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How far are we from a ‘Standard Model’ of the solar dynamo?
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Advances in Space Research 41 (2008) 868–873 www.elsevier.com/locate/asr
How far are we from a ‘Standard Model’ of the solar dynamo? Arnab Rai Choudhuri Department of Physics, Indian Institute of Science, Bangalore 560012, India Received 31 October 2006; received in revised form 31 March 2007; accepted 2 April 2007
Abstract Over the last few years, dynamo theorists seem to be converging on a basic scenario as to how the solar dynamo operates. The strong toroidal component of the magnetic field is produced in the tachocline, from where it rises due to magnetic buoyancy to produce active regions at the solar surface. The decay of tilted bipolar active regions at the surface gives rise to the poloidal component, which is first advected poleward by the meridional circulation and then taken below the surface to the tachocline where it can be stretched to produce the toroidal component. The mathematical formulation of this basic model, however, involves the specification of some parameters which are still uncertain. We review these remaining uncertainties which have resulted in disagreements amongst various research groups and have made it impossible to still arrive at something that can be called a standard model of the solar dynamo. 2007 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Sun: magnetic fields; Solar dynamo
1. Introduction There exist standard models in many subjects such as particle physics and cosmology. In our own field of solar physics, we have a standard model of the Sun’s internal structure, which has been triumphantly validated by helioseismology in the last couple of decades. Is it possible for us to have a standard model of the solar dynamo which is responsible for the generation of the cyclically evolving solar magnetic field? The main difficulty in arriving at a standard model of the solar dynamo is due to the fact that the solar dynamo is stochastically forced by the convective turbulence which we understand very little. However, underneath the various irregularities of the solar cycle, we can discern a reasonably regular mean periodic behaviour. We can think of the solar cycles as made up of an underlying mean periodic cycle and irregularities superposed on it. Can we at least have a standard model of the mean behaviour? Even that involves averaging over various fluctuating quantities. However, considerable progress has been made in the last few years, although we may not yet have a standard model on which E-mail address: [email protected]
all experts agree. My main purpose here is to review how much agreement at present exists amongst solar dynamo theorists and to highlight the still existing disagreements. A more detailed survey of the background can be found in Choudhuri (2003). 2. Basics of the solar dynamo In an axisymmetric model using spherical coordinates, the magnetic field can be written as B ¼ Bðr; h; tÞe/ þ r ½Aðr; h; tÞe/ :
ð1Þ
The magnetic field component in the / direction, denoted by B(r, h, t), is called the toroidal component. We believe that active regions on the solar surface arise from this component by magnetic buoyancy. The remaining part $ · [A(r, h, t)e/], which can be represented by drawing field lines in the poloidal plane, is called the poloidal component. We believe that the weak diffuse magnetic fields outside sunspots is associated with the poloidal component. Since the magnetic field is approximately frozen in the solar plasma, differential rotation can stretch out the poloidal component to produce the toroidal component. As differential rotation is mainly concentrated in the tachocline
0273-1177/$30 2007 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2007.04.006
A.R. Choudhuri / Advances in Space Research 41 (2008) 868–873
at the bottom of the solar convection zone, the strong toroidal field is expected to be produced there. To understand the formation of active regions on the solar surface, we have to study the dynamics of toroidal flux tubes rising through the solar convection zone (SCZ) due to magnetic buoyancy, starting from the bottom of SCZ. Three-dimensional simulations of such flux tube rise established that the Coriolis force plays a major role in this problem (Choudhuri and Gilman, 1987; Choudhuri, 1989; D’Silva and Choudhuri, 1993; Fan et al., 1993). The equipartition magnetic field at the bottom of SCZ would be of order 104 G. It was found that flux tubes with such field strengths are diverted by the Coriolis force to unrealistically high latitudes. Only if the magnetic field inside flux tubes is of order 105 G, various observational constraints seem to be satisfied. Let us now come to the question of poloidal field generation. A very influential idea, often called the a-effect, was first proposed by Parker (1955) and then developed further by Steenbeck et al. (1966). According to this idea, the helical turbulence in the SCZ twists the toroidal component to give rise to magnetic fields lying in the poloidal plane. However, if the toroidal component at the bottom of SCZ is as strong as 105 G, then turbulence will not be able to twist it and the a-effect will be suppressed. Hence many dynamo theorists have invoked an alternative idea of poloidal field generation proposed by Babcock (1961) and Leighton (1969). Bipolar sunspots appear tilted on the solar surface due to the action of the Coriolis force (D’Silva and Choudhuri, 1993). When such tilted bipolar regions decay, magnetic fields of opposite polarities get spread in slightly different latitudes, giving rise to a poloidal field. Thus bipolar sunspots, which originate from the toroidal component, act like a conduit through which the toroidal component gets transformed into the poloidal component. Putting together all these ideas, Choudhuri et al. (1995) and Durney (1995) succeeded in formulating a two-dimensional solar dynamo model in which the meridional circulation plays a very crucial role. Fig. 1 is a cartoon representation of this model. The toroidal field is produced by the differential rotation at the bottom of the SCZ and rises due to magnetic buoyancy. The Babcock–Leighton process at the surface gives rise to the poloidal field, which is advected by the meridional circulation first poleward and then down to the bottom of the SCZ where the strong differential rotation acts on it. A majority of the present-day dynamo theorists agree on this broad scenario. We thus seem to have at least a standard cartoon of the solar dynamo, if not a standard model. When we try to put this cartoon into a mathematical formulation, all kinds of disagreements crop up, as we shall see now.
