation of a nonlinear dynamo

AbstractEulerian and Lagrangian tools are used to detect coherent structures in the velocity and magnetic fields of a mean-field dynamo, produced by direct numerical simulations of the three-dimensional compressible magnetohydrodynamic equations with an isotropic helical forcing and moderate Reynolds number. Two distinct stages of the dynamo are studied: the kinematic stage, where a seed magnetic field undergoes exponential growth; and the saturated regime. It is shown that the Lagrangian analysis detects structures with greater detail, in addition to providing information on the chaotic mixing properties of the flow and the magnetic fields. The traditional way of detecting Lagrangian coherent structures using finite-time Lyapunov exponents is compared with a recently developed method called function $M$. The latter is shown to produce clearer pictures which readily permit the identification of hyperbolic regions in the magnetic field, where chaotic transport/dispersion of magnetic field lines is highly enhanced.

Galactic spiral patterns and dynamo action – II. Asymptotic solutions

The exploration of mean-field galactic dynamos affected by a galactic spiral pattern, begun in Chamandy et al. (2013, hereafter Paper I) with numerical simulations, is continued here with an asymptotic solution. The mean-field dynamo model used generalizes the standard theory to include the delayed response of the mean electromotive force to variations of the mean magnetic field and turbulence parameters (the temporal non-locality, or {\tau} effect). The effect of the spiral pattern on the dynamo considered is the enhancement of the {\alpha}-effect in spiral-shaped regions (which may overlap the gaseous spiral arms or be located in the interarm regions). The axisymmetric and enslaved non-axisymmetric modes of the mean magnetic field are studied semi-analytically to clarify and strengthen the numerical results. Good qualitative agreement is obtained between the asymptotic solution and numerical solutions of Paper I for a global, rigidly rotating material spiral (density wave). At all galactocentric distances except for the co-rotation radius, we find magnetic arms displaced in azimuth from the {\alpha}-arms, so that the ridges of magnetic field strength are more tightly wound than the {\alpha}-arms. Moreover, the effect of a finite dynamo relaxation time {\tau} (related to the turbulence correlation time) is to phase-shift the magnetic arms in the direction opposite to the galactic rotation even at the co-rotation radius. This mechanism can be used to explain the phase shifts between magnetic and material arms observed in some spiral galaxies.

A mean field dynamo from negative eddy diffusivity

Using direct numerical simulations, we verify that Roberts-IV flow exhibits dynamo action dominated by horizontally averaged large-scale magnetic field. With the test-field method, we compute the turbulent magnetic diffusivity and find that it is negative and overcomes the molecular diffusivity, thus explaining quantitatively the large-scale dynamo for magnetic Reynolds numbers above ≈8. As expected for a dynamo of this type, but contrary to α-effect dynamos, the two horizontal field components grow independently of each other and have arbitrary amplitude ratios and phase differences. Small length-scales of the mean magnetic field are shown to be stabilized by the turbulent magnetic diffusivity becoming positive at larger wavenumbers. Oscillatory decaying or growing solutions have also been found in certain wavenumber intervals and sufficiently large values of the magnetic Reynolds number. For magnetic Reynolds numbers below ≈0.5, the turbulent magnetic diffusivity is confirmed to be positive, as expected for all incompressible flows. Earlier claims of a dynamo driven by a modified Taylor–Green flow through negative eddy diffusivity could not be confirmed.

