The solar dynamo

The solar dynamo continues to pose a challenge to observers and theoreticians. Observations of the solar surface reveal a magnetic field with a complex, hierarchical structure consisting of widely different scales. Systematic features such as the solar cycle, the butterfly diagram, and Hale's polarity laws point to the existence of a deep-rooted large-scale magnetic field. At the other end of the scale are magnetic elements and small-scale mixed-polarity magnetic fields. In order to explain these phenomena, dynamo theory provides all the necessary ingredients including the effect, magnetic field amplification by differential rotation, magnetic pumping, turbulent diffusion, magnetic buoyancy, flux storage, stochastic variations and nonlinear dynamics. Due to advances in helioseismology, observations of stellar magnetic fields and computer capabilities, significant progress has been made in our understanding of these and other aspects such as the role of the tachocline, convective plumes and magnetic helicity conservation. However, remaining uncertainties about the nature of the deep-seated toroidal magnetic field and the effect, and the forbidding range of length scales of the magnetic field and the flow have thus far prevented the formulation of a coherent model for the solar dynamo. A preliminary evaluation of the various dynamo models that have been proposed seems to favor a buoyancy-driven or distributed scenario. The viewpoint proposed here is that progress in understanding the solar dynamo and explaining the observations can be achieved only through a combination of approaches including local numerical experiments and global mean-field modeling.Received: 5 May 2003, Published online: 15 July 2003     

Buoyant Magnetic Loops Generated by Global Convective Dynamo Action

Our global 3D simulations of convection and dynamo action in a Sun-like star reveal that persistent wreaths of strong magnetism can be built within the bulk of the convention zone. Here we examine the characteristics of buoyant magnetic structures that are self-consistently created by dynamo action and turbulent convective motions in a simulation with solar stratification but rotating at three times the current solar rate. These buoyant loops originate within sections of the magnetic wreaths in which turbulent flows amplify the fields to much higher values than is possible through laminar processes. These amplified portions can rise through the convective layer by a combination of magnetic buoyancy and advection by convective giant cells, forming buoyant loops. We measure statistical trends in the polarity, twist, and tilt of these loops. Loops are shown to preferentially arise in longitudinal patches somewhat reminiscent of active longitudes in the Sun, although broader in extent. We show that the strength of the axisymmetric toroidal field is not a good predictor of the production rate for buoyant loops or the amount of magnetic flux in the loops that are produced.     

Solar magnetic fields and convection

The solar magnetic fields observed in active regions and their residues are thought to be parts of toroidal field systems renewed every 11-yr cycle from a poloidal field. The latter may be either a reversing (dynamo) field or a non-reversing, primordial field. The latter view was held for some 70 yr, but the apparent reversals of the polar-cap fields in 1957–8 and the development of dynamo theory brought wide acceptance of the former. Here we consider evidence for and against each model, with these conclusions. (i) Several errors combine so that the non-spot measurements of gross magnetic fluxes are too low by factors of 10 or more. A permanent field of 2 G or more might remain unobserved. (ii) Measurements of average magnetic field strength are subject to various large errors. In particular, the reported reversals of the polar-cap fields are better explained in terms of tilts of toroidal field residues. (iii) Observations of new-cycle magnetic fields among old-cycle fields, of the gradual fading away of large unipolar regions, and the ubiquitous jumble of very small magnetic loop structures appear explicable only in terms of a primordial field. (iv) More positive evidence of a primordial field is found in the extreme order, symmetry and long-term stability of the polar cap streamers or rays. During one eclipse (1954) the primordial field was seen in the absence of all toroidal field residues. (v) A form of reversal of the interplanetary magnetic field is re-interpreted and shown to be consistent with a primordial, but not a dynamo, field. (vi) A test for a primordial field is that the fields below coronal holes should tend to be positive (outwards) in the northern hemisphere and negative in the southern hemisphere. (vii) Further evidence may be available by studying various plasma structures below coronal holes. An urgent requirement is a study of fibrils, faculae, macrospicules and rays in these regions.     