3. Mathematical formulation and the various parameters Representing the magnetic field by (1), it can be shown that the poloidal and toroidal components evolve according to the following equations
869
++ Strong Differential Rotation ..... Babcock–Leighton Process Magnetic Buoyancy Meridional Circulation
Fig. 1. A cartoon explaining how the solar dynamo works.
oA 1 1 þ ðv:rÞðsAÞ ¼ gp r2 2 A þ aB; ot s s oB 1 o o 1 þ ðrvr BÞ þ ðvh BÞ ¼ gt r2 2 B ot r or oh s 1 dgt o ðrBÞ; þ sðBp :rÞX þ r dr or
ð2Þ
ð3Þ
where s = rsin h. Here, v is the meridional flow, X is the internal angular velocity of the Sun and a is the coefficient which describes the generation of poloidal field at the solar surface from the decay of bipolar sunspots. The turbulent diffusivities for the poloidal and toroidal components are denoted by gp and gt. Since the toroidal component is more concentrated than the poloidal component and turbulence is suppressed in the regions of strong toroidal component, we allow for the turbulent diffusivities of the two components to be different in principle (for further discussion, see Choudhuri et al., 2005, Section 3.3). The dynamo model indicated in the cartoon form in Fig. 1 is achieved by using the helioseismically determined X which has a strong gradient at the bottom of the SCZ, by choosing a meridional circulation v as indicated in the cartoon and by making a concentrated near the solar surface. Additionally, we have to specify magnetic buoyancy in some suitable fashion. We have to solve the coupled partial differential Eqs. (2) and (3) numerically. For a large range of the parameter space, the numerical code is found to relax to periodic solutions. To the best of our knowledge, only about 25 papers have so far presented two-dimensional dynamo calculations of the kind sketched in the cartoon of Fig. 1 in more than a decade. We list below some of the major papers from different groups (Choudhuri et al., 1995, 2004; Dur-
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A.R. Choudhuri / Advances in Space Research 41 (2008) 868–873
ney, 1995, 1997, 2000; Dikpati and Charbonneau, 1999; Charbonneau and Dikpati, 2000; Ku¨ker et al., 2001; Nandy and Choudhuri, 2001, 2002; Dikpati and Gilman, 2001; Bonanno et al., 2002; Guerrero and Muno˜z, 2004; Chatterjee et al., 2004; Dikpati et al., 2004; Charbonneau et al., 2004, 2005). It may be mentioned that an early paper by Wang et al. (1991) presented a radially averaged onedimensional model with some of the ideas to be developed in these later papers. The number of scientists working on different aspects of solar cycle is quite large. In contrast, it always puzzles me why the number of scientists trying to understand the origin of the solar cycle is so small. One of the difficulties of getting into this field used to be that a new group had to develop a code from scratch and test the results of other groups before thinking of doing anything new. However, we have now made our dynamo code SURYA developed in Bangalore available with a detailed guide (Choudhuri, 2005) to anybody who sends me an email request. Just as there seems to be a broad agreement amongst dynamo theorists about the standard cartoon shown in Fig. 1, all agree that (2) and (3) are the basic equations we have to solve. However, disagreements begin when we have to specify the various parameters appearing in (2) and (3). Let us list all the parameters below. Differential rotation X: This is now provided by helioseismology. Meridional circulation v: Helioseismology throws light on its nature in the upper half of the SCZ, but we have no direct observations on its nature in the lower half of the SCZ. Poloidal field source parameter a: We observe the Babcock–Leighton process on the solar surface, but we are not sure if there are processes in the body or at bottom of the SCZ which also contribute to a.
