CONSTRAINTS ON THE APPLICABILITY OF AN INTERFACE DYNAMO TO THE SUN

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1 The Astrophysical Journal, 631: , 2005 September 20 # The American Astronomical Society. All rights reserved. Printed in U.S.A. CONSTRAINTS ON THE APPLICABILITY OF AN INTERFACE DYNAMO TO THE SUN Mausumi Dikpati, Peter A. Gilman, and Keith B. MacGregor High Altitude Observatory, National Center for Atmospheric Research, Mitchell Lane, Boulder, CO 80301; dikpati@ucar.edu, gilman@ucar.edu, kmac@ucar.edu Received 2005 March 7; accepted 2005 May 31 ABSTRACT Taking into account the helioseismically inferred interior structure, we show that a pure interface-type dynamo does not work for the Sun if the skin effect for poloidal fields does not allow them to penetrate the tachocline. Using a simple mean-field kinematic - dynamo model, we demonstrate that, in the absence of tachocline radial shear participating in the dynamo process, a latitudinal differential rotation can provide the necessary -effect to drive an oscillation in an interface dynamo, but it alone cannot produce the latitudinal migration. We show that to make an interface dynamo work with the constraints of interior structure and skin depth, a meridional circulation is essential. We conclude that a flux-transport dynamo driven by both the Babcock-Leighton and interface/bottom -effects is a robust large-scale dynamo for the Sun. Subject headings: Sun: activity Sun: interior Sun: magnetic fields 1. INTRODUCTION The cyclic evolution of the solar cycle magnetic features is most likely due to a magnetohydrodynamic dynamo operating inside the Sun. Over the past half century, many dynamo models have been proposed following Parker s - dynamo model (Parker 1955). Interface dynamos are a class of - dynamo model in which the -effect, responsible for producing a toroidal field by shearing a preexisting poloidal field by the strong radial differential rotation, works in the tachocline at the base of the convection zone, whereas the -effect, originating from helical turbulence and responsible for regenerating the poloidal fields, works in a thin layer just above the tachocline. The interaction between the two generating layers takes place through diffusion across the core-envelope interface. In the early 1990s, the interface dynamos were proposed ( Parker 1993) in order to provide solutions to the problem of flux storage in the convection zone, and also because the -effect is concentrated at the base of the convection zone rather than throughout its bulk (see also MacGregor & Charbonneau 1997; Markiel & Thomas 1999). In the interface dynamos, as in all other - dynamos, equatorward-propagating dynamo wave solutions can be obtained by satisfying the so-called Parker-Yoshimura sign rule, which states that : < ê hastobenegativeinthenorthern hemisphere. Since the solar internal rotation () is no longer a free parameter, having been fixed by helioseismology, only a suitable choice of the sign and the latitudinal profile of the interface -effect can satisfy the Parker-Yoshimura sign rule. Equatorward-propagating dynamo wave solutions were sought in - dynamo models in order to explain the equatorward migration of the sunspot belts. However, this sign rule is no longer a constraint in the more recent dynamo models of flux-transport type, in which the dynamo cycle period is primarily governed by the meridional circulation ( Wang & Sheeley 1991; Dikpati & Charbonneau 1999; Küker et al. 2001). Mean-field kinematic flux-transport dynamo models have also reproduced the majority of solar cycle features, including the appearance of sunspots in two belts at 35,the 1 The National Center for Atmospheric Research is sponsored by the National Science Foundation. 647 solar cycle period of 11 yr, and the correct phase relation between the equatorward-propagating sunspot belts and the polewarddrifting large-scale weak fields (Dikpati et al. 2004). In this paper, our motivation is to first investigate how well a mean-field kinematic interface dynamo driven by the conventional tachocline radial differential rotation can reproduce one or more of these features. Then we explore the viability of such a dynamo model for the Sun in the presence of the additional constraints on the solar internal structure from recent helioseismic results as well as the skin effect of poloidal fields posed by the theoretical calculation by Garaud (1999). Apart from fixing the solar internal rotation pattern, helioseismic observations and inferences now reveal hydrostatic and dynamical structures of the solar interior. The overshoot layer between the radiative zone boundary (r rz ) and the Schwarzschild boundary (r Sb ) is defined as a thin layer where the temperature gradient equals both its adiabatic and radiative value (9 ¼ 9 a ¼ 9 r ). In Figure 1, this thin layer has been denoted in gray. The helioseismically measured location for the base of the convection zone (r cz ; shown by the dashed line in Fig. 1), where the temperature gradient changes sharply, is approximately 0:713 0:001R (Christensen-Dalsgaard