Predicting solar ‘climate’ by assimilating magnetic data into a flux-transport dynamo

Observational and theoretical knowledge about global-scale solar dynamo ingredients have reached the stage that it is possible to calibrate a flux-transport dynamo for the Sun by adjusting only a few tunable parameters. The important ingredients in this class of model are differential rotation (Omega-effect), helical turbulence (alpha-effect), meridional circulation and turbulent diffusion. The meridional circulation works as a conveyor belt and governs the dynamo cycle period. Meridional circulation and magnetic diffusivity together govern the memory of the Sun's past magnetic fields. After describing the physical processes involved in a flux-transport dynamo, we will show that a predictive tool can be built from it to predict mean solar cycle features by assimilating magnetic field data from previous cycles. We will discuss the theoretical and observational connections among various predictors, such as dynamo-generated toroidal flux integral, cross-equatorial flux, polar fields and geomagnetic indices. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solar magnetic fields and convection

Recent developments in solar dynamo and other theories of magnetic fields and convection are discussed and extended. A basic requirement of these theories, that surplus fields are eliminated by turbulent or eddy diffusion, is shown to be invalid. A second basic requirement, that strong surface fields are created by granule or supergranule motions, is shown to be improbable. Parker's new thin-filament dynamo, based on the Petschek mechanism, is shown to provide the alternative possibilities: either the magnetic fields halt all convection or a steady state is reached in which the fields are a tangle of long, thin filaments. From the above and other considerations it is concluded that the dynamo and related diffuse-field theories are unacceptable, that solar magnetic fields are not dominated by convection, and that all the fields emerge as strong, concentrated fields (flux ropes) which were wound and twisted from a permanent, primordial field. The discussion may, incidentally, provide the physical elements of a deductive theory of hydromagnetic convection.     

On the generation of the large-scale and turbulent magnetic fields in solar-type stars

It is thought that the large-scale solar-cycle magnetic field is generated in a thin region at the interface of the radiative core (RC) and solar convection zone (SCZ). We show that the bulk of the SCZ virogoursly generates a small-scale turbulent magnetic field. Rotation, while not essential, increases the generation rate of this field.Thus, fully convective stars should have significant turbulent magnetic fields generated in their lower convection zones. In these stars the absence of a radiative core, i.e., the absence of a region of weak buoyancy, precludes the generation of a large-scale magnetic field, and as a consequence the angular momentum loss is reduced. This is, in our opinion, the explanation for the rapid rotation of the M-dwarfs in the Hyades cluster.Adopting the Utrecht's group terminology, we argue that the residual chromospheric emission should have three distinctive components: the basal emission, the emission due to the large-scale field, and the emission due to the turbulent field, with the last component being particularly strong for low mass stars.In the conventional dynamo equations, the dynamo frequencies and the propagation of the dynamo wave towards the equator are based on the highly questionable assumption of a constant . Furthermore, meridional motions, a necessary consequence of the interaction of rotation with convection, are ignored. In this context we discuss Stenflo's results about the global wave pattern decomposition of the solar magnetic field and conclude that it cannot be interpreted in the framework of the conventional dynamo equations.We discuss solar dynamo theories and argue that the surface layers could be essential for the generation of the poloidal field. If this is the case an -effect would not be needed at the RC-SCZ interface (where the toroidal field is generated). The two central problems facing solar dynamo theories may the transport of the surface poloidal field to the RC-SCZ interface and the uncertainty about the contributions to the global magnetic field by the small-scale magnetic features.Visitor, National Solar Observatory, National Optical Astronomy Observatories.The National Optical Astronomy Observatories are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.     

On the Connection Between Mean Field Dynamo Theory and Flux Tubes

Mean field dynamo theory deals with various mean quantities and does not directly throw any light on the question of existence of flux tubes. We can, however, draw important conclusions about flux tubes in the interior of the Sun by combining additional arguments with the insights gained from solar dynamo solutions. The polar magnetic field of the Sun is of order 10 G, whereas the toroidal magnetic field at the bottom of the convection zone has been estimated to be 100000 G. Simple order-of-magnitude estimates show that the shear in the tachocline is not sufficient to stretch a 10 G mean radial field into a 100000 G mean toroidal field. We argue that the polar field of the Sun must get concentrated into intermittent flux tubes before it is advected to the tachocline. We estimate the strengths and filling factors of these flux tubes. Stretching by shear in the tachocline is then expected to produce a highly intermittent magnetic configuration at the bottom of the convection zone. The meridional flow at the bottom of the convection zone should be able to carry this intermittent magnetic field equatorward, as suggested recently by Nandy and Choudhuri (2002). When a flux tube from the bottom of the convection zone rises to a region of pre-existing poloidal field at the surface, we point out that it picks up a twist in accordance with the observations of current helicities at the solar surface.     

