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RevMexAA (Serie de Conferencias), 36, CD252–CD261 (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 dınamo queopera en el sol. Este es un problema clasico en astrofısica que hasta el momento no se encuentra del todoresuelto. Aquı discutimos problemas actuales y limitantes del modelo del dınamo 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 dınamo.
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 Hale’s 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 Joy’s 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 π/2
1Astronomy Department, Instituto de Astronomia,Geofısica e Ciencias Atmosfericas, Universidade de SaoPaulo, Rua do Matao 1226, Sao Paulo, Brazil (guer-rero, [email protected]).
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 Ω 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 α effect: the first one isbased on the Parker’s 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 α 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 (BMR’s) 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 α 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 α, 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 solar parity anti-symmetric?
In the following paragraphs, we briefly summa-rize our model assumptions and, step by step, drawour approach to the questions above in the light ofthe numerical simulations.
2. MODEL
The equation that describes the temporal andspatial evolution of the magnetic field is the induc-tion equation:
∂B
∂t= ∇× [U × B + E − η∇× B] , (1)
where U = up + Ωr sin θ is the observed velocityfield, Ω is the angular velocity, B = ∇ × (Aeφ) +Bφeφ, where ∇× (Aeφ) and Bφ are the poloidal andtoroidal components of the magnetic field, respec-tively, η is the microscopic magnetic diffusivity and
E = αB + γ × B − β(∇× B) (2)
− δ × (∇× B) − κ(∇B) ,
corresponds to the first and second order terms ofthe expansion of the electromotive force, u × b, and
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CD254 GUERRERO & DE GOUVEIA DAL PINO
Fig. 1. Isorotation lines of the solar angular velocityas inferred from helioseismology observations (continu-ous lines) and the adopted meridional circulation streamlines. The red dot-dashed line shows the boundary be-tween the overshoot layer and the convection zone.
represents the action of the small-scale fluctuationsover the large scales. The coefficients of equation (3)are the so-called dynamo coefficients. The first termon the right hand side of equation (2) correspondsto the turbulent α effect coefficient, not consideredin our Babcock-Leighton formulation. The secondone is the turbulent magnetic pumping. The thirdcorresponds to the turbulent diffusivity, which in ourmodel is combined with the microscopic value (ηT =η + β). For the sake of simplicity, the other twoterms are neglected. We solve equation (1) for Aand Bφ with r and θ coordinates in the spatial ranges0.6R−R and 0−π, respectively, in a 200×200 gridesolution (see Guerrero & Munoz 2004, for detailsregarding the numerical model).
Fig. 2. Radial and latitudinal profiles for α (continu-ous line), ηT (dotted line) and for the pumping termsγr and γθ (dashed and dot-dashed lines, respectively).All the profiles are normalized to their maximum value.Extracted from Guerrero & de Gouveia Dal Pino (2008).
The profiles that we employ describe the re-sults of recent helioseismology inversions or numeri-cal simulations. For the differential rotation, we con-sider a profile mapped from helioseismology (see thecontinuous lines of Figure 1). For the meridionalflow, we consider one cell per meridional quadrant, asusually assumed (doted lines in Figure 1). The alphaterm (αBφ) is concentrated between 0.95 R and R
and at the latitudes where the sunspots appear (seethe continuous lines in Figure 2). Since it must resultthe emergence of magnetic flux tubes, we assume thisterm to be proportional to the toroidal field Bφ(rc, θ)at the overshoot interface between the radiative andthe convective regions, rc = 0.715 R. For the mag-netic diffusion, we consider only one gradient of dif-fusivity located at rc which separates the radiativestable region (with ηrz = 109 cm s−2) from the con-vective turbulent layer (with ηcz = 1011 cm s−2) (seethe dotted line in the upper panel of Figure 2).