et al. 1995; Basu 1997). The gray shaded region above the dashed line at r ¼ r cz is the helioseismically observed part of the overshoot region, which is slightly subadiabatic but may have positive convective flux. However, the region below r cz, containing negative convective flux, could store the superequipartition magnetic field (for a more detailed discussion, see x 2 of Corbard et al. 2001). While estimates of the locations of the convection zone base and the overshoot layer may change with progress in helioseismology, according to present knowledge of the helioseismically determined structure of the solar interior, we can construct Figure 1, which shows that the observed solar internal rotation pattern has only a small part of the tachocline located in the slightly subadiabatic overshoot layer above r cz (Charbonneau et al. 1999; Basu & Antia 2001). The rest of it is in the strongly subadiabatic radiative interior (extending below the gray shading). The diffusivity in the radiative region should approach a value corresponding to the molecular diffusivity. In such a situation, the skin depth of the oscillating poloidal fields that are generated in the outer edge of the tachocline is no more than a

2 648 DIKPATI, GILMAN, & MacGREGOR Vol. 631 presence of a depth-dependent but isotropic diffusivity. The governing ¼ 1 92 r 2 sin 2 A þ (r; ; B )B ; ¼ rsin B p =: : <:<B ê þ 9 2 r 2 sin 2 B ; ð2þ Fig. 1. Gray shaded region extending from radiative boundary (r rz )to Schwarzschild boundary (r Sb ) represents the overshoot layer; r cz represents the base of the convection zone where the temperature gradient changes abruptly to subadiabatic value. The gray shaded region above r cz is the helioseismically observed part of the overshoot layer. Solar rotation isocontours plotted in the meridional cut show that most of the tachocline is located in the subadiabatically stratified region. few hundredths of a percent of the solar radius (Garaud 1999). Using a simplified model calculation, Garaud (1999) has shown how deep the poloidal fields must penetrate to reach the tachocline region in order to be sheared to produce the significant toroidal fields. Our present knowledge about helioseismically determined solar internal structure and the theoretically obtained skin depth of the poloidal fields into the solar interior raises questions about the viability of interface-type dynamos for explaining the solar cycle. The purpose of this paper is to explore a kinematic mean-field - interface dynamo model in order to address the following questions. Can the latitudinal shear alone drive a dynamo when the skin depth effect prevents the poloidal fields from reaching the tachocline from an -effect layer at the interface? Since an equatorward-propagating dynamo wave solution is not possible without accessing the tachocline radial shear, how can an interface dynamo work for the Sun in producing the activity cycle? Given these difficulties in the applicability of an interface dynamo for the Sun, what role does the interface-type feature (which is already present in the Sun) play in solar dynamo processes? We describe an interface dynamo model in x 2 and present our answers to the above questions in x 3 by performing the numerical simulations of a simple mean-field kinematic - interface dynamo model in the partial and full absence of a tachocline, and then including a meridional circulation. We close with concluding remarks in x MODEL DESCRIPTION In order to run a typical interface dynamo simulation, we start from an axisymmetric mean-field - dynamo model in the where B p ¼ :<(Aê ) denotes the poloidal field, B the toroidal field, the depth-dependent diffusivity, the -effect, and the solar internal rotation. In the kinematic regime, we need to prescribe the dynamo ingredients. We adopt the following forms for and : ¼ core þ T 2 1 þ erf r 0:74R ; ð3þ 1 1 ¼ 0 cos 1 þ e ( =6) 4 1þ erf r 0:725R r 0:74R ; 1 erf for 0 <<=2; ð4þ 1 1 ¼ 0 cos 1 þ e ð 5=6 Þ 4 1þ erf r 0:725R r 0:74R ; 1 erf for =2 <<: ð5þ We use here a single-step diffusivity, as plotted in Figure 2, to include just the interface feature. We define the interface as a thin layer, the gray shaded region above the vertical solid line at r ¼ r cz ¼ 0:713R in Figure 2, through which the diffusivity drops rapidly from a higher diffusivity value to a few orders of magnitude lower. A two-step diffusivity profile, as used in Dikpati et al. (2002), could also be used in order to include the second diffusivity jump across the supergranular layer, but we omit that here for simplicity. We select an -effect peaked around the midlatitude instead of a simple classical cos profile because the classical convection is likely to be very weak near the interface. Therefore the -effect there should be dominated by the other sources of kinetic helicity, such as that generated by various Fig. 2. Diffusivity profile plotted as function depth. The gray shaded region above the solid vertical line at r ¼ 0:713R denotes the thin interface layer in our model. The entire gray shaded region denotes the tachocline.