On The Torsional Oscillations In Babcock–Leighton Solar Dynamo Models

The torsional oscillations at the solar surface have been interpreted by Schüssler and Yoshimura as being generated by the Lorentz force associated with the solar dynamo. It has been shown recently that they are also present in the upper half of the solar convection zone (SCZ). With the help of a solar dynamo model of the Babcock–Leighton type studied earlier, the longitudinal component of the Lorentz force, L , is calculated, and its sign or isocontours, are plotted vs. time, t, and polar angle, (the horizontal and vertical axis respectively). Two cases are considered, (1) differential rotation differs from zero only in the tachocline, (2) differential rotation as in (1) in the tachocline, and purely latitudinal and independent of depth in the bulk of the SCZ. In the first case the sign of L is roughly independent of latitude (corresponding to vertical bands in the t, plot), whereas in the second case the bands show a pole–equator slope of the correct sign. The pattern of the bands still differs, however, considerably from that of the helioseismic observations, and the values of the Lorentz force are too small at low latitudes. It is all but certain that the toroidal field that lies at the origin of the large bipolar magnetic regions observed at the surface, must be generated in the tachocline by differential rotation; the regeneration of the corresponding poloidal field, B _p has not yet been fully clarified. B _p could be regenerated, for example, at the surface (as in Babcock–Leighton models), or slightly above the tachocline, (as in interface dynamos). In the framework of the Babcock-Leighton models, the following scenario is suggested: the dynamo processes that give rise to the large bipolar magnetic regions are only part of the cyclic solar dynamo (to distinguish it from the turbulent dynamo). The toroidal field generated locally by differential rotation must contribute significantly to the torsional oscillations patterns. As this field becomes buoyant, it should give rise, at the surface, to the smaller bipolar magnetic regions as, e.g., to the ephemeral bipolar magnetic regions. These have a weak non-random orientation of magnetic axis, and must therefore also contribute to the source term for the poloidal field. Not only the ephemeral bipolar regions could be generated in the bulk of the SCZ, but many of the smaller bipolar regions as well (at depths that increase with their flux), all contributing to the source term for the poloidal field. In contrast to the butterfly diagram that provides only a very weak test of dynamo theories, the pattern of torsional oscillations has the potential of critically discriminating between different dynamo models.     

Flux-dominated solar dynamo model with a thin shear layer

Flux-dominated solar dynamo models have demonstrated to reproduce the main features of the large scale solar magnetic cycle, however the use of a solar like differential rotation profile implies in the the formation of strong toroidal magnetic fields at high latitudes where they are not observed. In this work, we invoke the hypothesis of a thin-width tachocline in order to confine the high-latitude toroidal magnetic fields to a small area below the overshoot layer, thus avoiding its influence on a Babcock-Leighton type dynamo process. Our results favor a dynamo operating inside the convection zone with a tachocline that essentially works as a storage region when it coincides with the overshoot layer. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solar and stellar activity: The theoretical approach

The unified sight of solar and stellar activity has revealed a worthwhile concept under several aspects, gaming in the last decade the increasing favour of observers and theorists, and the term solar-stellar connection has recently been introduced to point out the complementarity of solar and stellar observations in the background of the basic role played by the magnetic field.The great development of stellar activity observations suggests a much wider scenario than it were possible to imagine even a few years ago and stimulates theoretical work, most of which is in the framework of the - dynamo theory.Although dynamo theory seems to be plausible and successful in capturing the fundamental mechanism of solar and stellar activity, several uncertainties and intrinsic limits do still exist and are discussed together with alternative or complementary suggestions.Further, it is stressed the relevance of nonlinear problems in dynamo theory — as magnetoconvection, growth and stability of flux tubes against magnetic buoyancy, hydromagnetic global dynamos — to improve our understanding of both small and large scale interaction of rotation, turbulent convection and magnetic field, and of the transition from linear to nonlinear regime. Finally, recent dynamo models of stellar activity are critically reviewed, as to the dependence of activity indexes and cycles on rotation rate and spectral type.Open problems to be solved by future work are outlined, pointing out the role of ever increasing stellar data in widening out our comprehension of the dynamo operation modes, which seem to depend on stellar structure, rotation and age.     