Turbulent diffusivities gp and gt: Their surface values can be estimated from the study of magnetic field diffusion on the surface and then we have to make reasonable assumptions underneath the surface. Magnetic buoyancy: This has to be specified in some way when solving Eqs. (2) and (3). While specifying these parameters, one has to introduce something which nonlinearly limits any possible runaway growth of the magnetic field. This is often done by including a quenching in the a parameter. In some of our calculations (Chatterjee et al., 2004), magnetic buoyancy limits the magnetic field growth and it is not necessary to include a quenching. According to my judgment, major differences exist amongst solar dynamo theorists at the present time on the following questions: (i) the nature of meridional circulation near the base of the SCZ; (ii) the distribution of a in the lower portions of the SCZ; (iii) the most satisfactory way of treating magnetic buoyancy. These are the primary questions which have to be settled if we want to arrive at a standard model of the solar dynamo. It should not be assumed that these are merely matters of finer details. We shall point out in the next section that these uncertainties can affect the nature of dynamo solutions in quite drastic ways. While specifying the various parameters, we, of course, have some constraints when we try to match different aspects of observational data. The left plot in Fig. 2 is a butterfly diagram of sunspots superposed on a time–latitude plot of longitudinally averaged radial magnetic field at the solar surface. Any realistic solar dynamo model should satisfy the following characteristics. The solution should be cyclic with a period of about 22 years.
Fig. 2. Butterfly diagram of sunspots superposed on the time–latitude plot of Br. The observational plot is shown on the left. The comparable theoretical plot obtained by the dynamo model of Chatterjee et al. (2004) is on the right.
A.R. Choudhuri / Advances in Space Research 41 (2008) 868–873
The butterfly diagram of sunspots should be restricted to latitudes below 40. The weak fields outside active regions should drift poleward. Polar field reversal should take place at the time of sunspot maximum. The solution should be of dipolar nature. (The present solar magnetic field appears dipolar, but was it always so?) The model should provide an explanation why magnetic helicity of active regions tends to be negative (positive) in northern (southern) hemisphere. The right plot in Fig. 2 is obtained theoretically from our recent model (Chatterjee et al., 2004) and should be compared with the observational plot on the left side of Fig. 2. Our model is one of the first detailed models to be constructed that satisfies all the above requirements. However, the crucial question is whether the parameters we have used in our model correspond to the ‘reality’ inside the Sun. It may be possible that a completely different combination of parameters also gives a model satisfying all the requirements! 4. Disagreements amongst dynamo theorists The tachocline has a much stronger differential rotation at high latitudes (where oX/or is negative) compared to low latitudes (where oX/or is positive). Hence, helioseismically determined differential rotation tends to produce a strong toroidal field at high latitudes, whereas we see sunspots at low latitudes. This is clearly seen in the best theoretical butterfly diagrams obtained by Dikpati and Charbonneau (1999, Fig. 6) and Ku¨ker et al. (2001, Fig. 10), who were amongst the first to put the helioseismically determined X in their dynamo models. These theoretical butterfly diagrams indicate strong magnetic activity at high latitudes and look nothing like observational butterfly diagrams. Nandy and Choudhuri (2002) proposed a way out of this difficulty. Suppose the meridional circulation penetrates a little bit below the bottom of the SCZ into the region of convective overshooting. Then the strong toroidal field produced at high latitudes within the tachocline will be pushed into the stable layers underneath the SCZ by the meridional circulation and will not be able to rise up to form active regions at high latitudes. This toroidal field will then be advected to lower latitudes through the stable layers by the meridional circulation, which will ultimately bring the toroidal field within the SCZ at low latitudes where active regions will form. On using meridional circulation confined within the SCZ, Nandy and Choudhuri (2002) found a butterfly diagram with activity concentrated at high latitudes (Fig. 2A in their paper). On making the meridional circulation penetrate below the bottom of SCZ, however, the butterfly diagram looked completely different and appeared very solar-like (Nandy and Choudhuri, 2002, Fig. 2B). It should be kept in mind that,