3. THE LOCATION OF THE SOLAR DYNAMO
As remarked before, the differential rotation pat-tern which is responsible for the Ω effect, is revealedby high resolution helioseismology observations. It
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FLUX TRANSPORT SOLAR DYNAMO MODELS CD255
describes a solid-body rotation for the radiative core,and a differentially rotating convective layer with aretrograde velocity with respect to the radiative in-terior at higher latitudes and a pro-grade velocityat lower latitudes. The interface that bounds thesolid-body rotation zone is named tachocline − itsexact location and width have not been establishedyet. Another radial shear layer has been recentlyidentified just below the solar photosphere in the up-per 35 Mm of the sun (Corbard & Thompson 2002)(see the gray dashed line in Figure 1). With thisnewly discovered shear layer, it is even more difficultto define where the dynamo operates. There hasbeen so far, an apparent common agreement thatthe dynamo is operating at the tachocline. Howeverthis possibility has several problems (Brandenburg2005). One of the main difficulties is that toroidalflux ropes formed in the tachocline should have in-tensities ∼ 104−105 G in order to become buoyantlyunstable and to emerge at the surface to form a BMRthe appropriate tilt given by the Joy’s law (D’Silva& Choudhuri 1993; Fan et al. 1993; Caligari et al.1995, 1998; Fan & Fisher 1996; Fan 2004). One im-portant limitation of this scenario is that 105 G re-sults an energy density that is an order of magnitudelarger than the equipartition value, so that a stablelayer is required to store and amplify this magneticfield. This raises another question with regard to theway in which the magnetic flux is dragged down todeeper layers. In recent work (Guerrero & de Gou-veia Dal Pino 2007a), we have explored the contri-butions of the shear terms in the dynamo equation,(Bp · ∇)Ω = Br∂Ω/∂r + Bθ/r∂Ω∂θ with the aim ofdetermining where the most strong toroidal magneticfields are produced. We found that the radial shearcomponent is about two orders of magnitude smallerthan the latitudinal component. Therefore, whena new toroidal field begins to develop its growth isdominated by the latitudinal shear. Its amplificationbegins in the bulk of the convection zone and it istransported to the stable layer where it must reachthe desired magnitude (see Figure 3). How this fluxis transported is the subject of the next section.
If a near-surface shear layer is turned on in themodel (Guerrero & de Gouveia Dal Pino 2008, andreferences therein), two main branches appear in thebutterfly diagram (Figure 4). One is migrating pole-ward (at high latitudes) and another is migratingequatorward (below 45). This result is expectedif the Parker-Yoshimura sign rule (Parker 1955;Yoshimura 1975) applies. This contribution to thetoroidal field was previously explored in a Babcock-Leighton dynamo by Dikpati et al. (2002). They
Fig. 3. (a) Butterfly diagram and latitudinal snapshotsfor the toroidal (b) and the poloidal (c) fields. The dark(blue) and light (red) gray (color) scales represent posi-tive and negative toroidal fields, respectively; the contin-uous and dashed lines represent the positive and negativepoloidal fields. Only toroidal fields greater than 2×104 G(the most external contours) are shown in panels (a) and(b). This model started with an anti-symmetric initialcondition (see § 6 for details). Extracted from Guerrero& de Gouveia Dal Pino (2008).
have discarded the near-surface radial shear layerbecause it generates butterfly diagrams in which apositive toroidal field gives rise to a negative radialfield, which is exactly the opposite to the observed.Our results, on the other hand, present the correctphase lag between the fields. This difference prob-ably arises from the fact that we are using a lowermeridional circulation amplitude. Anyway, as canbe seen in Figure 4, the polar branches are strongenough to generate also undesirable sunspots closeto the poles. The period increases to 15.6 y due tothe fact that the dominant dynamo action at the sur-face goes in the opposite direction to the meridionalflow (i.e, the dynamo wave direction is dominatingover the meridional flow).