Resonances for activity waves in spherical mean field dynamos

We study activity waves of the kind that determine cyclic magnetic activity of various stars, including the Sun, as a more general physical rather than a purely astronomical problem. We try to identify resonances which are expected to occur when a mean-field dynamo excites waves of quasi-stationary magnetic field in two distinct spherical layers. We isolate some features that can be associated with resonances in the profiles of energy or frequency plotted versus a dynamo governing parameter. Rather unexpectedly however the resonances in spherical dynamos take a much less spectacular form than resonances in many more familiar branches of physics. In particular, we find that the magnitudes of resonant phenomena are much smaller than seem detectable by astronomical observations, and plausibly any related effects in laboratory dynamo experiments (which of course are not in gravitating spheres!) are also small. We discuss specific features relevant to resonant phenomena in spherical dynamos, and find parametric resonance to be the most pronounced type of resonance phenomena. Resonance conditions for these dynamo wave resonances are rather different from those for more conventional branches of physics. We suggest that the relative insignificance of the phenomenon in this case is because the phenomena of excitation and propagation of the activity waves are not well-separated from each other and this, together with the nonlinear nature of more-or-less realistic dynamos, suppress the resonances and makes them much less pronounced than resonant effects, for example in optics.

Solar Dynamo and Toroidal Field Instabilities

The possibility of non-axisymmetric (kink) instabilities of a toroidal field seated in the tachocline is much discussed in the literature. In this work, the basic properties of kink and quasi-interchange instabilities, produced by mixed toroidal and poloidal configuration, will be briefly reviewed. In particular, it will be shown that the unstable modes are strongly localized near the Equator and not near the Poles as often claimed in the literature. Based on the results of recent numerical simulations, it is argued that a non-zero helicity can already be produced at a non-linear level. A mean-field solar dynamo is then constructed with a positive α-effect in the overshoot layer localized near the Equator, and a meridional circulation with deep return flow. Finally, the possibility that the solar cycle is driven by an αΩ dynamo generated by the negative subsurface shear in the supergranulation layer will also be discussed.

Galactic spiral patterns and dynamo action – I. A new twist on magnetic arms

We generalise the theory of mean-field galactic dynamos by allowing for temporal non-locality in the mean electromotive force (emf). This arises in random flows due to a finite response time of the mean emf to changes in the mean magnetic field, and leads to the telegraph equation for the mean field. The resulting dynamo model also includes the nonlinear dynamo effects arising from magnetic helicity balance. Within this framework, coherent large-scale magnetic spiral arms superimposed on the dominant axially symmetric magnetic structure are considered. A non-axisymmetric forcing of the mean-field dynamo by a spiral pattern (either stationary or transient) is invoked, with the aim of explaining the phenomenon of magnetic arms. For a stationary dynamo forcing by a rigidly rotating material spiral, we find corotating non-axisymmetric magnetic modes enslaved to the axisymmetric modes and strongly peaked around the corotation radius. For a forcing by transient material arms wound up by the galactic differential rotation, the magnetic spiral is able to adjust to the winding so that it resembles the material spiral at all times. There are profound effects associated with the temporal non-locality, i.e. finite `dynamo relaxation time'. For the case of a rigidly rotating spiral, a finite relaxation time causes each magnetic arm to mostly lag the corresponding material arm with respect to the rotation. For a transient material spiral that winds up, the finite dynamo relaxation time leads to a large, negative (in the sense of the rotation) phase shift between the magnetic and material arms, similar to that observed in NGC 6946 and other galaxies. We confirm that sufficiently strong random seed fields can lead to global reversals of the regular field along the radius whose long-term survival depends on specific features of a given galaxy.

From large-scale to small-scale dynamos in a spherical shell

Kinematic dynamo action in a spherical shell is studied with a small-scale cellular prescribed velocity field. Three velocity fields are considered, all of which are axisymmetric and have a large-scale separation between the shell size and the dominant scale of the motion. The first flow is steady and strongly helical, so that a mean field dynamo might be expected. We find that indeed large-scale dynamo action is obtained at onset, but that the dynamo is of small-scale type at large magnetic Reynolds number Rm, where advection processes dominate and the magnetic field is generated on the scale of the cells in the flow. We study the transition between these two dynamo processes and find a gradual transition as Rm is increased, where the energy is slowly passed from the large scales to the small scales as the two dynamo processes morph into one another. The second flow, a time-dependent version of the first, produces almost identical results, but the transition appears to occur at smaller Rm than in the ste...

