Some recent developments in solar dynamo theory

We discuss the current status of solar dynamo theory and describe the dynamo model developed by our group. The toroidal magnetic field is generated in the tachocline by the strong differential rotation and rises to the solar surface due to magnetic buoyancy to create active regions. The decay of these active regions at the surface gives rise to the poloidal magnetic field by the Babcock-Leighton mechanism. This poloidal field is advected by the meridional circulation first to high latitudes and then down below to the tachocline. Dynamo models based on these ideas match different aspects of observational data reasonably well.     

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.     

Solar activity forecast with a dynamo model

Although systematic measurements of the Sun's polar magnetic field exist only from mid-1970s, other proxies can be used to infer the polar field at earlier times. The observational data indicate a strong correlation between the polar field at a sunspot minimum and the strength of the next cycle, although the strength of the cycle is not correlated well with the polar field produced at its end. This suggests that the Babcock–Leighton mechanism of poloidal field generation from decaying sunspots involves randomness, whereas the other aspects of the dynamo process must be reasonably ordered and deterministic. Only if the magnetic diffusivity within the convection zone is assumed to be high (of order  10¹² cm² s⁻¹  ), we can explain the correlation between the polar field at a minimum and the next cycle. We give several independent arguments that the diffusivity must be of this order. In a dynamo model with diffusivity like this, the poloidal field generated at the mid-latitudes is advected toward the poles by the meridional circulation and simultaneously diffuses towards the tachocline, where the toroidal field for the next cycle is produced. To model actual solar cycles with a dynamo model having such high diffusivity, we have to feed the observational data of the poloidal field at the minimum into the theoretical model. We develop a method of doing this in a systematic way. Our model predicts that cycle 24 will be a very weak cycle. Hemispheric asymmetry of solar activity is also calculated with our model and compared with observational data.     

P – ω Dynamo in Spherical Geometry

Based on the fundamental P – ω dynamo equation, using spherical polar coordinates, we carry out a study of turbulent plasma wave dynamo effect. For various rotation laws, different analytical solutions are derived. In the cases of no rotation and rigid rotation, the dynamo generates poloidal field only, while with differential rotation, regardless the differential rotation is radial or latitudinal, poloidal and toroidal fields are all generated. We may think that the solutions are the analytical forms of the magnetic field in a turbulent source region of celestial bodies. This revised version was published online in July 2006 with corrections to the Cover Date.     

A mean electromotive force induced by magnetic buoyancy instabilities

For a variety of reasons, based on results from magnetoconvection, self-consistent dynamo calculations and helioseismology, it seems plausible that the bulk of the solar magnetic field is located in the overshoot zone. Furthermore, it has also been suggested that the solar dynamo is operating in this region. The aim of this paper is then to show that it is possible to obtain a mean electromotive force (EMF), and hence an α -effect, in the convectively stable overshoot zone, which is driven by magnetic buoyancy instabilities.
By investigating the stability of a layer of magnetic field embedded between two non-magnetic layers of plasma we are able to show the following: first, that magnetic buoyancy instabilities indeed give rise to a mean EMF and, secondly, that the electromotive force is largest in the region where the magnetic layer is unstable, i.e. where the field strength decreases fastest with height.
Moreover, the influence of the rotation rate and the magnetic field strength on the magnetic buoyancy instability has been investigated in order to determine for which values of these parameters dynamo action might occur.     

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.     

How to Reach Superequipartition Field Strengths in Solar Magnetic Flux Tubes

A number of independent arguments indicate that the toroidal flux system responsible for the sunspot cycle is stored at the base of the convection zone in the form of flux tubes with field strength close to 10⁵ G. Although the evidence for such strong fields is quite compelling, how such field strength can be reached is still a topic of debate. Flux expulsion by convection should lead to about the equipartition field strength, but the magnetic energy density of a 10⁵-G field is two orders of magnitude larger than the mean kinetic energy density of convective motions. Line stretching by differential rotation (i.e., the “Ω effect” in the classical mean-field dynamo approach) probably plays an important role, but arguments based on energy considerations show that it does not seem feasible that a 10⁵-G field can be produced in this way. An alternative scenario for the intensification of the toroidal flux system in the overshoot layer is related to the explosion of rising, buoyantly unstable magnetic flux tubes, which opens a complementary mechanism for magnetic-field intensification. A parallelism is pointed out with the mechanism of “convective collapse” for the intensification of photospheric magnetic flux tubes up to field strengths well above equipartition; both mechanisms, which are fundamentally thermal processes, are reviewed.     