3 No. 1, 2005 INTERFACE DYNAMO FOR THE SUN 649 instabilities at the interface ( Ferriz-Mas et al. 1994; Thelen 2000; Dikpati & Gilman 2001a). Those kinetic helicity profiles are much more complicated than a simple cos profile, but they show the general trend of being maximum around the midlatitudes. These kinds of -effect profiles in the tachocline or at the interface have previously been used in solar dynamo simulations; the detailed studies indicate that the qualitative results are not too sensitive to the choice of tachocline or interface -effect profile in the fluxtransport dynamo models ( Dikpati & Gilman 2001b; Küker et al. 2001), but the results may be sensitive in the case of a conventional - type interface dynamo. We use the same -profile as given in Charbonneau & MacGregor (1997) and also by Dikpati & Charbonneau (1999). Throughout the calculation, we use an core of 5 ; 10 8 cm 2 s 1 and an T of 3 ; cm 2 s 1.Thevalueof is 2.5 in equations (4) and (5), and we choose the value of 0 so that the maximum amplitude of the -effect is about 25 cm s 1. Incorporating these mathematical forms of the dynamo ingredients, we solve equations (1) and (2) numerically. At the two poles and at the surface, we apply the same boundary conditions used by Dikpati & Gilman (2001b), but the bottom boundary is treated differently. Given the knowledge gained from the helioseismic inference (see the review in Corbard et al. 2001) and the theory of skin depth of poloidal fields (Garaud 1999), it is plausible to assume that the oscillating poloidal fields will not be able to penetrate below 0.7R. So we place the bottom boundary at 0.7R, at which the poloidal fields are zero and the toroidal fields follow the condition for the perfectly conducting core [@/@r(rb ) ¼ 0] there. We note that there may be other effects to include in the interface and flux-transport type dynamos, such as anisotropies in the -effect and diffusivity profiles. But those are beyond the scope of this paper and have not yet been systematically explored for their own sake yet. We are primarily concerned with comparing our results to those of the previously published work on solar dynamos in which such anisotropies have usually not been included. 3. RESULTS In order to compare our results with observations, we derive the usual sort of time-latitude plots from the model output to produce butterfly diagrams. Following the usual procedure (see, e.g., Dikpati & Charbonneau 1999), we extract the toroidal field strengths from the interface (B j r¼0:713r ) and radial fields from the surface (B r j r¼r ), and plot in the time-latitude planes as shown in the five panels of Figure 3 for five cases: (a) an interface dynamo with a tachocline; (b) an interface dynamo without a tachocline, but with the solar latitudinal differential rotation present; (c) an interface dynamo without a tachocline but with a solar-like meridional circulation; (d ) an interface dynamo with a meridional circulation having about 45 m s 1 surface flow speed (about 2 times a typical solar value); and (e) a dynamo driven by interface-type and Babcock-Leighton type -effects in the absence of a tachocline, but in the presence of a meridional circulation. In all cases, we include the interface-type -effect layer. The strengths of B j r¼0:713r are plotted in contours, and B r j r¼r is plotted in a gray-scale map in each frame. We discuss each panel of Figure 3 successively in the following three subsections A Typical Interface Dynamo Solution Not surprisingly, we can see in Figure 3a the equatorwardpropagating dynamo wave for B j r¼0:713r at the midlatitudes Fig. 3. Five frames showing interface dynamo solutions in time-latitude diagrams for (a) an interface-type - dynamo in the presence of a tachocline; (b) an interface dynamo without a tachocline, but with solar-like latitudinal differential rotation; (c) an interface dynamo without a tachocline, but with a solarlike meridional circulation as observed at the surface; (d) an interface dynamo without a tachocline and with a meridional flow having 2 times larger amplitude than that observed at the solar surface; and (e) a dynamo driven both by interfacetype and Babcock-Leighton type surface -effects in the presence of meridional circulation, but in absence of a tachocline. In all frames, gray-scale shades represent the surface radial fields, and contours represent the toroidal field strength at the interface. Spacing is logarithmic, with the innermost contour having a value of 100 kg, and three contours cover 1 order of magnitude of field strength. < 0. Along with the subsurface toroidal fields, their vector counterparts, namely the surface radial fields, in this model also drift equatorward, in contrast to the observed poleward drift of these fields. Butterfly wings are confined around the midlatitudes. We also see many overlapping butterfly wings that are also not solar-like. It is not possible to completely avoid this feature in a mean-field kinematic interface dynamo operating with isotropic diffusivity. A more solar-like cycle period and the confinement of the butterfly wings within 35 latitude can be obtained by suitably tuning the -effect and (given that is fixed by helioseismology), but we made no attempt to do that, because it is known that the interface-type dynamos are not robust enough to produce many other solar cycle features. For example, the poleward drift of the large-scale radial field branch in 11 yr with a quarter-period phase lag (Wang & Sheeley 1991) with the sunspot fields has not been reproduced by this class of dynamos. The evolution of the largescale surface radial fields is also important because these are the

