Mean-field Dynamo Theory Research Articles

ORIGIN OF SUNSPOTS

Sunspots, seen as cool regions on the surface of the Sun, are a thermal phenomenon. Sunspots are always associated with bipolar magnetic loops that break through the solar surface. Thus to explain the origin of sunspots we have to understand how the magnetic field originates inside the Sun and emerges at its surface. The field predicted by mean-field dynamo theories is too weak by itself to emerge at the surface of the Sun. However, because of the turbulent character of solar convection the fields generated by dynamo are intermittent – i.e., concentrated into ropes or sheets with large spaces in between. The intermittent fields are sufficiently strong to be able to emerge at the solar surface, in spite of the fact that their mean (average) value is weak. It is suggested here that magnetic fields emerge at the solar surface at those random times and places when the total magnetic field (mean field plus fluctuations) exceeds the threshold for buoyancy. The clustering of coherently emerged loops results in the formation of a sunspot. A non-axisymmetric enhancement of the underlying magnetic field causes in the clustering of sunspots forming sunspot groups, clusters of activity and active longitudes. The mean field, which is not directly observable, is also important, being responsible for the ensemble regularities of sunspots, such as Hale's law of sunspot polarities and the 11-year periodicity.

The Solar Dynamo: Old, Recent, and New Problems

A number of problems of solar and stellar dynamo theory are briefly reviewed and the current status of possible solutions is discussed. Results of direct numerical simulations are described in view of mean-field dynamo theory and the relation between the α-effect and the inverse cascade of magnetic helicity is highlighted. The possibility of ‘catastrophic’ quenching of the α-effect is explained in terms of the constraint placed by the conservation of magnetic helicity.

Mean Field Theory Interpretation of Solar Polarity Reversal

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.

Dynamo–generated turbulence and outflows from accretion discs

Local hydromagnetic simulations of accretion–disc turbulence currently provide the most convincing evidence that the origin of turbulence in discs could be the Balbus–Hawley magnetorotational instability. The main results of such calculations are highlighted with particular emphasis on the generation of large–scale magnetic fields. Comparison with mean–field dynamo theory is made. This theory is then used to address the question of the launching and collimation of winds emanating from the disc surfaces.

Physical Causes of Solar Activity

The magnetic fields that dominate the structure of the Sun's atmosphere are controlled by processes in the solar interior, which cannot be directly observed. Magnetic activity is found in all stars with deep convective envelopes: young and rapidly rotating stars are very active but cyclic activity only appears in slow rotators. The Sun's 11-year activity cycle corresponds to a 22-year magnetic cycle, since the sunspot fields (which are antisymmetric about the equator) reverse at each minimum. The record of magnetic activity is aperiodic and is interrupted by episodes of reduced activity, such as the Maunder Minimum in the seventeenth century, when sunspots almost completely disappeared. The proxy record from cosmogenic isotopes shows that similar grand minima recur at intervals of around 200 yr. The Sun's large-scale field is generated by dynamo action rather than by an oscillator. Systematic magnetic cycles are apparently produced by a dynamo located in a region of weak convective overshoot at the base of the convection zone, where there are strong radial gradients in the angular velocity Ω. The crucial parameter (the dynamo number) increases with increasing Ω and kinematic (linear) theory shows that dynamo action can set in at an oscillatory (Hopf) bifurcation that is probably subcritical. Although it has been demonstrated that the whole process works in a self-consistent model, most calculations have relied on mean-field dynamo theory. This approach is physically plausible but can only be justified under conditions that do not apply in the Sun. Still, mean-field dynamos do reproduce the butterfly diagram and other key features of the solar cycle. An alternative approach is to study generic behaviour in low-order models, which exhibit two forms of modulation, associated with symmetry-breaking and with reduced activity. Comparison with observed behaviour suggests that modulation of the solar cycle is indeed chaotic, i.e. deterministically rather than stochastically driven.

In Situ Origin of Large-Scale Galactic Magnetic Fields without Kinetic Helicity?

