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RevMexAA (Serie de Conferencias), 36, CD252CD261 (2009)

FLUX TRANSPORT SOLAR DYNAMO MODELS, CURRENT PROBLEMS

AND POSSIBLE SOLUTIONS

G. Guerrero1 and E. M. de Gouveia Dal Pino1

RESUMEN

El ciclo de manchas solares ha sido frecuentemente explicado como el resultado de un proceso de dnamo queopera en el sol. Este es un problema clasico en astrofsica que hasta el momento no se encuentra del todoresuelto. Aqu discutimos problemas actuales y limitantes del modelo del dnamo solar y posibles solucionesusando un modelo cinematico apoyados en una aproximacion de Babcock-Leighton. En particular se discute laimportancia del forzamiento por turbulencia magnetica versus la circulacion de flujo meridional en la operacionel dnamo.

ABSTRACT

The sunspot solar cycle has been usually explained as the result of a dynamo process operating in the sun.This is a classical problem in Astrophysics that until the present is not fully solved. Here we discuss currentproblems and limitations with the solar dynamo modeling and their possible solutions using the kinematicdynamo model with the Babcock-Leighton approximation as a tool. In particular, we discuss the importanceof the turbulent magnetic pumping versus the meridional flow circulation in the dynamo operation.

Key Words: Sun: magnetic fields

1. INTRODUCTION

The sunspot cycle is one of the most interestingmagnetic phenomenon in the Universe. It was dis-covered more than 150 years ago by Schwabe (1844),but until now it remains an open problem in as-trophysics. There are several large scale observedphenomena that evidence that the solar cycle cor-responds to a dynamo process operating inside thesun. These can be summarized as follows:

(1) The sunspots usually appear in pairs at bothsides of the solar equator; the leading spot of a pair(i.e. the one that points to the E-W direction) hasopposite polarity to the other one. Besides, leadingspots in the northern hemisphere have the oppositepolarity to that of the leading spots in the south-ern hemisphere. Sunspots invert their polarity ev-ery 11 years; the total period of the cycle is then22 years. This is known as the Hales law; (2) Astraight line connecting the leading and the com-panion spots of a pair has always an inclination of10 to 30 with respect to the equatorial line. Thisis known as the Joys law; (3) When the toroidalfield reaches its maximum, i.e., when the number ofsunspots is maximum, the global poloidal field in-verts its polarity, so that there is a phase lag of /2

1Astronomy Department, Instituto de Astronomia,Geofsica e Ciencias Atmosfericas, Universidade de SaoPaulo, Rua do Matao 1226, Sao Paulo, Brazil (guer-rero, dalpino@astro.iag.usp.br).

between the toroidal and poloidal components of themagnetic field; (4) The sunspots appear only in abelt of latitudes between 30 (at both sides of thesolar equator), these are known as the latitudes ofactivity; (5) The strength of the magnetic fields inthe sunspots is around 103 G. The magnitude of thediffuse poloidal field is of tens of G.

Parker (1955) was the first to try to explain thesolar cycle as a hydromagnetic phenomenon, sincethen although there has been important improve-ments in the observations, theory and simulations,a definitive model for the solar dynamo is still miss-ing. Helioseismology has mapped the solar internalrotation showing a detailed profile of the latitudi-nal and radial shear layers, which seems to confirmthe usually accepted idea that the first part of thedynamo process is the transformation of an initialpoloidal field into a toroidal one. This stage is knownas the effect. The second stage of the process, i.e.,the transformation of the toroidal field into a newpoloidal field of opposite polarity is a less understoodprocess, and has been the subject of intense debateand research. Two main hypotheses have been for-mulated in order to explain the nature of this effect,usually denominated the effect: the first one isbased on the Parkers idea of a turbulent mechanismwhere the poloidal field results from cyclonic convec-tive motions operating at small scales in the toroidalfield. These small loops should reconnect to form

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FLUX TRANSPORT SOLAR DYNAMO MODELS CD253

a large scale dipolar field. However, these modelsface an important problem: in the non-linear regime,i.e. when the back reaction of the toroidal field onthe motions becomes important, the effect can becatastrophically quenched (Vainshtein & Cattaneo1992) leading to an ineffective dynamo (Cattaneo &Hughes 1996)2.

The second one is based on the formulations ofBabcock (1961) and Leighton (1969) (BL). They pro-posed that the inclination observed in the bipolarmagnetic regions (BMRs) contains a net dipole mo-ment. The supergranular diffusion causes the driftof half of each of these active regions to the equa-tor and the drift of the other half in direction to thepoles. so that this large scale poloidal structure an-nihilates the previous dipolar field. The new dipolarfield is transported by the meridional circulations tothe higher latitudes in order to form the observed po-lar field. This second mechanism has the advantageof being directly observed at the surface (Wang etal. 1989, 1991), but it does not discard the existenceof other sources underneath.

Following the BL idea, the physical model for thesolar dynamo begins with a dipolar field. The differ-ential rotation stretches the poloidal lines and forma belt of toroidal field at some place within the so-lar interior, in the convective layer. This toroidalfield is somehow pushed through the turbulent con-vective eddies and forms strong and well organizedmagnetic flux tubes. When the magnetic field is in-tense enough and the density inside a tube is lowerthan the density of the surrounding plasma, it be-comes unstable and begins to emerge towards thesurface where it will form the BMRs. By diffusive de-cay, a BMR will form a net dipolar component withthe opposite orientation of the original one, this newdipolar field is amplified during the cycle evolutionuntil it reverses the previous dipolar field. In order tocomplete the cycle, it is necessary to transport thisnew poloidal flux first to the poles and then to theinternal layers where the toroidal field will be cre-ated again, and so on. Most of the models in the BLmechanism use the meridional circulation flow as themain agent of transport. Recent works have invokedthe turbulent pumping as an additional mechanismto advect the magnetic flux (see below).

In the absence of direct observations to confirm

2Nevertheless, it is noteworthy that when the magnetichelicity is included in dynamical computations of , it doesnot become catastrophycally quenched as long as the flux ofthe magnetic helicity remains non null (see Brandenburg &Subramanian 2005a, for a complete review of this subject).The shear in the fluid could be the way through which thehelicity flows to outside of the domain (Vishniac & Cho 2001).

the model above, several numerical studies have beenperformed in order to simulate the solar dynamo.These can be divided in two main classes: global dy-namical models (Brun et al. 2004) and mean fieldkinematic models (Dikpati & Charbonneau 1999;Chatterjee et al. 2004; Kuker et al. 2001; Bonannoet al. 2002; Guerrero & Munoz 2004; Kapyla et al.2006b). The first class integrates the full set of MHDequations in the solar convection zone and employthe inelastic approximation in order to overcome thenumerical constrain imposed by fully compressibleconvection on the time-step. These models are ableto reproduce the observed differential rotation pat-tern, but they do not generate a cyclic, and wellorganized pattern of toroidal magnetic field. Thesecond class of simulations solves the induction equa-tion only and uses observed and/or estimated profilesfor the velocity field and the diffusion terms. Thesemodels are relatively successful in reproducing thelarge scale features of the solar cycle, but the lack ofthe dynamical part of the problem has led to uncer-tainties in the dynamo mechanism.

We have carried out several numerical tests witha mean field dynamo model in the Babcock-Leightonapproximation in order to search for answers to fourmain issues:

(1) where is the solar dynamo located? (2) whatis the dominant flux transport mechanism? (3) howto explain the observed latitudes of the solar activ-ity? and (4) why is the so