4 650 DIKPATI, GILMAN, & MacGREGOR Vol. 631 fields that are being transported toward the pole, cancel the oldpolarity flux, and cause the polar field reversal every 11 yr. Their cyclic evolution, therefore, is not a mere surface phenomena, but is intimately tied to the solar cycle. In fact, all models that have so far been able to successfully explain the evolution of the large-scale surface radial fields include the cyclic information in them; the surface flux-transport models of Wang et al. (1989) and Schrijver & De Rosa (2003) have also implemented the cyclic source of the surface radial fields arising from the decay of active regions. We conclude for this subsection that a typical - interface dynamo, even without simulating the correct phase relationship between the cyclic polar fields and sunspot fields, can produce equatorward migration of the spot-producing toroidal fields if the tachocline radial shear is present. However, a more serious concern is posed by the helioseismically inferred structure of the solar interior and the theoretically obtained skin-depth value of the poloidal fields, namely whether an - interface-type dynamo can operate in the Sun if the tachocline is not available to it. We address this issue in x Can an Interface Dynamo Operate without Accessing the Tachocline? In all interface dynamos so far studied, the -effect, which is one of the necessary processes in the dynamo loop, has been provided by the radial shear in the tachocline regions. Above the outer edge of the tachocline and in the bulk of the convection zone, there is latitudinal shear but almost no radial shear. If the skin depth of the poloidal fields generated by the -effect layer at the interface is such as to prevent them from reaching the tachocline, the radial differential rotation cannot work on the poloidal fields to produce the toroidal fields. Then the question is whether a pure latitudinal shear can provide the necessary -effect to drive such a dynamo. Noting that the sharp drop in the diffusivity profile from a turbulent value to the molecular value should take place near the base of the convection zone (0.713R), we can perform the simulation for this situation in two different ways: (1) by raising the perfectly conducting bottom boundary of the dynamo domain above the top of the tachocline (taken at 0.72R in this calculation), or (2) by keeping the perfectly conducting bottom boundary at 0.7R, but pushing the top of the tachocline below that. Both of these cases are consistent with the helioseismic constraints on the solar internal rotation profile. Our calculations indicate that both methods produce qualitatively similar results; we present here only the results obtained usingmethod2.figure3bshows such a dynamo solution, but we see it is only an oscillation without any dynamo wave propagation in the latitudinal direction. The latitudinal shear alone is able to create the oscillation because the large-scale poloidal fields are predominantly horizontal; working on B can continuously produce the toroidal fields and hence sustain a dynamo oscillation. Dynamos migratory in latitude are not possible in this case because the Parker-Yoshimura sign rule cannot be satisfied; nor is there any other way of creating a latitudinal propagation of magnetic fields in a pure - type model, although there can be propagation of the dynamo wave in the radial direction. Therefore, from our results in xx 3.1 and 3.2, we conclude that an - interface dynamo without a tachocline does not work for the Sun Is There a Remedy? Given that we obtain a latitudinally nonpropagating dynamo oscillation in an interface dynamo driven by the solar latitudinal differential rotation in the absence of the radial differential rotation, the question naturally arises whether we can produce the latitudinal propagation of the dynamo-generated fields by some other means when the classical dynamo wave in the latitude direction is absent due to the absence of the tachocline. We hypothesize that the presence of a meridional circulation with a poleward surface flow such as that observed in the Sun (Duvall 1979; Komm et al. 1993; Hathaway et al. 1996; Haber et al. 2002; Basu & Antia 2003) and an equatorward subsurface return flow constructed from mass conservation may provide the necessary transport of magnetic fields in this dynamo. We already know that flux-transport dynamos with a tachocline work well for the Sun. But can flux-transport dynamos without any tachocline