Turbulent magnetic pumping in a Babcock-Leighton solar dynamo model
aa r X i v : . [ a s t r o - ph ] M a r Astronomy&Astrophysicsmanuscript no. guerrero c (cid:13)
ESO 2018October 28, 2018
Turbulent magnetic pumping in a Babcock-Leighton solar dynamomodel
G. Guerrero and E. M. de Gouveia Dal Pino ,⋆ Astronomy Department, Instituto de Astronomia, Geof´ısica e Ciˆencias AtmosfˆericasUniversidade de S˜ao Paulo, Rua do Mat˜ao 1226, S˜ao Paulo, Brazile-mail: guerrero,[email protected]
ABSTRACT
Context.
The turbulent pumping e ff ect corresponds to the transport of magnetic flux due to the presence of density and turbulencegradients in convectively unstable layers. In the induction equation it appears as an advective term and for this reason it is expected tobe important in the solar and stellar dynamo processes. Aims.
In this work, we have explored the e ff ects of the turbulent pumping in a flux-dominated Babcock-Leighton solar dynamo modelwith a solar-like rotation law. Methods.
In a first step, only vertical pumping has been considered through the inclusion of a radial diamagnetic term in the inductionequation. In a second step, a latitudinal pumping term has been also included and then, in a third step a near-surface shear has beenswitched on in the model.
Results.
The results reveal the importance of the pumping mechanism for solving current limitations in mean field dynamo modelingsuch as the storage of the magnetic flux and the latitudinal distribution of the sunspots. In the case that a meridional flow is assumed tobe present only in the upper part of the convective zone, it is the full turbulent pumping that regulates both the period of the solar cycleand the latitudinal distribution of the sunspots activity. In models that consider shear near the surface, a second shell of toroidal fieldis generated above r = . R ⊙ at all latitudes. If the full pumping is also switched on, the polar toroidal fields are e ffi ciently advectedinwards, and the toroidal magnetic activity survives only at the observed latitudes near the equator. With regard to the parity of themagnetic field, only models that combine turbulent pumping with near-surface shear always converge to the dipolar parity. Conclusions.
This result suggests that, under the Babcock-Leighton approach, the equartorward motion of the observed magneticactivity is governed by the latitudinal pumping of the toroidal magnetic field rather than by a large scale coherent meridional flow.Our results support the idea that the parity problem is related to the quadrupolar imprint of the meridional flow on the poloidalcomponent of the magnetic field and the turbulent pumping positively contributes to wash out this imprint.
Key words.
Sun: magnetic fields – Sun: activity
1. Introduction
Flux-dominated Babcock-Leighton (FDBL) solar dynamo aremean field models where the poloidal field is generated at thesurface by the transport and decay of bipolar magnetic regions(BMRs) which are formed by twisted buoyant magnetic fluxropes. For this process to occur, di ff erential rotation must beable to develop intense toroidal magnetic fields either at thetachocline or at the convection zone. Numerical simulations haveshown that magnetic flux tubes with intensity around 10 -10 Gare able to become buoyantly unstable and to emerge to the sur-face to form a bipolar magnetic region with the appropriate tilt,in agreement with the Joy’s law. One important limitation of thescenario above is that 10 G results an energy density that is anorder of magnitude larger than the equipartition value, so that astable layer is required to store and amplify the magnetic fields.This brings another problem with regard to the way by which themagnetic flux is dragged down to deeper layers.In the lack of accurate observations of the flow at the deeper lay-ers, numerical simulations have shown that the penetration of theplasma is restricted to only a few kilometers below the overshootlayer (Gilman & Miesch 2004; R¨udiger et al. 2005), neverthe-less, the magnetic fields can be transported, not only downwards,but also longitudinally and latitudinally when strong density ⋆ and turbulence gradients are present in the medium due to theturbulent pumping (Ziegler & R¨udiger 2003; Dorch & Nordlund2001).In axisymmetric mean field models of the solar cycle, the e ff ectsof the turbulent pumping have been rarely considered. A first ap-proach showing the importance of the pumping in the solar cyclewas made by Brandenburg et al. (1992), since then few workshave incorporated the diamagnetic pumping component in thedynamo equation as an extra di ff usive term which provides adownward velocity (K¨uker et al. 2001; Bonanno et al. 2002,2006). More recently, K¨apyl¨a et al. (2006b) have implementedsimulations of mean field dynamo in the distributed regime,including all the dynamo coe ffi cients, previously evaluatedin magneto-convection simulations (Ossendrijver et al. 2002;K¨apyl¨a et al. 2006a). They have produced butterfly diagramsthat approximately resemble the observations. However, toour knowledge no special e ff orts have been done to studythe pumping e ff ects in the meridional plane (i.e., inside theconvection zone) or in a FDBL description. The latter has beenfound to be particularly successful at reproducing most of thelarge scale features of the solar cycle (Dikpati & Charbonneau1999; Dikpati et al. 2004; Guerrero & de Gouveia Dal Pino2007a,b, hereafter GDPa, b). G. Guerrero and E. M. de Gouveia Dal Pino: Turbulent magnetic pumping in a Babcock-Leighton solar dynamo model
Fig. 1.
