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)     

Stability of toroidal magnetic fields in the solar tachocline and beneath

Stability of toroidal magnetic field in a stellar radiation zone is considered for the cases of uniform and differential rotation. In the rigidly rotating radiative core shortly below the tachocline, the critical magnetic field for instability is about 600 G. The unstable disturbances for slightly supercritical fields have short radial scales ∼1 Mm. Radial mixing produced by the instability is estimated to conclude that the internal field of the sun can exceed the critical value of 600 G only marginally. Otherwise, the mixing is too strong and not compatible with the observed lithium abundance. Analysis of joint instability of differential rotation and toroidal field leads to the conclusion that axisymmetric models of the laminar solar tachocline are stable to nonaxisymmetric disturbances. The question of whether sun-like stars can posses tachoclines is addressed with positive answer for stars with rotation periods shorter than about two months. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Steps for Building a Calibrated Flux-Transport Dynamo for the Sun

Given the complexity involved in a flux-transport-type dynamo driven by both Babcock – Leighton and tachocline α effects, we present here a step-by-step procedure for building a flux-transport dynamo model calibrated to the Sun as a guide for anyone who wishes to build this kind of model. We show that a plausible sequence of steps to reach a converged solution in such a dynamo consists of (i) numerical integration of a classical α – ω dynamo driven by a tachocline α effect, (ii) continued integration with inclusion of meridional circulation to convert the model into a flux-transport dynamo driven by only a tachocline α effect, (iii) final integration with inclusion of a Babcock – Leighton surface α effect, resulting in a flux-transport dynamo that can be calibrated to obtain a close fit of model output with solar observations.     

A dynamo model for axisymmetric and non-axisymmetric solar magnetic fields

More and more observations are showing a relatively weak, but persistent, non-axisymmetric magnetic field co-existing with the dominant axisymmetric field on the Sun. Its existence indicates that the non-axisymmetric magnetic field plays an important role in the origin of solar activity. A linear non-axisymmetric  α²– Ω  dynamo model is derived to explore the characteristics of the axisymmetric  ( m = 0)  and the first non-axisymmetric  ( m = 1)  modes and to provide a theoretical basis with which to explain the 'active longitude', 'flip-flop' and other non-axisymmetric phenomena. The model consists of an updated solar internal differential rotation, a turbulent diffusivity varying with depth, and an α-effect working at the tachocline in a rotating spherical system. The difference between the  α²–Ω  and the  α–Ω  models and the conditions that favour the non-axisymmetric modes under solar-like parameters are also presented.     

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)     

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)     

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.     

Self-excitation of motions in near-convection zones and the generation of solar magnetic field

In view of the recently discovered time variations in rotation velocity within the solar differentially rotating tachocline (Howe et al. 2000), we study conditions for the equilibrium and excitation of motions in nonrigidly rotating magnetized layers of the radiative zones located near the boundaries of the convection zone. The emphasis is on the possible relationship between quasi-periodic tachocline pulsations and the generation of a nonaxisymmetric magnetic field in the convection zone. This field generation is studied under the assumption that it results from a reduction in the expenditure of energy on convective heat transport. The (antisymmetric about the equator) field is shown to increase in strength if there are both a radial gradient in angular velocity and steady-state axisymmetric meridional circulation of matter. The sense of circulation is assumed to change (causing the sign of the generated field to change) after the maximum permissible field strength is reached. This is apparently attributable to the excitation of the corresponding turbulent viscosity of the medium. It is also important that the cyclic field variations under discussion are accompanied by variations in solar-type dipole magnetic field.     