The origin and sustenance of large scale galactic magnetic fields has been a long standing and controversial astrophysical problem. Here an alternative to the ``standard'' $\a-\Omega$ mean field dynamo and primordial theories is pursued. The steady supply of supernovae induced turbulence exponentiates the total field energy, providing a significant seed mean field that can be linearly stretched by shear. The observed micro-Gauss fields would be produced primarily within one vertical diffusion time since it is only during this time that linear stretching can compete with diffusion. This approach does not invoke exponential mean field dynamo growth from the helicity $\a$-effect but does employ turbulent diffusion, which limits the number of large scale reversals. The approach could be of interest if the helicity effect is suppressed independently of the turbulent diffusion. This is an important but presently unresolved issue.

Filaments and the Solar Dynamo

AbstractFilaments are a global phenomenon and their formation, structure and dynamics are determined by magnetic fields. So they are an important signature of the solar magnetism. The central mechanism in traditional mean-field dynamo theory is the alpha effect and it is a major result of this theory that the presence of kinetic or magnetic helicities is at least favourable for the effect. Recent studies of the magnetohydrodynamic equations by means of numerical bifurcation-analysis techniques have confirmed the decisive role of helicity for a dynamo effect. The alpha effect corresponds to the simultaneous generation of magnetic helicities in the mean field and in the fluctuations, the generation rates being equal in magnitude and opposite in sign. In the case of statistically stationary and homogeneous fluctuations, in particular, the alpha effect can increase the energy in the mean magnetic field only under the condition that also magnetic helicity is accumulated there. Generally, the two helicities generated by the alpha effect, that in the mean field and that in the fluctuations, have either to be dissipated in the generation region or to be transported out of this region. The latter may lead to the appearance of helicity in the atmosphere, in particular in filaments, and thus provide valuable information on dynamo processes inaccessible to in situ measurements.

The Karlsruhe Dynamo Experiment. A Mean Field Approach

At the Forschungszentrum Karlsruhe an experiment is in preparation which it is hoped, in view of the geodynamo and other cosmic dynamos, that a homogeneous dynamo will be demonstrated and investigated. This experiment is discussed within the framework of mean-field dynamo theory. Results are presented concerning kinematic cylindrical mean-field dynamo models reflecting some features of the experimental device, as well as results of detailed calculations of the α-effect that apply to arbitrarily high magnetic Reynolds numbers. On this basis estimates of the excitation conditions of the dynamo are given and predictions concerning the geometrical structure of the generated magnetic fields are made.

The Evolution of a Primordial Galactic Magnetic Field

We consider the hypothesis that galactic magnetic fields are primordial. We also discuss the various objections to this hypothesis. To test this hypothesis properly we assume that there was a magnetic field present in the galactic plasma before the galaxy formed and explore how such a field would evolve assuming a specific model for the interstellar medium in the galactic disk. After the galactic disk formed, the lines of force thread through it and remain connected to the external cosmic medium. They enter through one side of the disk, proceed horizontally a distance l in the disk, and then leave through the other side. We find that the lines of force are stretched by the differential rotation of the galactic disk, amplifying the toroidal component of the field and increasing l. When the magnetic field is strong enough, it produces ambipolar velocities that try to lift the line out of the galactic disk but in opposite directions on different parts of the line. The result is that, instead of the line being expelled from the disk, its horizontal length l is shortened, both in the radial and in the toroidal direction. This leads to a reduction of the rate of horizontal stretching and finally a reduction in the magnetic field strength. After a sufficient time, the magnetic field at all points goes through this reduction and the field strength approaches a universal function of time. This function is slowly decreasing and only depends on the ambipolar properties of the interstellar medium. At any given time the magnetic field is toroidal and has the same strength everywhere. On the other hand, it turns out that its direction varies rapidly with radius, changing sign every 100 parsecs or so. However, if the initial cosmic magnetic field is not uniform, the areas of one sign of the toroidal field dominate over the other. The resulting field has a net Faraday rotation. If such a field were observed with low resolution in an external galaxy, then the field would appear toroidal in between the spiral arms. The spiral density wave would turn it so that the lines appear to trace out the spiral arm, although the apparent lines really are the sum of pieces of magnetic lines as they cross the disk. They do not necessarily extend very far along the arms. We contend that this model of the magnetic field, which arises naturally from a primordial origin, can fit the observations as well as other models for the magnetic field, such as those arising from the mean field dynamo theory. Finally, because the field lines are topologically threaded through the disk, they cannot be expelled from the disk. This counters the objection against the primordial origin, namely that such a field could not survive very long in the galaxy.

