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.     

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.     

Formation of buoyant magnetic structures by a localized velocity shear

Motivated by considerations of the solar tachocline, we study the generation of strong buoyant magnetic structures by a sheared velocity field localized in a convectively stable background, using non-linear three-dimensional (3D) magnetohydrodynamic (MHD) simulations. The shear flow can spontaneously create strong tube-like toroidal (streamwise) magnetic structures from an imposed weak uniform poloidal (cross-stream) magnetic field. The structures are magnetically buoyant and therefore rise, and may evolve further to a rich variety of geometries, including kinked or arched shapes. The emergence process can repeat indefinitely with a characteristic period. These mechanisms may be relevant to the MHD processes in the solar tachocline and the creation and emergence of solar active regions.     

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.     

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.     

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     

Double magnetic cycle of solar activity

The source of the poloidal magnetic field was fixed using a uniform series of surface low-resolution magnetic field observations begun at Wilcox Solar Observatory at Stanford. The results obtained confirm the idea that low-frequency dynamo waves with a period approximately equal to 22 years and a high-frequency wave of a quasi-two-year period can coexist. It seems that an interaction between these components in the convection zone takes place on the Sun. Surface large-scale solar magnetic fields are analyzed using a two-dimensional Fourier method technique to study the poloidal field distribution. The first harmonic approximately equals the period of the magnetic cycle, appears at all latitudes, and reaches its the maximum value in the polar regions. Moreover, spectral analyses of axisymmetric magnetic field derivative in time found that the second important harmonic of a period approximately equal to two years appears at all latitudes. This second high-frequency harmonic dominates the polar latitude regions at the same time as the low-frequency one.     

A New Explanation for the Origin of Trans-equatorial Loops based on a Dynamo Model

Trans-equatorial loops (TLs) are one of the distinct magnetic structures in the solar corona and have a close relationship to solar activity. We present a systematic study of the origin of TLs linking with the Babcock?–?Leighton dynamo process based on the model of Chatterjee, Nandy, and Choudhuri (Astron. Astrophys. 427, 1019, 2004). We propose that TLs are visible signatures of poloidal field lines across the equator. The cycle variation of TL lengths obtained by the connectivities of poloidal field lines happens to be roughly in agreement with what one gets by considering the positions of sunspots. This explains why this cycle variation of TL lengths was found to be in conformity with Spörer’s law. The active regions always make the poloidal field configuration favorable to form TLs, which causes the conformity. The formation of TLs is a three-dimensional problem, which will require three-dimensional dynamo models for full investigation.     

On the possibility of a bimodal solar dynamo

A simple way to couple an interface dynamo model to a fast tachocline model is presented, under the assumption that the dynamo saturation is due to a quadratic process and that the effect of finite shear layer thickness on the dynamo wave frequency is analogous to the effect of finite water depth on surface gravity waves. The model contains one free parameter which is fixed by the requirement that a solution should reproduce the helioseismically determined thickness of the tachocline. In this case it is found that, in addition to this solution, another steady solution exists, characterized by a four times thicker tachocline and 4–5 times weaker magnetic fields. It is tempting to relate the existence of this second solution to the occurrence of grand minima in solar activity. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the out of phase appearance of large-scale diffuse magnetic field of the Sun with respect to sunspots

We assume the large-scale diffuse magnetic field of the Sun to originate from the poloidal component of a dynamo operating at the base of the convection zone, whereas the sunspots are due to the toroidal component. The evolution of the poloidal component is studied to model the poleward migration of the diffuse field seen on the solar surface and the polar reversal at the time of sunspots maxima (Dikpati and Choudhuri 1994, 1995).     

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     

On the large-scale diffuse magnetic field of the Sun – II. The Contribution of Active Regions

Dikpati and Choudhuri (1994, 1995) developed a model for the poleward migration of the weak diffuse magnetic field on the Sun's surface. This field was identified with the poloidal component produced by the solar dynamo operating at the base of the convection zone, and its evolution was studied by considering the effects of meridional circulation and turbulent diffusion. The earlier model is extended in this paper by incorporating the flux from, the decay of tilted active regions near the solar surface as an additional source of the poloidal field. This extended model can now explain various low-latitude features in the time-latitude diagram of the weak diffuse fields. These low-latitude features could not be accounted for in the earlier model, which was very successful in modeling the behavior at high latitudes. The time-latitude diagrams show that regions of a particular polarity often have tongues of opposite polarity. Such tongues can be produced in the theoretical model by incorporating fluctuations in the source term arising out of the decaying active regions.     

The Kinematic Theory of Solar Dynamo

Generation of the Sun‘s magnetic fields by self-inductive processes in the solar electrically conducting interior, the solar dynamo theory, is a fundamentally important subject in astrophysics. The kinematic dynamo theory concerns how the magnetic fields are produced by kinematically possible flows without being constrained by the dynamic equation. We review a number of basic aspects of the kinematic dynamo theory, including the magnetohydrodynamic approximation for the dynamo equation, the impossibility of dynamo action with the solar differential rotation, the Cowling‘s anti-dynamo theorem in the solar context, the turbulent alpha effect and recently constructed three-dimensional interface dynamos controlled by the solar tachocline at the base of the convection zone.     

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)     

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.     

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.     

Dynamo action in simulations of penetrative solar convection with an imposed tachocline

We summarize new and continuing three-dimensional spherical shell simulations of dynamo action by convection allowed to penetrate downward into a tachocline of rotational shear. The inclusion of an imposed tachocline allows us to examine several processes believed to be essential in the operation of the global solar dynamo, including differential rotation, magnetic pumping, and the stretching and organization of fields within the tachocline. In the stably stratified core, our simulations reveal that strong axisymmetric magnetic fields (of ∼ 3000 G strength) can be built, and that those fields generally exhibit a striking antisymmetric parity, with fields in the northern hemisphere largely of opposite polarity to those in the southern hemisphere. In the convection zone above, fluctuating fields dominate over weaker mean fields. New calculations indicate that the tendency toward toroidal fields of antisymmetric parity is relatively insensitive to initial magnetic field configurations; they also reveal that on decade-long timescales, the magnetic fields can briefly enter (and subsequently emerge from) states of symmetric parity.We have not yet observed any overall reversals of the field polarity, nor systematic latitudinal propagation. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

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.     

Stability of the solar tachocline with magnetic fields

The stability of magnetic fields in the solar tachocline is investigated. We present stability limits for higher azimuthal wave numbers and results on the dependence of the stability on the location of toroidal magnetic fields in latitude. While the dependence of the wave number with the largest growth rate on the magnetic field strength and the magnetic Prandtl number is small, the dependence on the magnetic Reynolds number Rm indicates that lowest azimuthal modes are excited for very high Rm. Upon varying the latitudinal position of the magnetic field belts, we find slightly lower stability limits for high latitudes, and very large stability limits at latitudes below 10°, with little dependence on latitude in between. An increase of the maximum possible field was achieved by adding a poloidal field. The upper limit for the toroidal field which can be stored in the radiative tachocline is then 1000 G, compared to about 100 G for a purely toroidal field as was found in an earlier work. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)