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CD256 GUERRERO & DE GOUVEIA DAL PINO
Fig. 4. Butterfly diagram for a model with the same pa-rameters as in Figure 3, but with near-surface shear layer.For this model T = 15.6 yr Bφmax
(r = 0.715) = 1.1×105,Bφmax
(r = 0.98) = 1.9 × 104 G and Brmax= 131.7
G. This model started with symmetric initial conditions.Extracted from Guerrero & de Gouveia Dal Pino (2008).
4. FLUX TRANSPORT MECHANISMS
For an αΩ dynamo, the Parker-Yoshimura signrule (Parker 1955; Yoshimura 1975) establishes thatthe direction of the dynamo wave is equatorward orpoleward if the product α ·∂Ω/∂r is < or > 0, respec-tively. Hence, models with a solar like rotation lawoperating at the tachocline, with a positive α effectin the northern hemisphere, as believed, and withoutmeridional circulation, should result a solution withmagnetic branches migrating in the opposite way tothat observed (Kuker et al. 2001). Models with onecell of meridional circulation, poleward at the sur-face and equatorward at the base of the convectionzone, produce results with the appropriate directionof propagation (Dikpati & Charbonneau 1999; Kukeret al. 2001; Bonanno et al. 2002). It was demon-strated also that the meridional circulation sets theperiod of the cycle (Dikpati & Charbonneau 1999)and that it is the most logical way to transport thenovel poloidal fields to the inner layers in the dynamoprocess. These models in which the time of the cyclefits better with the advective time than with the dif-fusive one are usually called advection dominated orflux-transport dynamos. There is, however, an im-portant problem with these models: they require alarge scale meridional flow and this is observed onlyat the surface. It is possible that the real meridionalflow is too weak at the inner regions to penetratethe overshoot layer and the tachocline (Gilman &Miesch 2004; Rudiger et al. 2005), or perhaps, it hasa multicell pattern (Mitra-Kraev & Thompson 2007;Bonanno et al. 2006; Jouve & Brun 2007). If this isthe case, it is very hard to explain the equatorwardmigration of the toroidal branches and it is neces-sary to find another flux-transport mechanism. Theturbulent pumping seems to be a good candidate.
The turbulent pumping effect corresponds to thetransport, in all directions, of magnetic flux due tothe presence of density (buoyancy) and turbulence(diamagnetism) gradients in convectively unstablelayers. In the FOSA (First Order Smoothing Ap-proximation, see e.g. Brandenburg & Subramanian2005b, and references therein), the radial componentof the diamagnetic pumping can be calculated as-suming a linear dependence with the variations inthe magnetic diffusivity Udia = −∇ηT /2; the buoy-ancy component depends on the density gradientsUbuo = −η∇ρ/ρ. In the boundary between the solarconvection zone and the overshoot layer, it is prob-able that the diamagnetic velocity is of the order of50 ms−1 (Kitchatinov & Rudiger 2008). This valuestrengthens the importance of the pumping relativeto the assumed radial meridional flow velocity whichis < 10 ms−1. For our model, we obtain Udia =' 47cm s−1 when a variation of two orders of magni-tude is considered in the diffusivity in a thin regionof 0.015 R (Guerrero & de Gouveia Dal Pino 2008)(the buoyancy component of the pumping is not con-sidered because the density is not a parameter inkinematic models).
The effects of turbulent pumping have beenrarely considered in mean field dynamo models. Afirst approach showing its importance in the solarcycle was made by Brandenburg et al. (1992); sincethen few works have incorporated the diamagneticpumping component in the dynamo equation as anextra diffusive term that can provide a downwardvelocity, as discussed above (Kuker et al. 2001; Bo-nanno et al. 2002, 2006). More recently, Kapyla et al.(2006b) have implemented simulations of the meanfield dynamo in the distributed regime, includingall the dynamo coefficients previously evaluated inmagneto-convection simulations (Ossendrijver et al.2002; Kapyla et al. 2006a). They produced butterflydiagrams that resemble the observations. However,to our knowledge no special efforts have been madeto study the pumping effects in the meridional plane(i.e., inside the convection zone) or in a BL descrip-tion.