Radial and latitudinal profiles for α (continuous line), η T (dotted line) and for the pumping terms γ r and γ θ (dashed anddot-dashed lines, respectively). All the profiles are normalizedto their maximum value.In this work, we explore the e ff ects of the turbulent pumpingon a FDBL model. In a first approximation, we include the ra-dial turbulent diamagnetism velocity term in the induction equa-tion as described by Kichatinov & Ruediger (1992), and thenin a second approach we add the pumping terms calculated inlocal magneto-convection simulations (Ossendrijver et al. 2002;K¨apyl¨a et al. 2006a). This latter approximation includes notonly the radial but also the latitudinal contribution of thepumping. Finally we will also discuss the implications of thepumping when the near-surface radial shear layer reported byCorbard & Thompson (2002) is considered. In the next, webriefly present the model, our results and then, we outline themain conclusions of this work.
2. The model
Our model solves the mean field induction equation: ∂ B ∂ t = ∇ × [ U × B + E − η T ∇ × B ] , (1)where U = u p + Ω r sin θ is the observed velocity field, Ω is the an-gular velocity, B = ∇ × ( A ˆ e φ ) + B φ ˆ e φ are the poloidal and toroidalcomponents of the magnetic field, respectively, η T is the mag-netic di ff usivity and E = α B + γ × B , (2)corresponds to the first order terms of the expansion of the elec-tromotive force, u × b , and represents the action of the small-scale fluctuations over the large scales. The coe ffi cients of (2)are the so called dynamo coe ffi cients. Normally, the mean fieldmodels do not consider the second term on the right hand sideof the equation, i.e, the turbulent pumping. The first term cor-responds to the alpha e ff ect that has been considered in severalways in the literature. A pure Babcock-Leighton model is an α Ω Fig. 2. (a) Butterfly diagram and latitudinal snapshots for thetoroidal (b) and the poloidal (c) fields. The dark (blue) and light(red) gray (color) scales represent positive and negative toroidalfields, respectively; while the continuous and dashed lines rep-resent the positive and negative poloidal fields, respectively. Forthis model T = . B φ max = . × G and B r max = . × (the most external con-tours) are shown in panels (a) and (b). This model has startedwith anti-symmetric initial condition (see § α φφ com-ponent of the α tensor. It should resemble the buoyant rising andtwisting of strong magnetic flux tubes, reproducing events of fastemergence of them (see the continuous lines in the diagrams ofFigure 1). We note that another source of poloidal field may ex-ist inside the convection zone and at the tachocline, but for ourpurpose we consider only the Babcock-Leighton α e ff ect since itis in fact, observed at the surface.We solve equation (1) for A and B φ with r and θ coordinates inthe spatial ranges 0 . R ⊙ − R ⊙ and 0 − π , respectively, in a 200 ×
200 grid resolution (Guerrero & Mu˜noz 2004, see details of thenumerical model in).
3. Results
The velocity field ( U ) considered in the calculations belowcorresponds to the analytical profiles of eqs. (4) and (5) ofDikpati & Charbonneau (1999). According to the Babcock-Leighton mechanism, the alpha term ( α B φ ) is concentrated be-tween 0 . R ⊙ and R ⊙ and at the latitudes where the sunspotsappear (see the continuous lines in Fig. 1). Since it must pro-duce the emergence of magnetic flux tubes, we consider thisterm as being proportional to the toroidal field B φ ( r c , θ ) at theovershoot interface r c = . R ⊙ . For the magnetic di ff usion, weconsider only one gradient of di ff usivity located at r c , which sep-arates the radiative stable region (with η rz = cm s − ) from the . Guerrero and E. M. de Gouveia Dal Pino: Turbulent magnetic pumping in a Babcock-Leighton solar dynamo model 3 Fig. 3.