On the origin and structure of stellar magnetic fields

Newly formed stars have magnetic fields provided by the compression of the interstellar field, and contrary to a widely accepted idea these fields are not destroyed by convective motions. For the same reason, the fallacy of ‘turbulent diffusion’, turbulent dynamo action is not possible in any star. Thus all stellar magnetic fields have a common origin, and persist throughout the lifetime of each star, including degenerate phases. This common origin, and a general similarity in stellar evolutionary processes, suggest that the fields may develop similar structural characteristics and MHD effects. This would open new possibilities of coordinating the studies of different types of stars and relating them to solar physics which has tended to become isolated from general stellar physics. As an initial step we consider three features of solar magnetic fields and their MHD effects. First, the solar magnetic field comprises two separate components: a poloidal field and a toroidal field. The former is a dipole field, permeating the entire Sun and closely aligned with the rotational axis; at the surface it is always concealed by much stronger elements of the toroidal field. The latter is probably wound from the former by differential rotation at latitudes below about 35°, where sections emerge through the solar surface and are then carried polewards. The second feature of solar magnetic fields is that all flux is concentrated into flux tubes of strength some kG, isolated within a much larger volume of non-magnetic plasma. The third feature is that the flux tubes are helically twisted into flux ropes (up to ?10²²Mx) and smaller elements ranging down to flux fibres (? 10¹⁸Mx). Some implications of similar features in other stars are discussed.     

A Rossby-wave dynamo for the Sun,II

To make the analysis more tractable, we simplify the equations of Part I to apply to two superposed layers of fluid, with horizontal variations in the motion and magnetic field represented by a small number of Fourier harmonics. The resulting set of eighteen ordinary nonlinear differential equations in time for the Fourier amplitudes is integrated numerically. We analyze in detail the dynamo action from a typical Rossby wave motion and compare it with the solar cycle.The field reversal process is similar in some respects to that put forth by Babcock. Toroidal fields are dragged up by vertical motions in the Rossby waves to form large-scale vertical fields, whose polarities alternate with longitude roughly like bipolar magnetic regions. Vertical fields of preferentially one polarity are carried toward the pole by the meridional motion in the wave to form an axisymmetric poloidal field. This poloidal field is then stretched out by the differential rotation into a new toroidal field of the opposite sign from the original. The poloidal field changes sign when the toroidal and bipolar region like fields are maximum, and vice versa.For the case studied, the reversal period is too short ( 2 years) and the poloidal fields too large ( 40 G) for the sun. Improvements for the model are discussed.Part I has been published in Solar Phys. 8, 316.     

Regularities in the growth of background magnetic fields

The cyclicity in the latitudinal distribution of the growth and decay rates of the total magnetic fluxes for weak magnetic fields is investigated. The synoptic maps of the line-of-sight solar magnetic field strength obtained at the Kitt Peak Observatory (USA) from January 1, 1977, to September 30, 2003, are used as the observational material. The latitudinal distributions of the growth rates of total magnetic fluxes with various strengths constructed from them and their evolution during three solar cycles have been compared with the analogous distribution of the total powers of rotation with various periods as well as the relative sunspot numbers and areas. The results obtained allow a unified picture of the development of solar cycles for weak and strong magnetic fields to be formulated. A new cycle begins with the growth of weak magnetic fields with a strength of 0–200 G at latitudes 20°–25° in both hemispheres. This occurs one year before the activity minimum determined from sunspots. Two years later, the growth rate of the total magnetic flux, which begins to propagate equatorward and poleward, reaches a maximum. This process coincides with the onset of the growth of strong sunspot magnetic fields at the corresponding latitudes and the formation of zones with a stable rotation. Subsequently, a fall-off in growth rate and then a flux decay for weak magnetic fields correspond to the growth of the sunspot areas. In light of the dynamo theory, the results obtained suggest that strong and weak magnetic fields are generated near the bottom of the convection zone, while the observed differences in their behavior are determined by the interaction of emerging magnetic flux tubes of various strengths with turbulent plasma motions inside the Sun.     