SOLAR MAGNETIC FIELD REVERSALS AND THE ROLE OF DYNAMO FAMILIES

The variable magnetic field of the solar photosphere exhibits periodic reversals as a result of dynamo activity occurring within the solar interior. We decompose the surface field as observed by both the Wilcox Solar Observatory and the Michelson Doppler Imager into its harmonic constituents, and present the time evolution of the mode coefficients for the past three sunspot cycles. The interplay between the various modes is then interpreted from the perspective of general dynamo theory, where the coupling between the primary and secondary families of modes is found to correlate with large-scale polarity reversals for many examples of cyclic dynamos. Mean-field dynamos based on the solar parameter regime are then used to explore how such couplings may result in the various long-term trends in the surface magnetic field observed to occur in the solar case.

Mean-field dynamo action in renovating shearing flows

We study mean-field dynamo action in renovating flows with finite and nonzero correlation time (τ) in the presence of shear. Previous results obtained when shear was absent are generalized to the case with shear. The question of whether the mean magnetic field can grow in the presence of shear and nonhelical turbulence, as seen in numerical simulations, is examined. We show in a general manner that, if the motions are strictly nonhelical, then such mean-field dynamo action is not possible. This result is not limited to low (fluid or magnetic) Reynolds numbers nor does it use any closure approximation; it only assumes that the flow renovates itself after each time interval τ. Specifying to a particular form of the renovating flow with helicity, we recover the standard dispersion relation of the α(2)Ω dynamo, in the small τ or large wavelength limit. Thus mean fields grow even in the presence of rapidly growing fluctuations, surprisingly, in a manner predicted by the standard quasilinear closure, even though such a closure is not strictly justified. Our work also suggests the possibility of obtaining mean-field dynamo growth in the presence of helicity fluctuations, although having a coherent helicity will be more efficient.

Waldmeier relations and the solar cycle dynamics by the mean-field dynamos

AbstractThe long-term variability of the sunspot cycle, as recorded by the Wolf numbers, are imprinted in different kinds of statistical relations which relate the cycle amplitudes, duration and shapes. This subject always gets a special attention because it is important for the solar activity forecast. We discuss statistical properties of the mean-field dynamo model with the fluctuating α-effect. Also, we estimate dynamical properties of the model for the long and short time-scale and compare it with the dynamics of the sunspot numbers data sets.

Galactic spiral patterns and dynamo action

AbstractThe theory of mean-field galactic dynamos is generalized by allowing for a finite response time of the mean electromotive force (emf) to variations in the mean magnetic field and small-scale turbulence. A non-axisymmetric forcing of the dynamo by a spiral pattern (either stationary or transient) is invoked. The resulting magnetic spiral arms are phase-shifted from the spiral arms of the pattern by an angle 15°–40°, opposite to the sense of galactic rotation. Our findings may help to explain the phase shift between material and magnetic arms observed in NGC 6946 and other galaxies.

On the role of asymmetries in the reversal of the solar magnetic field

AbstractWe study how the solar magnetic field evolves from antisymmetric (dipolar) to symmetric (quadrupolar) state during the course of its 11-yr cycle. We show that based on equatorial symmetries of the induction equation, flux transport solar mean field dynamo models excite mostly the antisymmetric (dipolar) family whereas a decomposition of the solar magnetic field data reveals that both families should be excited to similar amplitude levels. We propose an alternative solar dynamo solution based on North-South asymmetry of the meridional circulation to better reconcile models and observations.