The same as in Figs. 2 but considering here the radialdiamagnetic term. For this model T = . B φ max = . × Gand B r max = . η cz = cm s − ) (see the dottedline in the upper panel of Fig. 1). The non-dimensional parame-ters as defined in Dikpati & Charbonneau (1999) and employedin the models have the following values: R m = U R ⊙ /η cz = . C Ω =Ω eq R ⊙ /η cz = . × and C α = α R ⊙ /η cz = .
8, where R m isthe magnetic Reynolds number, U = is the maximum merid-ional flow velocity, Ω eq is the angular velocity at the equator, and α is the maximum amplitude of the α e ff ect. In all the cases, thetachocline thickness corresponds to 2% of the solar radii and isat a radius R = . R ⊙ .In Figure 2, the turbulent pumping is not considered. As it hasbeen reported in GDPa and GDPb, the most important contribu-tion to the toroidal field comes from the latitudinal shear termand therefore, the field that is responsible for the observed ac-tivity begins to be formed inside the convection zone. On onehand, it can be seen that the penetration into the stable layer isvery weak, on the other hand, the equartorward velocity is faster,so that the time that the toroidal field has to amplify to the val-ues required by the magnetic flux tube simulations is probablyshort. The upper panel of Fig. 2 shows the butterfly diagram af-ter a transient time of 3 . × years. At this time, the toroidalfield has almost reached a quadrupolar parity (i.e. the toroidalfield in both hemispheres has the same sign), which is in con-tradiction with the Hale’s law. In the solar dynamo modellingthis problem is known as the parity problem and we will discussthis subject in more detail in section §
5. In the next sections weconsider the turbulent magnetic pumping as an alternative mech-anism of penetration. The latitude of emergence of the toroidalfield depends on the stability criterion for the buoyancy (e.g. seeFerriz-Mas et al. 1994, and their Figs. 1 and 2 for details).
Fig. 4.
The same as in Fig. 2 for a model with the full turbulentpumping terms obtained from magneto-convection simulations.For this model T = . B φ max = . × G and B r max = .
92 G.This model has started with anti-symmetric initial condition.
A way to describe the diamagnetic behaviour of a non-homogeneous plasma based on a first order smoothing approxi-mation (FOSA) was outlined by Kichatinov & Ruediger (1992).If there is an inhomogeneous di ff usivity in a fluid, it causes trans-port of the magnetic field with an e ff ective velocity. We intro-duce this e ff ect in the model by changing the term η T ∇ × B to η T ∇ × B + ∇ η T × B / ff ect at the overshoot regionis U dia = −∇ η T / ≃
47 cm s − when a variation of two orders ofmagnitude is considered for the di ff usivity in a thin region of0 . R ⊙ . In a convectivelly unstable rotating plasma, the magnetic fieldis not advected in the vertical direction only. The diamagnetice ff ect may have components in all directions. Besides, anotherpumping e ff ect due to density gradients can develop and in someconditions can produce an upward transport that can balancethe diamagnetic e ff ect (Ziegler & R¨udiger 2003). Aiming at in-vestigating a more general pumping advection, we will con-sider in this section the integration of eq. (1) with E given G. Guerrero and E. M. de Gouveia Dal Pino: Turbulent magnetic pumping in a Babcock-Leighton solar dynamo model by eq. (2) and γ given by both contributions. The radial andlatitudinal components of this total γ were computed numer-ically from three-dimensional magneto-convection simulationsby Ossendrijver et al. (2002); K¨apyl¨a et al. (2006a). Similarlyto K¨apyl¨a et al. (2006b), we use the following profiles approx-imately fitted from the numerical simulations: γ θ = γ θ " + erf r − . . ! (3) × " − erf r − . . ! × cos θ sin θ.γ r = − γ r " + erf r − . . ! (4) × " − erf r − . . ! × exp r − . . ! cos θ + , where γ θ and γ θ define the maximum amplitudes of the pump-ing coe ffi cients. The latitudinal pumping is zero at the overshootlayer and assumes positive (negative) values at the convectionzone in the north (south) hemisphere; it is zero at the poles and atthe equator, with a maximum value of 100 cm s − around ∼ ◦ (see the dot-dashed lines in Fig. 1). The radial pumping, γ r , isnegative at the convection zone, indicating a downward transportuntil r = . − (see dashedlines in Fig. 1). We do not consider the longitudinal contributionof the pumping because it is small compared with the longitudi-nal velocity term u φ .Figure 4 shows that besides advecting the magnetic fields to-wards the stable regions, the pumping terms lead to a distinctlatitudinal distribution of the toroidal fields when compared withthe previous results of Figs. 2 and 3. The turbulent and densitygradient levels present in a convectively unstable layer cause thepumping of the magnetic field both down and equartorward, al-lowing its amplification within the stable layer and its later emer-gence at latitudes very near the equator. This result is importantfor the dynamo modeling because it suggests that the pumpingcan not only solve the problem of the storage of the toroidalfields in the stable layer, but it can also help to provide a latitu-dinal distribution that is in agreement with the observations. As the pumping and the meridional flow are both advective termsand in some regions inside the convection zone their radial andlatitudinal components have the same sign, when the total pump-ing is considered the period of the cycle is strongly a ff ected. Itgoes from 13 . . ff usivity at the convection zone.Another possibility is to decrease the depth of penetration of themeridional flow. This is supported by recent helioseismic results(Mitra-Kraev & Thompson (2007)) that suggest that the returnpoint of the meridional circulation can be at ∼ . R ⊙ . At lowerregions, beneath ∼ . R ⊙ , a second weaker convection cell oreven a null large scale meridional flow can exist. In Figure 5, wehave decreased the depth of penetration of the flow and foundthat the period increases to the observed value at the same time Fig. 5.