We have included the turbulent pumping terms(see the dashed and dot-dashed lines in Figure 2which correspond to the radial, γr, and latitudinal,γθ, pumping terms) calculated from local magneto-convection simulations (Ossendrijver et al. 2002;Kapyla et al. 2006a) into the induction equation(equation 1).
For a dynamo model operating at the tachocline,we find that the pumping terms lead to a distinctlatitudinal distribution of the toroidal fields when
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FLUX TRANSPORT SOLAR DYNAMO MODELS CD257
Fig. 5. Meridional flow streamlines and the butterflydiagram for a model with the full pumping term, butwith a shallow meridional flow penetration with a depthof only 0.8 R, U0 = 1300 cm s−1, γθ0 = 90 cm s−1 andγr0 = 30 cm s−1. For this model we obtain T = 10.8 yr,Bφmax
= 4.5 × 104 G and Brmax= 154.9 G. This model
started with anti-symmetric initial conditions. Extractedfrom Guerrero & de Gouveia Dal Pino (2008).
compared with the results of Figure 3. The turbu-lent and density gradient levels present in the con-vectively unstable layer cause the pumping of themagnetic field both down and equartorward, allow-ing its amplification within the stable layer and itslater emergence at latitudes very near the equator.
If we also include in this model recent helioseis-mic results (Mitra-Kraev & Thompson 2007) thatsuggest that the return point of the meridional cir-culation can be at ∼ 0.95 R, at lower regions, be-neath ∼ 0.8 R, a second weaker convection cell oreven a null large scale meridional flow can exist. InFigure 5, we obtain a butterfly diagram that agreeswith the main features of the solar cycle, besides, wefind that in this case, it is the pumping terms thatregulate the period of the cycle, leading to a differ-ent class of dynamo that is advection-dominated byturbulent pumping rather than by a deep meridionalflow.
For the dynamo model operating at the near-surface shear layer, we find that the toroidal fieldscreated at high latitudes are efficiently pushed downbefore reaching a significant amplitude, so that onlythe equatorial branches below 45 survive (see Fig-ure 6). This result is explained by the fact that theradial pumping component has its maximum ampli-tude close to the poles (see the dashed line in Fig-ure 2). These results also agree with the observa-tions.
5. HOW TO EXPLAIN THE OBSERVEDLATITUDES OF THE SOLAR ACTIVITY?
The radial shear at the tachocline has its maxi-mum amplitude in regions close to the poles, for thisreason, it is a common problem in mean field dynamomodels to present large undesirable toroidal mag-netic fields in the polar regions. Nandy & Choudhuri(2002) proposed a deep meridional flow as a way toavoid the formation of strong toroidal fields at highlatitudes. Under this assumption, the strong toroidalmagnetic fields formed at high latitudes are pusheddown, inside the radiative zone, by the meridionalflow and are stored there until they reach latitudesbelow 30. However this assumption may lead to un-desirable mixing of the chemical elements betweenthe radiative and convective zones and to problemsregarding the angular momentum transfer. Besides,some results of numerical simulations show that themeridional flow is unable to penetrate the tachocline(Gilman & Miesch 2004; Rudiger et al. 2005).
We have explored this subject in two ways. First,we built a hybrid model in which we combined theprofiles used by Nandy & Choudhuri (2002) with
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CD258 GUERRERO & DE GOUVEIA DAL PINO
Fig. 6. The same as in Figure 5, but for a model withnear-surface shear action. For this model T = 16.3 yrBφmax
(r = 0.715) = 9.7 × 104, Bφmax(r = 0.98) = 1.9 ×
104 G and Brmax= 164.4 G. This model started with
symmetric initial conditions. Extracted from Guerrero& de Gouveia Dal Pino (2008).