radial differential rotation also work for the Sun? By including a meridional flow, which closes at the base of the convection zone (r ¼ r cz ; see Gilman & Miesch 2004) and has a counterclockwise (clockwise) flow cell in the north (south) hemisphere, we solve dynamo equations (1) and (2) with the additional terms and 1 r 1 (u =:)(r sin A) r rb u B ; ð7þ respectively, in the left-hand sides of equations (1) and (2). Figure 3c shows only a faint signature of latitudinal migration of the dynamo-generated toroidal fields at the interface. A solarlike surface flow speed of 20 m s 1 maximum produces long cycles with a full-cycle period of about 50 yr, much longer than the solar cycle (note the compressed time axis in Fig. 3c). By increasing the surface flow speed of the meridional circulation up to a value of 45 m s 1 (too large compared to solar observations by a factor of 2), we construct another time-latitude diagram, as shown in Figure 3d. We see in this frame a period of reversal that is close to solar but only a slightly improved migratory pattern in latitude, certainly still not very solar-like. Nevertheless, migratory dynamo solutions are obtained in the absence of a tachocline when a meridional circulation is included. It is easy to see that it also works when the tachocline is partially accessed (no time-latitude diagram is presented for this case). Furthermore, the poleward drift of the surface radial fields (see the gray-scale map in Figs. 3c and 3d ) can be reproduced. Because the interface poloidal fields first drift with the meridional flow toward the equator at the base of the convection zone, they reach the surface with the upwelling flow there, and finally they drift poleward with the poleward surface flow. The dynamo cycle period in a flux-transport dynamo driven by an interface -effect is also determined by meridional flow speed, as in other flux-transport dynamos (Wang & Sheeley 1991; Dikpati & Charbonneau 1999; Küker et al. 2001). But we find here that the flow speed necessary to produce a solar-like cycle period is higher than that observed at the surface. This is because a major part of the dynamo operates in the low-diffusivity regions, which keep the fields frozen long enough; the flow speed necessary to produce a solar-like cycle period is higher than that observed at the surface. This problem goes away if two additional mechanisms, namely, the buoyant rise of the toroidal fields to the surface and the Babcock-Leighton type surface poloidal source (Babcock 1959; Leighton 1964) that originates from the decay of these erupted fields, are included in the model. The result is shown in

5 No. 1, 2005 INTERFACE DYNAMO FOR THE SUN 651 Figure 3e. These additional processes contribute to the transport of magnetic fields. Therefore, in a kinematic regime and under the mean-field approximation, a dynamo driven by both the surface and bottom -effects and in the presence of meridional circulation is a robust solar dynamo whether the tachocline participates or not. In fact, the inclusion of the tachocline did not lead to a butterfly diagram (in the calibrated flux-transport dynamo model of Dikpati et al. 2004) significantly different than Figure 3d; the tachocline shear plays a minor role when the poloidal fields are primarily horizontal. The addition of a Babcock-Leighton source term to the dynamo makes it work much better for the Sun, yielding a shorter compared to what we obtain in Figure 3c and more solar-like dynamo period, and a more solar-like butterfly diagram with a better phase relation between the phase of sunspot number and polar field. The period is shorter for the same meridional flow because the influence of the newly induced toroidal field is felt instantly at the top, due to the inclusion of a parametric representation of rising flux tubes. The poloidal fields from this surface source are then swept quickly and coherently to the poles in just 2 to 3 yr, and carried below to help produce the next cycle. Without the Babcock-Leighton effect, both diffusion and advection by meridional flow bring poloidal flux to the surface from below more slowly. In the flux-transport scenario, it is the slow branch of the meridional circulation that sets the period, determining how quickly poloidal flux at the bottom, produced from all sources, is swept equatorward to midlatitudes and sheared into toroidal fields to be the source for the next sunspot cycle. All of this goes on whether or not the poloidal field is sheared by the radial differential rotation at the bottom. The action of latitudinal differential rotation is enough. But why is the cycle period longer in Figure 3c than in Figure 3e, or why do we need a larger flow speed in Figure 3d to produce a solar-like cycle period in the flux-transport dynamo with only the interface -effect? The answer is that the surface poloidal fields generated by the Babcock-Leighton mechanism no longer contain the signature of any classical speed of radial or latitudinal propagation when they reach the bottom via the conveyor belt of the meridional circulation. By