Meridional flow streamlines and the butterfly diagram fora model with the full pumping term, but with a shallow merid-ional flow penetration with a depth of only 0 . R ⊙ , U = − , γ θ =
90 cm s − and γ r =
30 cm s − . For this model we obtain T = . B φ max = . × G and B r max = . T ≃ . U − . γ − . r γ − . θ , ≤ U ≤ − , ≤ γ θ ≤
140 cm s − , ≤ γ r ≤
120 cm s − . (5)This indicates that the pumping terms regulate the period of thecycle, leading to a di ff erent class of dynamo that is advection-dominated not by a deep meridional flow but by turbulent pump-ing. Figure 5 shows a butterfly diagram with fiducial values forthe meridional flow which result a period of 10 . Ω effect at the near-surface shear layer Helioseismology inversions have identified a second radial shearlayer located below the solar photosphere in the upper 35 Mmof the sun (Corbard & Thompson 2002). It is possible, that thesolar dynamo is operating in this region, as has been criticallydiscussed by Brandenburg (2005). The more attractive featuresof an α Ω dynamo operating in this region are, among others, . Guerrero and E. M. de Gouveia Dal Pino: Turbulent magnetic pumping in a Babcock-Leighton solar dynamo model 5 Fig. 6.
Butterfly diagram for a model with the same parame-ters of Fig. 2, but with near-surface shear layer. For this model T = . B φ max ( r = . = . × , B φ max ( r = . = . × Gand B r max = . G in order to formsunspots with the observed magnitudes, but 10 G is su ffi cient;and (ii) with near-surface Ω e ff ect it is possible to explain thecoincidence of the angular velocity of the sunspots in the pho-tosphere with the rotation velocity at R = . R ⊙ (see Fig. 2 ofBrandenburg 2005), as well as the apparent disconnection be-tween the sunspot and its roots (Kosovichev 2002). The con-tribution of a near-surface radial shear has been investigated ininterface-like dynamos (Mason et al. 2002), in distributed dy-namos with a turbulent α e ff ect (K¨apyl¨a et al. 2006b), and alsoin advection dominated dynamos (Dikpati et al. 2002). The lat-ter authors have discarded the radial shear layer since it gener-ates butterfly diagrams in which a positive toroidal field givesrise to a negative radial field, which is exactly the opposite to theobserved. In this section, we include the radial shear term in ourFDBL model in order to explore the contribution of the pump-ing to this new configuration. We use the analytical expressiongiven in eqs. (1)-(3) of Dikpati et al. (2002). The near-surfaceshear described by these equations relates a negative shear be-low 45 ◦ with a positive shear above this latitude (see Fig. 1 ofDikpati et al. 2002) .With the assumption that the sunspots are formed in the upperlayers, the Babcock-Leighton poloidal source term, which isconcentrated in the same region (above r = . R ⊙ ), does nothave to be non-local anymore. For the same reason, the valuesof both, the radial and the toroidal fields in the butterfly diagramcan be taken in the same radial point ( r = . R ⊙ ). Using thesame parameters as in the model of Fig. 2, but consideringa near-surface shear, the results of Figure 6 show two mainbranches in the butterfly diagram. One is migrating poleward (atthe high latitudes) and one is migrating equatorward (below 45 ◦ .This result is expected if the Parker-Yoshimura sign rule (Parker1955; Yoshimura 1975) is considered. We note that the resultingparity is quadrupolar but with the correct phase lag between thefields, which is opposite to the obtained in Dikpati et al. (2002).This di ff erence probably arises from the fact that we are usinga lower meridional circulation amplitude. Anyway, the polarbranches are strong enough to generate undesirable sunspotsclose to the poles. The period increases to 15 . This profile is slightly di ff erent from that used by K¨apyl¨a et al.(2006b) since the latter consider a negative radial shear at all latitudes. Fig. 7.