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unless there is some amount of overshooting turbulence below the bottom of SCZ, it is physically not possible for the meridional circulation to penetrate much (Gilman and Miesch, 2004). Dikpati et al. (2005) criticized this model on the ground that it is unphysical to assume that the meridional circulation penetrates below the bottom of SCZ. Choudhuri et al. (2005) respond that the bottom of SCZ is the least understood region of the Sun and one should be open-minded. They also point out that Dikpati and her colleagues themselves have been using deeply penetrating meridional circulation in their dynamo models without clearly mentioning it (Choudhuri et al., 2005, Fig. 2). Dikpati et al. (2004) is the first paper which claims to produce solar-like butterfly diagrams with non-penetrating meridional circulation. Surprisingly, there is no discussion in this paper how the authors solve the problem of magnetic buoyancy of toroidal field in high latitudes. Moreover, we and other groups (Jie Jiang, private communication) who tried to reproduce this model with other codes have not been able to reproduce the results of Dikpati et al. (2004). A second important uncertainty is regarding the distribution of a. If a is concentrated near the surface as expected for the Babcock–Leighton mechanism, Dikpati and Gilman (2001) found that the dynamo solution tends to be quadrupolar, whereas solar magnetic fields appear dipolar. Only on including an a at the bottom of SCZ, they were able to get dipolar solutions. This result was confirmed by Bonanno et al. (2002). Based on this result, Dikpati and Gilman (2001) argued that poloidal field generation should take place at the bottom of the SCZ. In a dipolar solution, poloidal field lines connect across the equator. Chatterjee et al. (2004) found that it is possible for this to happen and give dipolar solutions if the diffusivity of the poloidal field is sufficiently high, even when a is concentrated near the surface. Chatterjee et al. (2004) do not claim that a has to be strictly zero well below the surface. However, the argument that we must have a non-zero a at the base of SCZ to make the solutions dipolar is clearly incorrect. Finally we point out that there are two popular ways of handling magnetic buoyancy in two-dimensional dynamo models. 1. If B inside the SCZ is larger than a critical value, we move a part of it to the surface. Our group in Bangalore is following this recipe. 2. In the poloidal field source term aB in (2), we multiply a at the surface by the value of B at the bottom of SCZ rather than at the surface. This recipe was proposed by Choudhuri and Dikpati (1999) and is used in the dynamo models of the group in Boulder. Unfortunately, these two recipes can give solutions of totally different nature. Fig. 3 shows solutions obtained by treating magnetic buoyancy by these two different recipes in otherwise identical dynamo models. After Dikpati
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A.R. Choudhuri / Advances in Space Research 41 (2008) 868–873
Fig. 3. Plots of the contours of toroidal field and the poloidal field lines. (a) and (b) correspond to dynamo solutions in which magnetic buoyancy is treated by Recipe 1 and Recipe 2, respectively.
et al. (2005) pointed out that they cannot reproduce the results of Chatterjee et al. (2004), we realized that they are getting different results merely because they treat magnetic buoyancy differently (Choudhuri et al., 2005). Certainly more careful studies are needed to ascertain the best way of treating magnetic buoyancy in two-dimensional dynamo models. 5. Conclusion After Parker (1955) wrote his fundamental paper on the solar dynamo, many theorists in 1960s and 1970s attempted to build detailed models of the solar dynamo. Not much was known about the interior of the SCZ at that time and theorists were making assumptions which seemed ‘reasonable’ at that time, but now appear like wild guesses completely off the mark. The new era of solar dynamo research dawned after (i) helioseismology mapped out the angular velocity distribution in the interior of the Sun; (ii) detailed flux tube simulations made it clear that the toroidal field at the base of the SCZ should be as strong as 105 G; and (iii) it became increasingly clear that meridional circulation must play an