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.     

The Alpha Effect and the Observed Twist and Current Helicity of Solar Magnetic Fields

We present a straightforward comparison of model calculations for the α-effect, helicities, and magnetic field line twist in the solar convection zone with magnetic field observations at atmospheric levels. The model calculations are carried out in a mixing-length approximation for the turbulence with a profile of the solar internal rotation rate obtained from helioseismic inversions. The magnetic field data consist of photospheric vector magnetograms of 422 active regions for which spatially-averaged values of the force-free twist parameter and of the current helicity density are calculated, which are then used to determine latitudinal profiles of these quantities. The comparison of the model calculations with the observations suggests that the observed twist and helicity are generated in the bulk of the convection zone, rather than in a layer close to the bottom. This supports two-layer dynamo models where the large-scale toroidal field is generated by differential rotation in a thin layer at the bottom while the α-effect is operating in the bulk of the convection zone. Our previous observational finding was that the moduli of the twist factor and of the current helicity density increase rather steeply from zero at the equator towards higher latitudes and attain a certain saturation at about 12 – 15^∘. In our dynamo model with algebraic nonlinearity, the increase continues, however, to higher latitudes and is more gradual. This could be due to the neglect of the coupling between small-scale and large-scale current and magnetic helicities and of the latitudinal drift of the activity belts in the model.     

Solar cycle according to mean magnetic field data

To investigate the shape of the solar cycle, we have performed a wavelet analysis of the large–scale magnetic field data for 1960–2000 for several latitudinal belts and have isolated the following quasi-periodic components: ∼22, 7 and 2 yr. The main 22-yr oscillation dominates all latitudinal belts except the latitudes of ±30° from the equator. The butterfly diagram for the nominal 22-yr oscillation shows a standing dipole wave in the low-latitude domain  (∣θ∣≤ 30°)  and another wave in the sub-polar domain  (∣θ∣≥ 35°)  , which migrates slowly polewards. The phase shift between these waves is about π. The nominal 7-yr oscillation yields a butterfly diagram with two domains. In the low-latitude domain  (∣θ∣≤ 35°)  , the dipole wave propagates equatorwards and in the sub-polar region, polewards. The nominal 2-yr oscillation is much more chaotic than the other two modes; however the waves propagate polewards whenever they can be isolated.
We conclude that the shape of the solar cycle inferred from the large-scale magnetic field data differs significantly from that inferred from sunspot data. Obviously, the dynamo models for a solar cycle must be generalized to include large-scale magnetic field data. We believe that sunspot data give adequate information concerning the magnetic field configuration deep inside the convection zone (say, in overshoot later), while the large-scale magnetic field is strongly affected by meridional circulation in its upper layer. This interpretation suggests that the poloidal magnetic field is affected by the polewards meridional circulation, whose velocity is comparable with that of the dynamo wave in the overshoot layer. The 7- and 2-yr oscillations could be explained as a contribution of two sub-critical dynamo modes with the corresponding frequencies.     