The same as in Fig. 5 for a model with near-surfaceshear action. For this model T = . B φ max ( r = . = . × , B φ max ( r = . = . × G and B r max = . ffi ciently pushed down before reaching a significantamplitude, so that only the equatorial branches below 45 ◦ survive. This scenario requires that the pumping be dominantover the buoyancy at such latitudes. Also, the phase relationof B r B φ , obtained in the model of Fig. 7 seems to be the oneobserved, at least at the latitude of activity, however, there issome overlapping between one cycle and the next. Resultswhich are in better agreement with the observations may beachieved if the parameters are finely tuned.We note that the introduction of the radial shear close to the sur-face when a meridional flow cell penetrating down to r = . R ⊙ is considered, as in the model of Fig. 4, requires an increase ofthe amplitude of α . This result is in agreement with that foundby K¨apyl¨a et al. (2006b)
5. Brief remarks on the parity problem
Despite that it has been already explored by several authors,the anti-symmetry (dipolar parity) or symmetry (quadrupolarparity) of the toroidal magnetic fields across the solar equatorstill constitutes one of the most challenging questions in thesolar dynamo theory. This is mainly because the resultingparity in a model is very sensitive to a huge parameter space.The solar-like (antisymmetric) solution could result fromthe e ff ective di ff usive coupling of the poloidal field in bothhemispheres (Chatterjee et al. 2004), but it may also depend on G. Guerrero and E. M. de Gouveia Dal Pino: Turbulent magnetic pumping in a Babcock-Leighton solar dynamo model the position and amplitude of the α e ff ect (Dikpati & Gilman2001; Bonanno et al. 2002), or be the result of the imprint of thequadrupolar form of the meridional flow on the poloidal mag-netic field, as argued by Charbonneau (2007). Small variationsin the parameter space can switch one solution from a dipolarto a quadrupolar one. Although the main goal of this work wasnot to study the parity problem itself, but the contribution of theturbulent magnetic pumping, it is interesting to take advantageof the full sphere integration in order to see how the pumpinga ff ects the parity.All the simulations presented in the previous sections evolved10 time steps up to ∼ years. All started with antisymetric(A) or with symmetric (S) toroidal magnetic field, but we havealso performed tests with random (R) fields. The parity of the so-lution is calculated, following Chatterjee et al. (2004), with theequation below: P ( t ) = R T / − T / ( B N ( t ) − B N )( B S ( t ) − B S ) dt qR T / − T / ( B N ( t ) − B N ) dt qR T / − T / ( B S ( t ) − B S ) dt , (6)where B N and B S are the values of the toroidal magnetic fieldat r = . R ⊙ , and θ = ◦ and − ◦ , respectively, B N and B S aretheir respective temporal averages over one period. The value of P should be between + − – The models without pumping (e.g., Fig. 2), and those withdiamagnetic pumping (Fig. 3) result in quadrupolar solu-tion. When beginning with a dipolar initial condition theytake several years before switching to a quadrupolar solu-tion (see Fig. 8a). This result diverges from the one obtainedby Dikpati & Gilman (2001) or Chatterjee et al. (2004) inwhich the change begins only after around 500 yr. This resultindicates the strong sensitivity of the parity to the initial pa-rameters, in such a way that, for example, the present parityobserved in the sun could be temporary, at least in the casethat the turbulent pumping is not relevant for the dynamo. – The models with full pumping (e.g., Fig. 5) conserve the ini-tial parity if this is symmetric or anti-symmetric (see Fig 8b).When initialized with a random field, the system tends firstto choose the quadrupolar parity, but then it tends to migrateto the anti-symmetric (dipolar) parity (dot-dashed line of Fig8b), suggesting that the strong quadrupolar imprint due tomeridional circulation could be whashed out when the fullturbulent pumping is switched on. – Models with full pumping plus near-surface shear layer (e.g.Fig. 7) tend to the dipolar parity since the first years of in-tegration (see continuous line of Fig. 8c). In these modelswe find that the coupling of the polodial fields in both hemi-spheres is more e ff ective. This is probably due to the employ-ment of a local α term. See, for example also Chatterjee et al.(2004), where a near-surface α e ff ect is combined with abuoyancy numerical mechanism. They find a similar cou-pling. However this coupling alone is not enough to ensurea dipolar parity (see Fig. 6 and the dashed line of Fig. 8c).It is also necessary to eliminate the e ff ect of the quadrupo-lar shape of the meridional flow upon the poloidal magneticcomponent. This can be done by the action of the pumpingat the entire convection zone (as indicated by the continuousline of Fig. 8c). Fig. 8.