important role. The solar dynamo models constructed in the last few years appear much more realistic than the earlier models which, however, played historically important roles and provided us with insight that was crucial in building the recent models – a fact which should not be forgotten. We now have a broad understanding of how the solar dynamo operates. I have called it the standard cartoon. However, the mathematical formulation of this cartoon requires specification of several parameters, some of which seem to have large uncertainties at the present time. It is possible to construct theoretical models which explain various aspects of observational data reasonably well. But such theoretical models may not be unique, as the models developed in Bangalore and Boulder use very different combinations of parameters. We urgently need the induction of new groups in this field who may be able to bring
fresh outlook and look at the existing controversies from an impartial point of view. Acknowledgments This review was presented in the COSPAR Scientific Assembly in Beijing when I was a Visiting Professor at the National Astronomical Observatories of China. I am grateful to Jingxiu Wang and his colleagues in the National Astronomical Observatories for the very warm hospitality I received. I thank Piyali Chatterjee for valuable discussions and for help in producing the figures. References Babcock, H.W. The topology of the Sun’s magnetic field and the 22-year cycle. Astrophys. J. 133, 572–587, 1961. Bonanno, A., Elstner, D., Ru¨diger, G., Belvedere, G. Parity properties of an advection-dominated solar a2X-dynamo. Astron. Astrophys. 390, 673–680, 2002. Charbonneau, P., Blais-Laurier, G., St.Jean, C. Intermittency and phase persistence in a Babcock–Leighton model of the solar cycle. Astrophys. J. 616, L183–L186, 2004. Charbonneau, P., Dikpati, M. Stochastic fluctuations in a Babcock– Leighton model of the solar cycle. Astrophys. J. 543, 1027–1043, 2000. Charbonneau, P., St-Jean, C., Zacharias, P. Fluctuations in Babcock– Leighton dynamos. I. Period doubling and transition to chaos. Astrophys. J. 619, 613–622, 2005. Chatterjee, P., Nandy, D., Choudhuri, A.R. Full-sphere simulations of a circulation-dominated solar dynamo: exploring the parity issue. Astron. Astrophys. 427, 1019–1030, 2004. Choudhuri, A.R. The influence of the Coriolis force on flux tubes rising through the solar convection zone. Sol. Phys. 123, 217–239, 1989. Choudhuri, A.R. The solar dynamo as a model of the solar cycle, in: Dwivedi, B.N. (Ed.), Dynamic Sun. Cambridge University Press, pp. 103–127, 2003. Choudhuri, A.R. The User’s Guide to the Solar Dynamo Code SURYA (available on request), 2005. Choudhuri, A.R., Chatterjee, P., Nandy, D. Helicity of solar active regions from a dynamo model. Astrophys. J. 615, L57–L60, 2004. Choudhuri, A.R., Dikpati, M. On the large-scale diffuse magnetic field of the Sun. II. The contribution of active regions. Sol. Phys. 184, 61–76, 1999.
A.R. Choudhuri / Advances in Space Research 41 (2008) 868–873 Choudhuri, A.R., Gilman, P.A. The influence of the Coriolis force on flux tubes rising through the solar convection zone. Astrophys. J. 316, 788– 800, 1987. Choudhuri, A.R., Nandy, D., Chatterjee, P. Reply to the comments of Dikpati et al. Astron. Astrophys. 437, 703–704, 2005. Choudhuri, A.R., Schu¨ssler, M., Dikpati, M. The solar dynamo with meridional circulation. Astron. Astrophys. 303, L29–L32, 1995. Dikpati, M., Charbonneau, P. A Babcock–Leighton flux transport dynamo with solar-like differential rotation. Astrophys. J. 518, 508– 520, 1999. Dikpati, M., de Toma, G., Gilman, P.A., et al. Diagnostics of polar field reversal in solar cycle 23 using a flux transport dynamo model. Astrophys. J. 601, 1136–1151, 2004. Dikpati, M., Gilman, P.A. Flux-transport dynamos with a-effect from global instability of tachocline differential rotation: a solution for magnetic parity selection in the Sun. Astrophys. J. 559, 428–442, 2001. Dikpati, M., Rempel, M., Gilman, P.A., MacGregor, K.B. Comments on ‘‘Full-sphere simulations of circulation-dominated solar dynamo: exploring the parity issue’’. Astron. Astrophys. 437, 699–702, 2005. D’Silva, S., Choudhuri, A.R. A theoretical model for the tilts of bipolar magnetic regions. Astron. Astrophys. 272, 621–633, 1993. Durney, B.R. On a Babcock–Leighton dynamo model with a deep-seated generating layer for the toroidal magnetic field. Sol. Phys. 160, 213– 235, 1995. Durney, B.R. On a Babcock–Leighton solar dynamo model with a deepseated generating layer for the toroidal magnetic field. IV. Astrophys. J. 486, 1065–1077, 1997.