A dynamo driven by zonal jets at the upper surface: Applications to giant planets

We present a dynamo mechanism arising from the presence of barotropically unstable zonal jet currents in a rotating spherical shell. The shear instability of the zonal flow develops in the form of a global Rossby mode, whose azimuthal wavenumber depends on the width of the zonal jets. We obtain self-sustained magnetic fields at magnetic Reynolds numbers greater than 10³. We show that the propagation of the Rossby waves is crucial for dynamo action. The amplitude of the axisymmetric poloidal magnetic field depends on the wavenumber of the Rossby mode, and hence on the width of the zonal jets. We discuss the plausibility of this dynamo mechanism for generating the magnetic field of the giant planets. Our results suggest a possible link between the topology of the magnetic field and the profile of the zonal winds observed at the surface of the giant planets. For narrow Jupiter-like jets, the poloidal magnetic field is dominated by an axial dipole whereas for wide Neptune-like jets, the axisymmetric poloidal field is weak.     

Transport Effects in the Evolution of the Global Solar Magnetic Field

The axisymmetric component of the large-scale solar magnetic fields has a pronounced poleward branch at higher latitudes. In order to clarify the origin of this branch we construct an axisymmetric model of the passive transport of the mean poloidal magnetic field in the convective zone, including meridional circulation, anisotropic diffusivity, turbulent pumping and density pumping. For realistic values of the transport coefficients we find that diffusivity is prevalent, and the latitudinal distribution of the field at the surface simply reflects the conditions at the bottom of the convective zone. Pumping effects concentrate the field to the bottom of the convective zone; a significant part of this pumping occurs in a shallow subsurface layer, normally not resolved in dynamo models. The phase delay of the surface poloidal field relative to the bottom poloidal field is found to be small. These results support the double dynamo wave models, may be compatible with some form of a mixed transport scenario, and exclude the passive transport theory for the origin of the polar branch.     

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)     

The dual role of shear in large‐scale dynamos

The role of shear in alleviating catastrophic quenching by shedding small‐scale magnetic helicity through fluxes along contours of constant shear is discussed. The level of quenching of the dynamo effect depends on the quenched value of the turbulent magnetic diffusivity. Earlier estimates that might have suffered from the force‐free degeneracy of Beltrami fields are now confirmed for shear flows where this degeneracy is lifted. For a dynamo that is saturated near equipartition field strength those estimates result in a 5‐fold decrease of the magnetic diffusivity as the magnetic Reynolds number based on the wavenumber of the energy‐carrying eddies is increased from 2 to 600. Finally, the role of shear in driving turbulence and large‐scale fields by the magneto‐rotational instability is emphasized. New simulations are presented and the 3π /4 phase shift between poloidal and toroidal fields is confirmed. It is suggested that this phase shift might be a useful diagnostic tool in identifying mean‐field dynamo action in simulations and to distinguish this from other scenarios invoking magnetic buoyancy as a means to explain migration away from the midplane. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Observations of magnetic flux compression in jet impact experiments

The astrophysical jet experiment at Caltech generates a T=2–5 eV, n=10²¹–10²² m⁻³ plasma jet using coplanar disk electrodes linked by a poloidal magnetic field. A 100 kA current generates a toroidal magnetic field; the toroidal field pressure inflates the poloidal flux surface, magnetically driving the jet. The jet travels at up to 50 km/s for ∼20–25 cm before colliding with a cloud of initially neutral gas. We study the interaction of the jet and the cloud in analogy to an astrophysical jet impacting a molecular cloud. Diagnostics include magnetic probe arrays, a 12-channel spectroscopic system and a fast camera with optical filters. When a hydrogen plasma jet collides with an argon target cloud, magnetic measurements show the magnetic flux compressing as the plasma jet deforms. As the plasma jet front slows and the plasma piles up, the density of the frozen-in magnetic flux increases.     