contrast, the poloidal fields generated at the interface contain a classical propagation speed radially downward in this case, because (@/@) > 0in the northern hemisphere. So the poloidal fields generated by this interface -effect propagate downward and thereby partially escape the equatorward transport by meridional flow, causing the dynamo period to get longer. Some have argued that the Babcock-Leighton effect and surface transport processes may be decoupled from and incidental to the basic solar global dynamo that produces the 11 yr period. We see no physical basis within mean-field or any other solar dynamo theory for assuming this separation. And mean-field theory, as modified to include observed meridional circulation and observed surface poloidal source terms, has been demonstrated in the references cited above to be fully capable of accommodating these processes, so why not include them and see what effect they have? We are able to estimate their amplitudes and profiles much better than we can for other sources of -effect, amplitudes and profiles of turbulent magnetic diffusivity, or either of their possible anisotropies. 4. COMMENTS AND CONCLUSIONS We show here that a conventional interface-type dynamo will not be able to produce an equatorward-propagating dynamo wave(asshowninfig.3b) if the poloidal fields from the interface cannot penetrate the tachocline (Garaud 1999). Without the tachocline radial shear, the -effect provided by the latitudinal shear at the interface can drive the equatorward-propagating dynamo if a solar-like meridional circulation is included, but it cannot produce the correct cycle period (see Figs. 3c and 3d). By contrast, addition of a dominant Babcock-Leighton -effect into such an - interface-type flux-transport dynamo works well whether the poloidal fields penetrate the tachocline or not. This is because the radial differential rotation is not needed to produce a dynamo wave propagating equatorward in this class of model; instead, a meridional circulation consistent with observations transports magnetic fields toward the equator at the rate necessary to generate the observed solar cycle period (see Fig. 3e and the physical explanation in x 3.3). While the weak latitudinal differential rotation can provide the necessary -effect, working on a B much stronger than the B r component of the poloidal fields, and can produce dynamo alone cannot produce the latitudinal migration of the dynamo-generated magnetic fields. In such a situation, the inclusion of a meridional circulation into an interface dynamo provides a plausible remedy for producing some equatorward transport of the dynamo-generated magnetic fields. This is true both in cases when the poloidal fields cannot reach the tachocline at all and when they can partially reach the tachocline. But for a solar-like surface meridional flow speed and the equatorward return flow predicted from mass conservation, as we have already seen in Figure 3c, tilts of the butterfly wings differ significantly from those in the observed butterfly diagrams. Until we gain further knowledge about the subadiabaticity, diffusivity, and skin depth of the poloidal fields across the interface, it is fair to assume that the major part of the tachocline does not participate in the large-scale solar dynamo processes. One of our findings, that the inclusion of a Babcock-Leighton type surface -effect along with a relatively small interface -effect can reproduce the majority of the solar cycle features even in the absence of a tachocline, also reinforces the conclusion from previous calculations that a flux-transport type dynamo driven primarily by a Babcock-Leighton type surface poloidal source and a small bottom -effect (Dikpati et al. 2004) is a robust large-scale kinematic solar dynamo. We close by discussing a remaining issue: is the interface feature necessary at all for the solar dynamo process, or can we locate the large-scale solar dynamo entirely at the surface where the Babcock-Leighton -effect and the near-surface radial shear layer together can close the dynamo loop near the photosphere? If the latter were the case, we could not explain the observed tilts of the bipolar active regions at the surface. Such tilts can be acquired by the action of the Coriolis force working on the toroidal flux tubes in the course of their buoyant rise from the base of the convection zone. And such strong toroidal fields can be produced at the bottom not due to the tachocline, but due to the fact that there is an interface across which the turbulent diffusivity of the convection zone sharply drops to a value several orders of magnitude lower and, hence, helps amplify the toroidal fields against their ohmic dissipation. We thank Mark Miesch for reviewing the manuscript and for his helpful comments. We extend our thanks to an anonymous referee whose constructive criticisms on the earlier versions of the manuscript have helped improve the paper. We acknowledge the support from NASA through awards W and W

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