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Durney, B.R. On the torsional oscillations in Babcock–Leighton solar dynamo models. Sol. Phys. 196, 1–18, 2000. Fan, Y., Fisher, G.H., DeLuca, E.E. The origin of morphological asymmetries in bipolar active regions. Astrophys. J. 405, 390–401, 1993. Gilman, P.A., Miesch, M.S. Limits to penetration of meridional circulation below the solar convection zone. Astrophys. J. 611, 568–574, 2004. Guerrero, G.A., Muno˜z, J.D. Kinematic solar dynamo models with a deep meridional flow. Mon. Not. R. Astron. Soc. 350, 317–322, 2004. Ku¨ker, M., Ru¨diger, G., Schultz, M. Circulation-dominated solar shell dynamo models with positive alpha-effect. Astron. Astrophys. 374, 301–308, 2001. Leighton, R.B. A magneto-kinematic model of the solar cycle. Astrophys. J. 156, 1–26, 1969. Nandy, D., Choudhuri, A.R. Towards a mean field formulation of the Babcock–Leighton type solar dynamo. I. a-coefficient versus Durney’s double ring approach. Astrophys. J. 551, 576–585, 2001. Nandy, D., Choudhuri, A.R. Explaining the latitudinal distribution of sunspots with deep meridional flow. Science 296, 1671–1673, 2002. Parker, E.N. Hydromagnetic dynamo models. Astrophys. J. 122, 293–314, 1955. Steenbeck, M., Krause, F., Ra¨dler, K.H. A calculation of the mean electromotive force in an electrically conducting fluid in turbulent motion under the influence of Coriolis forces. Z. Naturforsch. 21a, 369–376, 1966. Wang, Y.-M., Sheeley, N.R., Nash, A.G. A new solar cycle model including meridional circulation. Astrophys. J. 383, 431–442, 1991.
Advances in Space Research 41 (2008) 868–873 www.elsevier.com/locate/asr
How far are we from a ‘Standard Model’ of the solar dynamo? Arnab Rai Choudhuri Department of Physics, Indian Institute of Science, Bangalore 560012, India Received 31 October 2006; received in revised form 31 March 2007; accepted 2 April 2007
Abstract Over the last few years, dynamo theorists seem to be converging on a basic scenario as to how the solar dynamo operates. The strong toroidal component of the magnetic field is produced in the tachocline, from where it rises due to magnetic buoyancy to produce active regions at the solar surface. The decay of tilted bipolar active regions at the surface gives rise to the poloidal component, which is first advected poleward by the meridional circulation and then taken below the surface to the tachocline where it can be stretched to produce the toroidal component. The mathematical formulation of this basic model, however, involves the specification of some parameters which are still uncertain. We review these remaining uncertainties which have resulted in disagreements amongst various research groups and have made it impossible to still arrive at something that can be called a standard model of the solar dynamo. 2007 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Sun: magnetic fields; Solar dynamo
1. Introduction There exist standard models in many subjects such as particle physics and cosmology. In our own field of solar physics, we have a standard model of the Sun’s internal structure, which has been triumphantly validated by helioseismology in the last couple of decades. Is it possible for us to have a standard model of the solar dynamo which is responsible for the generation of the cyclically evolving solar magnetic field? The main difficulty in arriving at a standard model of the solar dynamo is due to the fact that the solar dynamo is stochastically forced by the convective turbulence which we understand very little. However, underneath the various irregularities of the solar cycle, we can discern a reasonably regular mean periodic behaviour. We can think of the solar cycles as made up of an underlying mean periodic cycle and irregularities superposed on it. Can we at least have a standard model of the mean behaviour? Even that involves averaging over various fluctuating quantities. However, considerable progress has been made in the last few years, although we may not yet have a standard model on which E-mail address: [email protected]
all experts agree. My main purpose here is to review how much agreement at present exists amongst solar dynamo theorists and to highlight the still existing disagreements. A more detailed survey of the background can be found in Choudhuri (2003). 2. Basics of the solar dynamo In an axisymmetric model using spherical coordinates, the magnetic field can be written as B ¼ Bðr; h; tÞe/ þ r ½Aðr; h; tÞe/ :
ð1Þ
The magnetic field component in the / direction, denoted by B(r, h, t), is called the toroidal component. We believe that active regions on the solar surface arise from this component by magnetic buoyancy. The remaining part $ · [A(r, h, t)e/], which can be represented by drawing field lines in the poloidal plane, is called the poloidal component. We believe that the weak diffuse magnetic fields outside sunspots is associated with the poloidal component. Since the magnetic field is approximately frozen in the solar plasma, differential rotation can stretch out the poloidal component to produce the toroidal component. As differential rotation is mainly concentrated in the tachocline
0273-1177/$30 2007 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2007.04.006