Ram pressure effects in the galactic plane and galactic dynamos in the no-zapproximation

The magnetic field of galaxies is believed to be produced by internal dynamo action, but can be affected by motion of the galaxy through the surrounding medium. Observations of polarized radio emission of galaxies located in galaxy clusters have revealed noticeable features of large-scale magnetic configurations, including displacements of the magnetic structures from the optical images and tails, which are possible imprints of ram pressure effects arising from motion of the galaxies through the intracluster medium. We present a quantitative dynamo model which attempts to describe the above effects. In contrast to the traditional problem of a wind affecting a body with a prescribed magnetic field, we investigate how a non-magnetized wind flow affects a magnetic field that is being self-excited by galactic dynamo action. In order to isolate the leading physical effects we exploit a simple dynamo model that can describe relevant effects. In particular, we use what is known as the 'no-z' approximation for the mean-field dynamo equations. In a suitable parametric range we obtain displacements of the large-scale magnetic field, as well as magnetic tails. However, the specific details of their locations are quite counterintuitive. The direction of displacement is perpendicular to, rather than parallel to, the wind direction. The point at which the tail emerges from the galaxy depends on details of the model. The tail is eventually directed downstream. In the simplest case the magnetic tail begins in the region where the wind decreases the total gas velocity. Any wind that penetrates the galaxy modifies the intrinsic dynamo action. These features are different from those found in ram-pressure models. Any determination of galactic motion through the cluster medium from observational data needs to take the effects of dynamo action into account.

Effect of stochasticity in mean field dynamo models

We present a comprehensive investigation into the effect of choosing the stochastic control parameters in a simplified—Parker dynamo model. Through considering the manifold of marginal stability, i.e., the region of parameter space where the mean growth rate is zero, we show that stochastic fluctuations are not prohibitive to dynamo. Furthermore, by directly comparing results obtained by periodic and Gaussian coloured noise alpha with identical characteristic time-scales and fluctuating amplitudes, we find that the transition to dynamo is significantly eased for stochastically fluctuating alpha. The effect of stochasticity in magnetic diffusion is also investigated, highlighting the importance of resonance between poloidal and toroidal magnetic fields on the growth rate. Furthermore, we show that probability density functions of the growth-rate, magnetic field, and magnetic energy can provide a wealth of useful information regarding the dynamo behaviour/intermittency. Finally, the statistical properties of the dynamo such as temporal correlation and fluctuating amplitude are found to be dependent on the distribution of the fluctuations in stocastic parameters.

Variations of the solar cycle profile in a solar dynamo with fluctuating dynamo governing parameters

Solar cycles vary in their amplitude and shape. There are several empirical relations between various parameters linking cycle's shape and amplitude, in particular the Waldmeier relations. As solar cycle is believed to be a result of the solar dynamo action, these relations require explanation in the framework of this theory.Here we aim to present a possible explanation of such kind. We relate the cycle-to-cycle variability of solar activity to fluctuations of solar dynamo drivers and primarily to fluctuations in the parameter responsible for recovery of the poloidal magnetic field from the toroidal one. To be specific, we develop such a model in the framework of the mean-field dynamo based on the differential rotation and $\alpha$-effect. We demonstrate that the mean-field dynamo based on a realistic rotation curve and nonlinearity associated with the magnetic helicity balance reproduces both qualitatively and quantitatively the Waldmeier relations observed in sunspot data since 1750 (SIDC data). The model also reproduces more or less successfully other relations between the parameters under discussion, in particular, the link between odd and even cycles (Gnevyshev-Ohl rule). We conclude that the contemporary solar dynamo theory provides a way to explain the cycle-to-cycle variability of solar activity as recorded in sunspots. We discuss the importance of the model for stellar activity cycles which, as known from the data of HK project, demonstrate the cycle-to-cycle variability similar to solar cycles.

Fluctuating αO dynamos by iterated matrices

Abstract Recent publications have suggested various scaling laws for dynamo growth rates when dynamo action is made possible or enhanced by shear. In order to understand the relation between the different scalings, we study a simple model mean field dynamo, in which the α-effect is piecewise constant in time, allowing a simple derivation of the growth rates in terms of iterated matrices. It is found that all the different scaling laws that appear in earlier work can be seen in different parameter regimes.