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)     

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)     

A two-layer αω-dynamo model with dynamic feedback on the ω-effect

In an attempt to produce a simple representation of an interface dynamo, I examine a dynamo model composed of two one-dimensional (radially averaged) pseudo-spherical layers, one in the convection zone and possessing an α-effect, and the other in the tachocline and possessing an ω-effect. The two layers communicate by means of an analogue of Newton's law of cooling, and a dynamical back-reaction of the magnetic field on ω is provided. Extensive bifurcation diagrams are calculated for three separate values of η, the ratio of magnetic diffusivities of the two layers. I find recognizable similarities to, but also dramatic differences from, the comparable one-layer model examined by Roald &38; Thomas. In particular, the solar-like dynamo mode found previously is no longer stable in the two-layer version; in its place there is a sequence of periodic, quasi-periodic and chaotic modes probably created in a homoclinic bifurcation. These differences are important enough to provide support for the view that the solar dynamo cannot be meaningfully modelled in one dimension.     

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.     

Evidence for interchange reconnection between a coronal hole and an adjacent emerging flux region

Coronal holes are regions of dominantly monopolar magnetic field on the Sun where the field is considered to be ‘open’ towards interplanetary space. Magnetic bipoles emerging in proximity to a coronal hole boundary naturally interact with this surrounding open magnetic field. In the case of oppositely aligned polarities between the active region and the coronal hole, we expect interchange reconnection to take place, driven by the coronal expansion of the emerging bipole as well as occasional eruptive events. Using SOHO/EIT and SOHO/MDI data, we present observational evidence of such interchange reconnection by studying AR 10869 which emerged close to a coronal hole. We find closed loops forming between the active region and the coronal hole leading to the retreat of the hole. At the same time, on the far side of the active region, we see dimming of the corona which we interpret as a signature of field line ‘opening’ there, as a consequence of a topological displacement of the ‘open’ field lines of the coronal hole. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

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.     

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     

Analytical theory of tachocline transport at mid latitudes

We provide a theory of magnetic diffusion, momentum transport, and mixing in the solar tachocline by considering magnetohydrodynamics (MHD) turbulence on a β plane subject to a large scale shear (provided by the latitudinal differential rotation). In the strong magnetic field regime, we find that the turbulent viscosity and diffusivity are reduced by magnetic fields only, similarly to the two-dimensional MHD case (without Rossby waves). In the weak magnetic field regime, we find a crossover scale (L_R) from a Alfvén dominated regime (on small scales) to a Rossby dominated regime (on large scales). For parameter values typical of the tachocline, L_R is larger than the solar radius so that Rossby waves are unlikely to play an important role in the transport of magnetic field and angular momentum. This is mainly due to the enhancement of magnetic back-reaction by shearing which efficiently generates small scales, thus strong currents. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamical processes in flux tabes and their role in chromospheric heating

We model the dynamical interaction between magnetic flux tubes and granules in the solar photosphere which leads to the excitation of transverse (kink) and longitudinal (sausage) tube waves. The investigation is motivated by the interpretation of network oscillations in terms of flux tube waves. The calculations show that for magnetic field strengths typical of the network, the energy flux in transverse waves is higher than in longitudinal waves by an order of magnitude. But for weaker fields, such as those that might be found in internetwork regions, the energy fluxes in the two modes are comparable. Using observations of footpoint motions, the energy flux in transverse waves is calculated and the implications for chromospheric heating are pointed out.     

Axial rotation, orbital revolution and solar spin–orbit coupling

The orbital motion of the Sun has been linked with solar variability, but the underlying physics remains unknown. A coupling of the solar axial rotation and the barycentric orbital revolution might account for the relationships found. Some recent published studies addressing the physics of this problem have made use of equations from rotational physics in order to model particle motions. However, our standard equations for rotational velocity do not accurately describe particle motions due to orbital revolution. The Sun's orbital motion is a state of free fall; in consequence, aside from very small tidal motions, the associated particle velocities do not vary as a function of position on or within the body of the Sun. In this note, I describe and illustrate the fundamental difference between particle motions in rotation and revolution, in order to dispel some part of the confusion that has arisen in the past and that which may yet arise in the future. This discussion highlights the principal physical difficulty that must be addressed and overcome by future dynamical spin–orbit coupling hypotheses.