A.R. Choudhuri / Advances in Space Research 41 (2008) 868–873
at the bottom of the solar convection zone, the strong toroidal field is expected to be produced there. To understand the formation of active regions on the solar surface, we have to study the dynamics of toroidal flux tubes rising through the solar convection zone (SCZ) due to magnetic buoyancy, starting from the bottom of SCZ. Three-dimensional simulations of such flux tube rise established that the Coriolis force plays a major role in this problem (Choudhuri and Gilman, 1987; Choudhuri, 1989; D’Silva and Choudhuri, 1993; Fan et al., 1993). The equipartition magnetic field at the bottom of SCZ would be of order 104 G. It was found that flux tubes with such field strengths are diverted by the Coriolis force to unrealistically high latitudes. Only if the magnetic field inside flux tubes is of order 105 G, various observational constraints seem to be satisfied. Let us now come to the question of poloidal field generation. A very influential idea, often called the a-effect, was first proposed by Parker (1955) and then developed further by Steenbeck et al. (1966). According to this idea, the helical turbulence in the SCZ twists the toroidal component to give rise to magnetic fields lying in the poloidal plane. However, if the toroidal component at the bottom of SCZ is as strong as 105 G, then turbulence will not be able to twist it and the a-effect will be suppressed. Hence many dynamo theorists have invoked an alternative idea of poloidal field generation proposed by Babcock (1961) and Leighton (1969). Bipolar sunspots appear tilted on the solar surface due to the action of the Coriolis force (D’Silva and Choudhuri, 1993). When such tilted bipolar regions decay, magnetic fields of opposite polarities get spread in slightly different latitudes, giving rise to a poloidal field. Thus bipolar sunspots, which originate from the toroidal component, act like a conduit through which the toroidal component gets transformed into the poloidal component. Putting together all these ideas, Choudhuri et al. (1995) and Durney (1995) succeeded in formulating a two-dimensional solar dynamo model in which the meridional circulation plays a very crucial role. Fig. 1 is a cartoon representation of this model. The toroidal field is produced by the differential rotation at the bottom of the SCZ and rises due to magnetic buoyancy. The Babcock–Leighton process at the surface gives rise to the poloidal field, which is advected by the meridional circulation first poleward and then down to the bottom of the SCZ where the strong differential rotation acts on it. A majority of the present-day dynamo theorists agree on this broad scenario. We thus seem to have at least a standard cartoon of the solar dynamo, if not a standard model. When we try to put this cartoon into a mathematical formulation, all kinds of disagreements crop up, as we shall see now.
3. Mathematical formulation and the various parameters Representing the magnetic field by (1), it can be shown that the poloidal and toroidal components evolve according to the following equations
869
++ Strong Differential Rotation ..... Babcock–Leighton Process Magnetic Buoyancy Meridional Circulation
Fig. 1. A cartoon explaining how the solar dynamo works.
oA 1 1 þ ðv:rÞðsAÞ ¼ gp r2 2 A þ aB; ot s s oB 1 o o 1 þ ðrvr BÞ þ ðvh BÞ ¼ gt r2 2 B ot r or oh s 1 dgt o ðrBÞ; þ sðBp :rÞX þ r dr or
ð2Þ
ð3Þ
where s = rsin h. Here, v is the meridional flow, X is the internal angular velocity of the Sun and a is the coefficient which describes the generation of poloidal field at the solar surface from the decay of bipolar sunspots. The turbulent diffusivities for the poloidal and toroidal components are denoted by gp and gt. Since the toroidal component is more concentrated than the poloidal component and turbulence is suppressed in the regions of strong toroidal component, we allow for the turbulent diffusivities of the two components to be different in principle (for further discussion, see Choudhuri et al., 2005, Section 3.3). The dynamo model indicated in the cartoon form in Fig. 1 is achieved by using the helioseismically determined X which has a strong gradient at the bottom of the SCZ, by choosing a meridional circulation v as indicated in the cartoon and by making a concentrated near the solar surface. Additionally, we have to specify magnetic buoyancy in some suitable fashion. We have to solve the coupled partial differential Eqs. (2) and (3) numerically. For a large range of the parameter space, the numerical code is found to relax to periodic solutions. To the best of our knowledge, only about 25 papers have so far presented two-dimensional dynamo calculations of the kind sketched in the cartoon of Fig. 1 in more than a decade. We list below some of the major papers from different groups (Choudhuri et al., 1995, 2004; Dur-