Turbulent dynamo near tachocline and reconstruction of azimuthal magnetic field in the solar convection zone

In order to extend the abilities of the αΩ dynamo model to explain the observed regularities and anomalies of the solar magnetic activity, the negative buoyancy phenomenon and the magnetic quenching of the α effect were included in the model, as well as newest helioseismically determined inner rotation of the Sun were used. Magnetic buoyancy constrains the magnitude of toroidal field produced by the Ω effect near the bottom of the solar convection zone (SCZ). Therefore, we examined two “antibuoyancy” effects: i) macroscopic turbulent diamagnetism and ii) magnetic advection caused by vertical inhomogeneity of fluid density in the SCZ, which we call the ∇ρ effect. The Sun's rotation substantially modifies the ∇ρ effect. The reconstruction of the toroidal field was examined assuming the balance between mean‐field magnetic buoyancy, turbulent diamagnetism and the rotationally modified ∇ρ effect. It is shown that at high latitudes antibuoyancy effects block the magnetic fields in the deep layers of the SCZ, and so the most likely these deep‐rooted fields could not become apparent at the surface as sunspots. In the near‐equatorial region, however, the upward ∇ρ effect can facilitate magnetic fields of about 3000 – 4000 G to emerge through the surface at the sunspot belt. Allowance for the radial inhomogeneity of turbulent velocity in derivations of the helicity parameter resulted in a change of sign of the α effect from positive to negative in the northern hemisphere near the bottom of the SCZ. The change of sign is very important for direction of the Parker's dynamo‐waves propagation and for parity of excited magnetic fields. The period of the dynamo‐wave calculated with allowance for the magnetic quenching is about seven years, that agrees by order of magnitude with the observed mean duration of the sunspot cycles. Using the modern helioseismology data to define dynamo‐parameters, we conclude that north‐south asymmetry should exist in the meridional field. At low latitudes in deep layers of the SCZ, the αΩ dynamo excites most efficiency the dipolar mode of the meridional field. Meanwhile, in high‐latitude regions a quadrupolar mode dominates in the meridional field. The obtained configuration of the net meridional field is likely to explain the magnetic anomaly of polar fields (the apparent magnetic “monopole”) observed near the maxima of solar cycles. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Magnetic field bending in accretion discs with dynamos

We consider the problem of poloidal magnetic field advection and bending of an initially vertical field owing to radial inflow in thin accretion discs. For a ratio of kinematic viscosity to magnetic diffusivity of order unity, significant bending of an externally applied vertical field cannot occur in a disc with no internal dynamo. However, we show that if poloidal field is generated by a dynamo operating near its critical state, then significant field bending may be possible. Our results are of particular relevance to wind launching from accretion discs.     

Flux expulsion by inhomogeneous turbulence

Flux expulsion is an important consequence of the interaction of magnetic fields with fluid convection and has been well studied for particular cases of steady, single-cell flows. Here we examine a related phenomenon in inhomogeneous turbulence using direct numerical simulations. To understand our numerical results, we analyse average properties of our model, and obtain mean transport coefficients which can be used to describe the approach of the system to its final state. For the kinematic problem these transport coefficients give an excellent prediction of the expulsion process; however, the enhanced transport is suppressed by dynamical back-reaction of the Lorentz force. Finally, we discuss the astrophysical implications for magnetic fields in stellar convection zones. Segregation of magnetic fields from turbulent motion not only allows strong toroidal fields to accumulate in regions of convective overshoot but also permits significant poloidal fields to be maintained by dynamo action in stars like the Sun.     

Tachocline Confinement by an Oscillatory Magnetic Field

Helioseismic measurements indicate that the solar tachocline is very thin, its full thickness not exceeding 4% of the solar radius. The mechanism that inhibits differential rotation to propagate from the convective zone to deeper into the radiative zone is not known, though several propositions have been made. In this paper we demonstrate by numerical models and analytic estimates that the tachocline can be confined to its observed thickness by a poloidal magnetic field B _p of about one kilogauss, penetrating below the convective zone and oscillating with a period of 22 years, if the tachocline region is turbulent with a diffusivity of η∼10¹⁰ cm² s⁻¹ (for a turbulent magnetic Prandtl number of unity). We also show that a similar confinement may be produced for other pairs of the parameter values (B _p, η). The assumption of the dynamo field penetrating into the tachocline is consistent whenever η≳10⁹ cm² s⁻¹. Supplementary material to this paper is available in electronic form at http://dx.doi.org/10.1023/A:1013389631585