those used by Dikpati & Charbonneau (1999) andallowed a deep meridional flow (Guerrero & Munoz2004). We found that the high-latitude toroidal fieldis sensitive to the model. Then, we explored thisproblem by changing the shape and the thicknessof the solar tachocline (Guerrero & de Gouveia DalPino 2007a,b) and have found that the thinner thetachocline, the smaller the intensity of the toroidalmagnetic field at high latitudes (see Figure 7). Athin tachocline must be fully contained inside theovershoot zone, in such a way that only part ofthe poloidal magnetic field is able to reach it andthen produce a small quantity of toroidal field. Thetoroidal field generated there is not strong enoughto emerge. In Figure 8, we compare the toroidalfields produced for a thin tachocline with those pro-duced with an intermediate one. This result is alsodependent of the magnetic diffusivity in the convec-tion zone, as it can be seen in Figure 7, however,
Fig. 7. Maximum of the toroidal magnetic field at thetop of the tacholine as a function of the diffusivity (inlog-scale) at a latitude of 60. The different line stylesrepresent different widths of the tachocline d1. The dot-ted line represents the limit between buoyant and non-buoyant magnetic fields 5×104 G. Only the values belowthis line will appear at the desired latitudes. Extractedfrom Guerrero & de Gouveia Dal Pino (2007a).
for a thin tachocline the models always result weaktoroidal fields above 60.
With regard to this question of the distributionof the toroidal field, the role of the pumping is alsoimportant. While it pushes poloidal fields inside thetachocline, it also pumps all the field equatorward, insuch a way that the toroidal fields formed inside theconvection zone due to latitudinal shear will go fastto the stable region where they are stored and am-plified before the eruption (Guerrero & de GouveiaDal Pino 2008).
In the case that we consider the sunspots as theproduct of toroidal fields being formed at the near-surface layer, the latitude of activity is easily ex-plained by the pumping, as described in the previoussection (see Figure 6).
6. THE PARITY PROBLEM
The anti-symmetric parity observed in the so-lar cycle is one of the most challenging questions inthe solar dynamo theory. The magnetic parity in amodel may depend on the location of the α effect(Dikpati & Gilman 2001; Bonanno et al. 2002), oron the diffusive coupling between the poloidal field inboth hemispheres (Chatterjee et al. 2004), but it may
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Fig. 8. Snapshots of the positive (negative) toroidal field contours are shown in blue (red) scale together with the fieldlines of the positive (negative) poloidal field in continuous (dashed) lines, for four different times (T/8, T/4, 3T/8 andT/2) along a half 11-year cycle. The upper (a) and middle (b) panels present the same snapshots for models with athin (0.02 R) and an intermediate (0.06 R) tachocline. Extracted from Guerrero & de Gouveia Dal Pino (2007b).
also be the result of the imprint of the quadrupolarform of the meridional flow on the poloidal magneticfield, as argued by Charbonneau (2007). This couldexplain why models with the α effect located at theupper layers (where the magnetic Reynols numberand the quadrupolar imprint of the meridional floware larger) tend to a quadrupolar parity faster thanmodels with the α effect located at the tachocline.
We have explored a little this problem in order tosee whether the turbulent pumping plays some rolein it. We have found that models without pump-ing result a quadrupolar solution. When beginningwith a dipolar initial condition, they spend severalthousand years before switching to a quadrupolar so-lution (see Figure 9a). This result diverges from theone obtained by Dikpati & Gilman (2001) or Chat-terjee et al. (2004) in which the change begins onlyafter around 500 yr. This result indicates the strongsensitivity of the parity to the initial conditions, insuch a way that, for example, the present parity ob-served in the sun could be temporary.
The models with full pumping (e.g., Figure 5)conserve the initial parity either it is symmetric oranti-symmetric (see Figure 9b). This suggests thatthe strong quadrupolar imprint due to meridionalcirculation can be washed out when the full turbulentpumping is switched on.