Parity curves for the three classes of models considered,i.e., (a) for models without pumping (as e.g., in Fig. 2); (b) formodels with full pumping (as e.g., in Fig. 5); and (c) for modelswith near-surface shear (as e.g., in Figs. 6 and 7). In the panels(a) and (b), the continuous, dashed and dot-dashed lines corre-spond to symmetric, anti-symmetric and random initial condi-tions, respectively. In the bottom, (c), panel, the continous lineis used for the model with turbulent pumping while the dashedline is for the model without pumping.
6. Discussion and conclusions
We have performed 2D numerical simulations of BLFD solardynamo models including the turbulent pumping. Our first setof simulations include a solar rotation profile but without the anear-surface radial shear layer. The results show that the pump-ing transport e ff ect is, in fact, relevant in solar dynamo mod-elling, since it can solve two important problems widely dis-cussed in the literature: the storage of the toroidal field at the sta-ble layer and its latitudinal distribution. A new class of dynamois proposed in which the meridional flow is important only nearthe surface layer in order to make the Babcock-Leighton mecha-nism to operate over the toroidal fields, while in the inner layers,the advection is dominated by the pumping velocity. Our resultssupport the idea that the equatorward migration of the sunspotactivity is related to the latitudinal pumping velocity at the over-shoot layer and the convection zone. Another attractive featureof this model is that a large coherent meridional flow is not anymore required.In a second set of simulations, we have included the shear layerfound by Corbard & Thompson (2002) at the upper 35 Mm ofthe sun. The results show the formation of a second shell ofstrong toroidal field just below the photosphere when the fullpumping is absent. The branches of this field obey the Parker- . Guerrero and E. M. de Gouveia Dal Pino: Turbulent magnetic pumping in a Babcock-Leighton solar dynamo model 7 Yoshimura sign rule for a positive α e ff ect, i.e., they move pole-ward at high latitudes and equatorward below 45 ◦ . The role ofthe pumping on this kind of models is also interesting since itreduces the amplitude of the polar toroidal fields pushing theminwards (Fig. 7). It should be noted that these models only workfine if a shallow meridional circulation profile is used. When adeeper meridional flow going down to the tachocline is consid-ered, a strong α e ff ect is required in order to excite the dynamo.With regard to the parity problem, our results show that a sim-ple α Ω dynamo with the α e ff ect concentrated near the surfaceleads to a quadrupolar parity, although the switch from dipolarto quadrupolar parity takes longer than in previous studies. Themodels with full pumping conserve the initial parity, and whenthe initial condition is random, the system tends to switch to adipolar parity. All the models that combine full pumping withnear-surface shear prefer dipolar parity solutions too.In summary, our results have demonstrated the importance of thepumping in the solar dynamo, and suggest that this e ff ect mustbe included in forthcoming studies, even in those that employmultiple convection cells (Bonanno et al. 2006; Jouve & Brun2007). Besides it decrease the influence of the meridional flowin two important aspects: on the period of the cycle and on thelatitudinal distribution of the toroidal fields. On the other hand,our results indicate that in the presence of full pumping thereare two possible solutions to the question on where the dynamooperates: it could be either at the convection zone with the mag-netic flux tubes emerging from the overshoot layer, or it could beat the layers near the surface. Both possibilities have their prosand cons, however a kinematic dynamo model only is not su ffi -cient to conjecture a definitive answer and we will explore thisin more detail in forthcoming work. Acknowledgements.
This work was supported by CNPq and FAPESP grants.G. Guerrero thanks the MPI in Garching and the ALFA project for their kindhospitality and support during the production of part of the present paper. Wewould like to thank also to the anonymous referee for his / her suggestions thathave enriched this work. References
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