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.     

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)     

Helioseismology and the solar cycle: Past, present and future

A major goal of helioseismology is to understand the mechanism of the solar cycle. In this paper, some results of helioseismic observations relevant to the cycle are briefly reviewed, the current state-of-the-art is discussed, and near-term future directions are sketched out. Topics covered include the internal rotation rate; activity-related parameter variations; the tachocline; far-side imaging; the torsional oscillation; and meridional flows.     

A study of possible temporal and latitudinal variations in the properties of the solar tachocline

Temporal variations of the structure and the rotation rate of the solar tachocline region are studied using helioseismic data from the Global Oscillation Network Group (GONG) and the Michelson Doppler Imager (MDI) obtained during the period 1995–2000. We do not find any significant temporal variation in the depth of the convection zone, the position of the tachocline or the extent of overshoot below the convection zone. No systematic variation in any other properties of the tachocline, like width, etc., is found either. The possibility of periodic variations in these properties is also investigated. Time-averaged results show that the tachocline is prolate with a variation of about 0.02 R_⊙ in its position. Neither the depth of the convection zone nor the extent of overshoot shows any significant variation with latitude.     

Fluctuations of the Solar Dynamo Observed in the Solar Wind and Interplanetary Magnetic Field at 1 AU and in the Outer Heliosphere

Recent helioseismic observations have found strong fluctuations at a period of about 1.3 years in the rotation speed around the tachocline in the deep solar convection layer. Similar mid-term quasi-periodicities (MTQP; periods between 1–2 years) are known to occur in various solar atmospheric and heliospheric parameters for centuries. Since the deep convection layer is the expected location of the solar magnetic dynamo, its fluctuations could modulate magnetic flux generation and cause related MTQP fluctuations at the solar surface and beyond. Accordingly, it is likely that the heliospheric MTQP periodicities reflect similar changes in solar dynamo activity. Here we study the occurrence of the MTQP periodicities in the near and distant heliosphere in the solar wind speed and interplanetary magnetic field observed by several satellites at 1 AU and by four interplanetary probes (Pioneer 10 and 11 and Voyager 1 and 2) in the outer heliosphere. The overall structure of MTQP fluctuations in the different locations of the heliosphere is very consistent, verifying the solar (not heliospheric) origin of these periodicities. We find that the mid-term periodicities were particularly strong during solar cycle 22 and were observed at two different periods of 1.3 and 1.7 years simultaneously. These periodicities were latitudinally organized so that the 1.3-year periodicity was found in solar wind speed at low latitudes and the 1.7-year periodicity in IMF intensity at mid-latitudes. While all heliospheric results on the 1.3-year periodicity are in a good agreement with helioseismic observations, the 1.7-year periodicity has so far not been detected in helioseismic observations. This may be due to temporal changes or due to the helioseismic method where hemispherically antisymmetric fluctuations would so far have remained hidden. In fact, there is evidence that MTQP fluctuations may occur antisymmetrically in the northern and southern solar hemisphere. Moreover, we note that the MTQP pattern was quite different during solar cycles 21 and 22, implying fundamental differences in solar dynamo action between the two halves of the magnetic cycle.     

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)     

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)     

Variations of helioseismic parameters due to magnetic field generated by a flux transport model

The change of sound speed has been found at the base of the convection during the solar cycles,which can be used to constrain the solar internal magnetic field.We aim to check whether the magnetic field generated by the solar dynamo can lead to the cyclic variation of the sound speed detected through helioseismology.The basic configuration of magnetic field in the solar interior was obtained by using a Babcock-Leighton(BL) type flux transport dynamo.We reconstructed one-dimensional solar models by assimilating magnetic field generated by an established dynamo and examined their influences on the structural variables.The results show that magnetic field generated by the dynamo is able to cause noticeable change of the sound speed profile at the base of the convective zone during a solar cycle.Detailed features of this theoretical prediction are also similar to those of the helioseismic results in solar cycle 23 by adjusting the free parameters of the dynamo model.     

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.     

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     

Global solar dynamo models: Simulations and predictions

Flux-transport type solar dynamos have achieved considerable success in correctly simulating many solar cycle features, and are now being used for prediction of solar cycle timing and amplitude. We first define flux-transport dynamos and demonstrate how they work. The essential added ingredient in this class of models is meridional circulation, which governs the dynamo period and also plays a crucial role in determining the Sun’s memory about its past magnetic fields. We show that flux-transport dynamo models can explain many key features of solar cycles. Then we show that a predictive tool can be built from this class of dynamo that can be used to predict mean solar cycle features by assimilating magnetic field data from previous cycles.     