On the other hand, the full pumping in modelsthat include near-surface shear tend to the dipolarparity from the first years of integration (see contin-uous line of Figure 9c). We explain this result asa product of a better coupling between the poloidalfields in both hemispheres (this coupling is due tothe local α effect considered in these cases), plus theaction of the pumping eliminating the effect of thequadrupolar component of the meridional flow onthe poloidal magnetic field.
7. SUMMARY AND DISCUSSION
We have used a mean field dynamo model in theBL approach in order to look for answers to fourcurrent problems widely reported in the literature ofsolar dynamo modeling.
Our results confirm the idea that there should bea magnetic layer below the convection zone wheremagnetic fields which are mainly produced in theconvection zone are stored. We have found that it ispossible to have a flux-transport dynamo without awell defined meridional flow pattern, dominated bythe pumping advection. However, efforts in orderto obtain a more realistic profile for the meridionalvelocity profile are still required, as well as to obtaina better comprehension of the real contribution ofthe pumping velocities.
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CD260 GUERRERO & DE GOUVEIA DAL PINO
Fig. 9. Parity curves for the three classes of models con-sidered, i.e., (a) for models without pumping (as, e.g.,in Figure 3); (b) for models with full pumping (as, e.g.,in Figure 5); and (c) for models with near-surface shear(as, e.g., in Figures 4 and 6). In the panels (a) and(b), the continuous, dashed and dot-dashed lines corre-spond to symmetric, anti-symmetric and random initialconditions, respectively. In panel (c), the continuous lineis used for the model with turbulent pumping while thedashed line is for the model without pumping. Extractedfrom Guerrero & de Gouveia Dal Pino (2008).
Nevertheless, numerical simulations includingnear-surface shear as observed, provide also supportfor a near-surface magnetic layer, since toroidal fieldswith intensities between 103 - 104 G are formed thereand since the radial differential rotation is negativeat lower latitudes, the direction of migration of thebutterfly wings constructed with these fields repro-duces the one observed (e.g., Figure 6).
The latitude of activity established by the ob-servations (between ±30) can be explained by bothscenarios above for the magnetic layer. For magneticfields being stored in the overshoot region, a thin(∼
< 0.2 R) tachocline could be the solution to avoidstrong toroidal fields at the polar regions. On theother hand, if the near-surface shear is considered,then the magnetic pumping provides the required
downwards flux of the weak polar fields, letting onlythe toroidal fields to survive at the active latitudes.
The parity in a dynamo solution is a problem thatrequires especial attention. We have investigatedthis problem looking for the role that the pumpingcould play in the solutions. Our simulations supportthe idea that the quadrupolar solution in most of themodels is due to the strong quadrupolar imprint dueto the one-cell meridional flow pattern. This imprintis larger at the surface than at the bottom of theconvection zone, therefore the results of Dikpati &Gilman (2001) and Bonanno et al. (2002) which sug-gest that an α effect operating at the tachocline re-sults an anti-symmetric solution, could be explainedby this fact. We suggest that models with an α ef-fect operating close to the surface are also able togenerate anti-symmetric solutions, as observed, if amechanism such as pumping cleans the quadrupolarimprint.
More observational and theoretical efforts arenecessary in order to determine where the feet ofthe magnetic flux tubes responsible for the sunspotsare located. This is an issue that needs to be solvedbefore a more realistic coherent dynamo model canbe constructed.
REFERENCES
Babcock, H. W. 1961, ApJ, 133, 572Bonanno, A., Elstner, D., & Belvedere, G. 2006, Astron.
Nachr., 327, 680Bonanno, A., Elstner, D., Rudiger, G., & Belvedere, G.