On the Generation of Equipartition-Strength Magnetic Fields by Turbulent Hydromagnetic Dynamos

The generation of a mean magnetic field by the action of small-scale turbulent fluid motions, the α-effect, is a fundamental ingredient of mean-field dynamo theory. However, recent mathematical models and numerical experiments are providing increasingly strong support to the notion that at high magnetic Reynolds numbers, the α-effect is strongly impeded long before the mean magnetic field has reached energy equipartition with the driving fluid motions. Taken at face value, this raises serious doubt as to whether the solar magnetic field is produced by a turbulent hydromagnetic dynamo after all, since it is an observed fact that the Sun does possess a structured, large-scale mean magnetic field of strength comparable to equipartition. In this Letter we demonstrate that the class of mean-field turbulent hydromagnetic models known as interface dynamos can produce equipartition-strength mean magnetic fields even in the presence of strong α-quenching.

Overcoming the Backreaction on Turbulent Motions in the Presence of Magnetic Fields

Standard magnetohydrodynamic theories, such as the mean field dynamo theory, have been criticized when the back reaction of the magnetic field on turbulent motions is neglected. For the dynamo, this back reaction has been argued to suppress the turbulent motions required for optimal mean field production. Here it is suggested that if the magnetic field is spatially intermittent, for example residing in flux tubes, the back reaction on turbulent flows may be significantly reduced.

Models for the magnetic field of M81

We study several mean field dynamo models in disk geometry in an attempt to understand the origin of the nonaxisymmetric magnetic field present in M81. There appear to be three (at least) relevant mechanisms, which are not mutually exclusive. Because field growth times are not very short compared to galactic ages, a predominantly nonaxisymmetric seed field may still give a significantly nonaxisymmetric field after times of order 10 10 yr, even if the stable field configuration is axisymmetric. The spiral structure may give a non-axisymmetric structure to the disk turbulence, and thus to the turbulent coefficients appearing in mean field dynamo theory

Existence and Energy Balance of the Solar Dynamo

The idea behind the use of ensemble averaging and the finite magnetic energy method of van Geffen and Hoyng (1992) is briefly discussed. Applying this method to the solar dynamo shows that the turbulence — an essential ingredient of traditional mean field dynamo theory — poses grave problems: the turbulence makes the magnetic field so unstable that it becomes impossible to recognize any period.

Flux Tubes and Dynamos

The structure of solar surface magnetic fields, the way they erupt from the the convection zone below, and processes like flux expulsion and fragmentation instabilities support the view that magnetic flux in a stellar convection zone is in an intermittent, fragmented state which can be described as an ensemble of magnetic flux tubes. Depending on size and field strength, the dynamics of magnetic flux tubes can strongly differ from the behavior of a passive, diffuse field which is often assumed in conventional mean-field dynamo theory. Observed properties of active regions like emergence in low latitudes, Hale's polarity rules, tilt angles, and the process of sunspot formation from smaller fragments, together with theoretical considerations of the dynamics of buoyant flux tubes indicate that the magnetic structures which erupt in an emerging active region are not passive to convection and originate in a source region (presumably an overshoot layer below the convection zone proper) with a field strength of at least 105 G, far beyond the equipartition field strength with respect to convective flows. We discuss the consequences of such a situation for dynamo theory of the solar cycle and consider the possibility of dynamo models on the basis of flux tubes. A simple, illustrative example of a flux tube dynamo is presented.