The flux ejection dynamo with small diffusivity

A number of examples are worked out to illustrate the consequences of reverse flux ejection from the surface of a convective layer of conducting fluid. Generally the reverse flux ejection has the opposite effect of magnetic buoyancy, tending to bury the fields rather than bringing them through the surface. Even a weak flux ejection effect prevents the excape of magnetic field through the surface. Reverse flux ejection at the surface of an -dynamo profoundly alters the character of the solutions of the dynamo equations. Altogether, flux ejection serves to obscure the interpretation of magnetic observations. The outstanding problem now is to determine under what circumstances there exists cyclonic convection with rotations in excess of ±1/2 in the rising columns of fluid. Negative turbulent diffusion is expected to be a close companion of the flux ejection effect.This work was supported by the National Aeronautics and Space Administration under grant NGL 14-001-001.     

HALF-WIDTH OF A SOLAR DYNAMO WAVE IN PARKER'S MIGRATORY DYNAMO

A kinematic -dynamo model of magnetic field generation in a thin convection shell with nonuniform helicity for large dynamo numbers is considered in the framework of Parker's migratory dynamo. The asymptotic solution obtained of equations governing the magnetic field has the form of an anharmonic travelling dynamo wave. This wave propagates over most latitudes of the solar hemisphere from high latitudes to the equator, and the amplitude of the magnetic field first increases and then decreases with propagation. Over the subpolar latitudes, the dynamo wave reverses; there the dynamo wave propagates polewards and decays with latitude. The half-width of the maximum of the magnetic field localisation and the phase velocity of the dynamo wave are calculated. Butterfly diagrams are plotted and analysed and these show that even a simple model may reveal some properties of the solar magnetic fields.     

Dynamo action in the presence of an imposed magnetic field

Dynamo action within the cores of Ap stars may offer intriguing possibilities for understanding the persistent magnetic fields observed on the surfaces of these stars. Deep within the cores of Ap stars, the coupling of convection with rotation likely yields magnetic dynamo action, generating strong magnetic fields. However, the surface fields of the magnetic Ap stars are generally thought to be of primordial origin. Recent numerical models suggest that a primordial field in the radiative envelope may possess a highly twisted toroidal shape. We have used detailed 3-D simulations to study the interaction of such a twisted magnetic field in the radiative envelope with the core-dynamo operating in the interior of a 2 solar mass A-type star. The resulting dynamo action is much more vigorous than in the absence of such a fossil field, yielding magnetic field strengths (of order 100 kG) much higher than their equipartition values relative to the convective velocities. We examine the generation of these fields, as well as the growth of large-scale magnetic structure that results from imposing a fossil magnetic field. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Constraints on the Solar Internal Magnetic Field from a Buoyancy Driven Solar Dynamo

We report here results from a dynamo model developed on the lines of the Babcock-Leighton idea that the poloidal field is generated at the surface of the Sun from the decay of active regions. In this model magnetic buoyancy is handled with a realistic recipe – wherein toroidal flux is made to erupt from the overshoot layer wherever it exceeds a specified critical field B _c (10⁵ G). The erupted toroidal field is then acted upon by the α-effect near the surface to give rise to the poloidal field. In this paper we study the effect of buoyancy on the dynamo generated magnetic fields. Specifically, we show that the mechanism of buoyant eruption and the subsequent depletion of the toroidal field inside the overshoot layer, is capable of constraining the magnitude and distribution of the magnetic field there. We also believe that a critical study of this mechanism may give us new information regarding the solar interior and end with an example, where we propose a method for estimating an upper limit of the difusivity within the overshoot layer. This revised version was published online in July 2006 with corrections to the Cover Date.     