2002, A&A, 390, 673Brandenburg, A. 2005, ApJ, 625, 539Brandenburg, A., Moss, D., & Tuominen, I. 1992, in ASP
Conf. Ser. 27, The Solar Cycle, ed. K. L. Harvey (SanFrancisco: ASP), 536
Brandenburg, A., & Subramanian, K. 2005a, Phys. Rep.,417, 1
. 2005b, Phys. Rep., 417, 1Brun, A. S., Miesch, M. S., & Toomre, J. 2004, ApJ, 614,
1073Caligari, P., Moreno-Insertis, F., & Schussler, M. 1995,
ApJ, 441, 886. 1998, ApJ, 502, 481
Cattaneo, F., & Hughes, D. W. 1996, Phys. Rev. E, 54,4532
Charbonneau, P. 2007, Adv. Space Res., 39, 1661Chatterjee, P., Nandy, D., & Choudhuri, A. R. 2004,
A&A, 427, 1019Corbard, T., & Thompson, M. J. 2002, Sol. Phys., 205,
211Dikpati, M., & Charbonneau, P. 1999, ApJ, 518, 508Dikpati, M., Corbard, T., Thompson, M. J., & Gilman,
P. A. 2002, ApJ, 575, L41Dikpati, M., & Gilman, P. A. 2001, ApJ, 559, 428D’Silva, S., & Choudhuri, A. R. 1993, A&A, 272, 621
© 2
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Inst
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Ast
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mía
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II: F
rom
La
bo
rato
ry a
nd S
tars
to th
e P
rimo
rdia
l Uni
vers
e -
Sup
ple
me
nta
ry C
DEd
. A. E
squi
vel,
J. F
ranc
o, G
. Ga
rcía
-Se
gur
a, E
. M. d
e G
ouv
eia
Da
l Pin
o, A
. La
zaria
n, S
. Liz
ano
, & A
. Ra
ga
FLUX TRANSPORT SOLAR DYNAMO MODELS CD261
Fan, Y. 2004, Living Rev. Solar Phys., 1, 1Fan, Y., & Fisher, G. H. 1996, Sol. Phys., 166, 17Fan, Y., Fisher, G. H., & Deluca, E. E. 1993, ApJ, 405,
390Gilman, P. A., & Miesch, M. S. 2004, ApJ, 611, 568Guerrero, G. A., & de Gouveia Dal Pino, E. M. 2007a,
A&A, 464, 341. 2007b, Astron. Nachr., 328, 1122. 2008, A&A, 485, 267
Guerrero, G. A., & Munoz, J. D. 2004, MNRAS, 350, 317Jouve, L., & Brun, A. S. 2007, A&A, 474, 239Kapyla, P. J., Korpi, M. J., Ossendrijver, M., & Stix, M.
2006a, A&A, 455, 401Kapyla, P. J., Korpi, M. J., & Tuominen, I. 2006b, As-
tron. Nachr., 327, 884Kitchatinov, L. L., & Rudiger, G. 2008, Astron. Nachr.,
329, 372Kuker, M., Rudiger, G., & Schultz, M. 2001, A&A,
374, 301Leighton, R. B. 1969, ApJ, 156, 1Mitra-Kraev, U., & Thompson, M. J. 2007, As-
tron.Nachr., 328, 1009Nandy, D., & Choudhuri, A. R. 2002, Science, 296, 1671Ossendrijver, M., Stix, M., Brandenburg, A., & Rudiger,
G. 2002, A&A, 394, 735Parker, E. N. 1955, ApJ, 122, 293Rudiger, G., Kitchatinov, L. L., & Arlt, R. 2005, A&A,
444, L53Schwabe, M. 1844, Astron. Nachr., 21, 233Vainshtein, S. I. & Cattaneo, F. 1992, ApJ, 393, 165Vishniac, E. T., & Cho, J. 2001, ApJ, 550, 752Wang, Y.-M., Nash, A. G., & Sheeley, Jr., N. R. 1989,
Science, 245, 712Wang, Y.-M., Sheeley, Jr., N. R., & Nash, A. G. 1991,
ApJ, 383, 431Yoshimura, H. 1975, ApJ, 201, 740
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