The spectrum of random magnetic fields in the mean field dynamo theory of the Galactic magnetic field

view Abstract Citations (405) References (10) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Spectrum of Random Magnetic Fields in the Mean Field Dynamo Theory of the Galactic Magnetic Field Kulsrud, Russell M. ; Anderson, Stephen W. Abstract The fluctuation spectrum that must arise in a mean field dynamo generation of galactic fields if the initial field is weak is considered. A kinetic equation for its evolution is derived and solved. The spectrum evolves by transfer of energy from one magnetic mode to another by interaction with turbulent velocity modes. This kinetic equation is valid in the limit that the rate of evolution of the magnetic modes is slower than the reciprocal decorrelation time of the turbulent modes. This turns out to be the case by a factor greater than 3. Most of the fluctuation energy concentrates on small scales, shorter than the hydrodynamic turbulent scales. The fluctuation energy builds up to equipartition with the turbulent energy in times that are short compared to the e-folding time of the mean field. The turbulence becomes strongly modified before the dynamo amplification starts. Thus, the kinematic assumption of the mean dynamo theory is invalid. Thus, the galactic field must have a primordial origin, although it may subsequently be modified by dynamo action. Publication: The Astrophysical Journal Pub Date: September 1992 DOI: 10.1086/171743 Bibcode: 1992ApJ...396..606K Keywords: Dynamo Theory; Interstellar Magnetic Fields; Interstellar Matter; Cosmology; Galactic Evolution; Magnetic Flux; Magnetohydrodynamic Turbulence; Astrophysics; ISM: MAGNETIC FIELDS; MAGNETOHYDRODYNAMICS: MHD; TURBULENCE full text sources ADS |

Dynamo theory

Recent developments in dynamo theory are discussed both from a kinematic and hydromagnetic point of view. The relative importance of the α- and ω-effects of mean field dynamo theory is examined. In the case of the Earth, recent results point to a model of αω-type in which the primary force balance is between Coriolis, pressure, Lorentz forces with motion driven by buoyancy forces. Accordingly, the magnetic field is strong of order ( ρΩ σ ) 1 2 , where ρ is fluid density, σ is the electrical conductivity and Ω is the rotation rate, and predominantly azimuthal. It is large compared with the meridional magnetic field emerging at the Earth's surface. Within the wider framework of planetary magnetic fields detailed issues include the nature of the convection leading to the α-effect, the generation of azimuthal zonal motion (thermal winds, geostrophic flow and Taylor's condition) responsible for the ω-effect, and the steady or oscillatory nature of the ensuing dynamo.

The solar dynamo

Abstract This review begins with remarks on contributions by T. G. Cowling to the problem of the solar magnetism. Then I discuss problems connected to first-order smoothing in mean-field dynamo theory. Next, the reasons for a solar dynamo operating near the base of the outer convection zone will be recalled, and models will be reviewed. Finally, I address the variety of possible solutions to the dynamo equations: Stationary and oscillatory, symmetric and antisymmetric with respect to the equator, axisymmetric and longitude-dependent. Limit cycles and chaotic fields which appear in the non-linear regime will be mentioned in this context.

The Alpha-Effect in Galaxies is Highly Anisotropic

Besides the mean flow the alpha is the other input quantity for any mean-field dynamo model. It describes the generation of turbulent electromotive force <u × B> from a large-scale field <B> for a given turbulence. The necessary helicity of the turbulence results from the joint action of Coriolis force and density stratification. The standard estimate of 1 km/s for alpha in galaxies is a surely well-established approximation. One of the essentials, however, remains open. Due to the extremely anisotropic structure of disks the tensorial character of alpha can no longer be ignored. In stellar applications anisotropy in the α-tensor leads to a preferred excitation of non-axisymmetric magnetic fields. That is true for α2 -dynamos if the alpha parallel to the rotation axis, α||, is much smaller than that in the equatorial plane, α⊥. The idea is that also for disk-like configurations a similar behaviour makes the existence of the observed large-scale non-axisymmetric magnetic BSS modes understandable within the frame of the mean-field dynamo theory.

Activity cycle periods in late-type stars

Period of magnetic activity versus the stellar angular velocity for stars of given spectral type having extended convective shells is estimated within the framework of mean field dynamo theory. The dependence appears to be not monotonous, and can be checked by observations.

A review of: “Mean-field magnetohydrodynamics and dynamo theory”

Abstract By F. Krause and K.-H. Rädler. Pergamon, Oxford, 1981. 271 pp. (Cloth £15.00 ($36.00) (ISBN 0 08025 0416))