Wound magnetostatics and flaring

We discuss the winding of a force-free axisymmetric magnetic field rooted on a heavy conductor onz=0. In quadrupolar symmetry the field expands in the half-spacez>0 and the toroidal flux concentrates on a conical surface. After a mean twist of 208°, the conical layer hosts large toroidal current loops with reversal of the magnetic flux on either side. The evolution of the field structure is described by scale-free static solutionsBr ^–(p+2), withp taking values between 0 and 2. The large expansion factor of the field structure is suggestive of flaring originating on the solar photosphere.     

Coronal loops and active region structure

We intercompared synoptic H, Ca K, magnetograph and Skylab soft X-ray and EUV data for the purpose of identifying the basic coronal magnetic structure of loops in a typical active region and studying its evolution. We focussed on a complex of activity in July 1973, especially McMath 12417. Our principal results are: (1) Most of the brightest loops connected the bright f plage to either the sunspot penumbra or to p satellite spots; no non-flaring X-ray loops end in umbrae; (2) short, bright loops had one or both ends in regions of emergent flux, strong fields or high field gradients; (3) stable, strongly sheared loop arcades formed over filaments; (4) EFRs were always associated with compact X-ray arcades; and (5) loops connecting to other active regions had their bases in outlying plage of weak field strength in McM 417 where H fibrils marked the direction of the loops. We conclude that a typical loop brightens in response to magnetic field activity at its feet, which heats the plasma. This suggests that the loop acts as a trap for gas convected from its base.     

The flux-rope-fibre theory of solar magnetic fields

The flux-rope theory of solar magnetic fields is reviewed briefly and, together with the dynamo theory, compared with various observational results. Dynamo and related theories are based on fields controlled by the plasma, and it is shown that such fields cannot account for the strong surface fields or even emerge without becoming tangled. Observations which appear uniquely explicable in terms of powerful (4000 G), helically twisted flux ropes and their many twisted flux fibres (3×10¹⁸ Mx) are listed as follows. (i) Emerging magnetic flux is seen first as pairs of small, closely spaced flux concentrations whose motions suggest magnetic control to provide bipolar regions of extent10⁵ km. The associated system of arch filaments rotates on the disk as would a series of emerging flux fibres twisted into a rope. (ii) Sunspots form by the accretion of pores and magnetic knots of like polarity, sometimes moving along curved paths between stationary elements of opposite polarity. (iii) Fluxes of10²² Mx in large sunspots must have been concentrated to strengths of4000 G before emerging, and also strongly helically twisted to avoid the flute instability. (iv) The trumpet-shaped flux-rope-fibre sunspot model (Figure 6) accounts readily for the phenomena of the moat convection, the sunspot energy deficit, the complex Evershed flow, penumbral filaments (flux 3×10¹⁸ Mx) and temporary light bridges. (v) Asymmetries in sunspot groups (in spot size, lifetime and proper motion) show that the spot fields are extensions of two submerged magnetic structures comprising strong fields. (vi) Sunspots decay by the loss of magnetic knots with strong fields and flux 5×10¹⁸ Mx. These must be isolated flux tubes, twisted to account for their stability. (vii) Flux fibres leaving a spot are prone to the kink instability, thus accounting for their sudden appearance in pairs, the transport of total flux several times that of the spot and net flux equal to that of the spot. (viii) Ephemeral active regions and X-ray bright points are explained similarly without invoking improbably huge quantities of new flux. (ix) Atmospheric structures show a high prevalence of helical twists (force-free fields) and rotary motions on all scales from spicules to large prominences. It is difficult to account for these twists unless they are present in emerging flux. (x) In and above the photosphere the flux fibres (3×10¹⁸ Mx) fray into loose associations of flux threads (3×10¹⁷ Mx) to provide a simple, selfconsistent model of the solar filigree and the chromospheric rosette (bush) with its group of mottles (spicules). (xi) Global patterns of surface and coronal magnetic fields reveal puzzling features such as the migration of large unipolar regions and the freedom from differential rotation of some structures. Submerged flux ropes peeling out of the Sun provide a starting point for explaining these effects. These results provide a strong case for the flux-rope theory against the entrenched dynamo theory, and suggest that more observations should be made of the above ten phenomena. Where possible, simultaneous observations should be made of Zeeman effects and of plasma distributions and velocity field seen in white light and spectral lines.