Modeling spatio‐temporal nonlocality in mean‐field dynamos

AbstractWhen scale separation in space or time is poor, the mean‐field α effect and turbulent diffusivity have to be replaced by integral kernels by which the dependence of the mean electromotive force on the mean magnetic field becomes nonlocal. Earlier work in computing these kernels using the test‐field method is now generalized to the case in which both spatial and temporal scale separations are poor. The approximate form of the kernel for isotropic stationary turbulence is such that it can be treated in a straightforward manner by solving a partial differential equation for the mean electromotive force. The resulting mean‐field equations are solved for oscillatory α –shear dynamos as well as α2 dynamos with α linearly depending on position, which makes this dynamo oscillatory, too. In both cases, the critical values of the dynamo number is lowered due to spatio‐temporal nonlocality.When scale separation in space or time is poor, the mean‐field α effect and turbulent diffusivity have to be replaced by integral kernels by which the dependence of the mean electromotive force on the mean magnetic field becomes nonlocal. Earlier work in computing these kernels using the test‐field method is now generalized to the case in which both spatial and temporal scale separations are poor. The approximate form of the kernel for isotropic stationary turbulence is such that it can be treated in a straightforward manner by solving a partial differential equation for the mean electromotive force. The resulting mean‐field equations are solved for oscillatory α –shear dynamos as well as α2 dynamos

Multiscale magnetic fields in spiral galaxies: evolution and reversals

Magnetic fields in nearby, star-forming galaxies reveal both large-scale patterns and small-scale structures. A large-scale field reversal may exist in the Milky Way but no such reversals have been observed so far in external galaxies. The effects of star-forming regions of galaxies need to be included when modelling the evolution of their magnetic fields, which can then be compared to future radio polarization observations. The causes of large-scale field reversals also need clarification. Our model of field evolution in isolated disc galaxies includes a standard mean-field dynamo and continuous injection of turbulent fields (the effect of supernova explosions) in discrete star forming regions by implicit small-scale dynamo action. Synthetic maps of radio synchrotronemission and Faraday rotation measures are computed. A large-scale dynamo is essential to obtain regular large-scale spiral magnetic fields, observed in many galaxies. These appear, on kpc scales in near energy equilibrium with the turbulence, after 1-2 Gyr (redshift 4-3). Turbulent field injection generates small-scale field structures. Depending on model parameters, large-scale field reversals may persist over many Gyrs and can survive until the present epoch. Significant polarized radio synchrotron emission from young galaxies is expected at redshifts less than 4. Faraday rotation measures (RM) are crucial to detect field reversals. Large-scale patterns ofrotation measures can be observed at redshifts less than 3. Our model can explain the general form of axisymmetric spiral fields with many local distortions, as observed in nearby galaxies. For a slightly different choice of parameters, large-scale field reversals can persist over the lifetime of a galaxy. Comparison of our synthetic maps with future observations of distant galaxies with the planned Square Kilometre Array (SKA) will allow refinement of models.

The fluctuating α-effect and Waldmeier relations in the nonlinear dynamo models

We study the possibility of reproducing the statistical relations of the sunspot activity cycle, such as the so-called Waldmeier relations, the cycle period–amplitude and the cycle rise rate–amplitude relations, by means of the mean-field dynamo models with fluctuating α-effect. The dynamo model takes into account the long-term fluctuations of the α-effect and two types of nonlinear feedback of the mean field on the α-effect, namely, the algebraic quenching and the dynamic quenching due to the magnetic helicity generation. We found that the models are able to reproduce qualitatively and quantitatively the inclination and dispersion across the Waldmeier relations with 20% fluctuations of the α-effect. The models with dynamic quenching are in better agreement with observations than the models with algebraic α-quenching. We compare the statistical distributions of the modelled parameters, such as the amplitude, period, and the rise and decay rates of the sunspot cycles with observations.

MAGNETOROTATIONAL TURBULENCE TRANSPORTS ANGULAR MOMENTUM IN STRATIFIED DISKS WITH LOW MAGNETIC PRANDTL NUMBER BUT MAGNETIC REYNOLDS NUMBER ABOVE A CRITICAL VALUE

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.