A Consistent one-Dimensional Model for the Turbulent Tachocline

The first consistent model for the turbulent tachocline is presented, with the turbulent diffusivity computed within the model instead of being specified arbitrarily. For the origin of the 3D turbulence a new mechanism is proposed. Owing to the strongly stable stratification, the mean radial shear is stable, while the horizontal shear is expected to drive predominantly horizontal, quasi-2D motions in thin slabs. Here I suggest that a major source of 3D overturning turbulent motions in the tachocline is the secondary shear instability due to the strong, random vertical shear arising between the uncorrelated horizontal flows in neighboring slabs. A formula for the vertical diffusivity due to this turbulence, Equation (9), is derived and applied in a simplified 1D model of the tachocline. It is found that Maxwell stresses due to an oscillatory poloidal magnetic field of a few hundred gauss are able to confine the tachocline to a thickness less than 5 Mm. The integral scale of the 3D overturning turbulence is the buoyancy scale, on the order of 10 km, and its velocity amplitude is a few m s^–1, yielding a vertical turbulent diffusivity on the order of 10⁸ cm² s^–1.     

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.     

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.     

Long Term Evolution of Solar Meridional Circulation and Phase Synchronization Viewed Through a Symmetrical Kuramoto Model

The solar-cycle oscillations of the toroidal and poloidal components of the solar magnetic field in the northern solar hemisphere have a persistent phase difference of about \(\pi \). We propose a symmetrical Kuramoto model with three coupled oscillators as a simple way to understand this anti-synchronization. We solve an inverse problem and reconstruct natural frequencies of the top and bottom oscillators under the conditions of a constant coupling strength and a non-delayed coupling. These natural frequencies are associated with angular velocities of the meridional flow circulation near the solar surface and in the deep layer of the solar convection zone. A relationship between our reconstructions of the shallow and the deep meridional flow speed during recent Solar Cycles 21?–?23 is in agreement with estimates obtained in helioseismology and flux-transport dynamo modeling. The reconstructed top oscillator speed presents significant solar-cycle like variations that agree with recent helioseismical reconstructions. The evolution of reconstructed natural frequencies strongly depends on the coupling strength. We find two stable regimes in the case of strong coupling with a change of regime during anomalous solar cycles. We see the onset of a new transition in Solar Cycle 24. We estimate the admitted range of coupling values and find evidence of cross-equatorial coupling between solar hemispheres not accounted for by the model.     

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.     

Meridional circulation and the period of the solar dynamo cycle

We consider the conditions in the transition from the tachocline to the solar convective zone with changing diffusion coefficient. The topology of the magnetic fields involved in the solar dynamo is revised under the assumption that intermediate fields (of the order of 10 mT) have a dominant role in generating the fields in new cycle. The inclusion of meridional circulation is found to increase the dynamo wave period in comparison to the observed period. This suggests that the αΩ-effects are unimportant in calculating the solar cycle period but hold significance in determining the cycle amplitude.     

Does the tachocline show solar cycle related changes?

The tachocline at the base of the convection zone is generally believed to be the seat of the solar dynamo. Here we investigate whether the tachocline shows any detectable change using several 72 day time-series of the Michelson Doppler Imager (MDI) Medium-l data. We do not find any clear evidence of change with time.     

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)     

On helioseismic tests of basic physics

An important goal of helioseismology is to provide information about the basic physics and parameters that determine the structure of the solar interior. Here we discuss the procedures applied in such analyses, using as an example attempts to obtain significant constraints on the value of Newton's gravitational constant G from helioseismology. The analysis is based on complete direct and inverse helioseismic analysis of a set of accurate observed acoustic frequencies. We confirm, as found by previous investigations based on different approaches, that the actual level of precision of the helioseismic inferences does not allow us to constrain G with a precision better than that which can be reached with direct experimental measurements. The conclusion emphasizes the importance in helioseismic inferences of considering not only the accuracy with which solar oscillations are measured, but also the effect of uncertainties in other aspects of the model computation and helioseismic analysis.