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ney, 1995, 1997, 2000; Dikpati and Charbonneau, 1999; Charbonneau and Dikpati, 2000; Ku¨ker et al., 2001; Nandy and Choudhuri, 2001, 2002; Dikpati and Gilman, 2001; Bonanno et al., 2002; Guerrero and Muno˜z, 2004; Chatterjee et al., 2004; Dikpati et al., 2004; Charbonneau et al., 2004, 2005). It may be mentioned that an early paper by Wang et al. (1991) presented a radially averaged onedimensional model with some of the ideas to be developed in these later papers. The number of scientists working on different aspects of solar cycle is quite large. In contrast, it always puzzles me why the number of scientists trying to understand the origin of the solar cycle is so small. One of the difficulties of getting into this field used to be that a new group had to develop a code from scratch and test the results of other groups before thinking of doing anything new. However, we have now made our dynamo code SURYA developed in Bangalore available with a detailed guide (Choudhuri, 2005) to anybody who sends me an email request. Just as there seems to be a broad agreement amongst dynamo theorists about the standard cartoon shown in Fig. 1, all agree that (2) and (3) are the basic equations we have to solve. However, disagreements begin when we have to specify the various parameters appearing in (2) and (3). Let us list all the parameters below. Differential rotation X: This is now provided by helioseismology. Meridional circulation v: Helioseismology throws light on its nature in the upper half of the SCZ, but we have no direct observations on its nature in the lower half of the SCZ. Poloidal field source parameter a: We observe the Babcock–Leighton process on the solar surface, but we are not sure if there are processes in the body or at bottom of the SCZ which also contribute to a.
Turbulent diffusivities gp and gt: Their surface values can be estimated from the study of magnetic field diffusion on the surface and then we have to make reasonable assumptions underneath the surface. Magnetic buoyancy: This has to be specified in some way when solving Eqs. (2) and (3). While specifying these parameters, one has to introduce something which nonlinearly limits any possible runaway growth of the magnetic field. This is often done by including a quenching in the a parameter. In some of our calculations (Chatterjee et al., 2004), magnetic buoyancy limits the magnetic field growth and it is not necessary to include a quenching. According to my judgment, major differences exist amongst solar dynamo theorists at the present time on the following questions: (i) the nature of meridional circulation near the base of the SCZ; (ii) the distribution of a in the lower portions of the SCZ; (iii) the most satisfactory way of treating magnetic buoyancy. These are the primary questions which have to be settled if we want to arrive at a standard model of the solar dynamo. It should not be assumed that these are merely matters of finer details. We shall point out in the next section that these uncertainties can affect the nature of dynamo solutions in quite drastic ways. While specifying the various parameters, we, of course, have some constraints when we try to match different aspects of observational data. The left plot in Fig. 2 is a butterfly diagram of sunspots superposed on a time–latitude plot of longitudinally averaged radial magnetic field at the solar surface. Any realistic solar dynamo model should satisfy the following characteristics. The solution should be cyclic with a period of about 22 years.
Fig. 2. Butterfly diagram of sunspots superposed on the time–latitude plot of Br. The observational plot is shown on the left. The comparable theoretical plot obtained by the dynamo model of Chatterjee et al. (2004) is on the right.
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The butterfly diagram of sunspots should be restricted to latitudes below 40. The weak fields outside active regions should drift poleward. Polar field reversal should take place at the time of sunspot maximum. The solution should be of dipolar nature. (The present solar magnetic field appears dipolar, but was it always so?) The model should provide an explanation why magnetic helicity of active regions tends to be negative (positive) in northern (southern) hemisphere. The right plot in Fig. 2 is obtained theoretically from our recent model (Chatterjee et al., 2004) and should be compared with the observational plot on the left side of Fig. 2. Our model is one of the first detailed models to be constructed that satisfies all the above requirements. However, the crucial question is whether the parameters we have used in our model correspond to the ‘reality’ inside the Sun. It may be possible that a completely different combination of parameters also gives a model satisfying all the requirements! 4. Disagreements amongst dynamo theorists The tachocline has a much stronger differential rotation at high latitudes (where oX/or is negative) compared to low latitudes (where oX/or is positive). Hence, helioseismically determined differential rotation tends to produce a strong toroidal field at high latitudes, whereas we see sunspots at low latitudes. This is clearly seen in the best theoretical butterfly diagrams obtained by Dikpati and Charbonneau (1999, Fig. 6) and Ku¨ker et al. (2001, Fig. 10), who were amongst the first to put the helioseismically determined X in their dynamo models. These theoretical butterfly diagrams indicate strong magnetic activity at high latitudes and look nothing like observational butterfly diagrams. Nandy and Choudhuri (2002) proposed a way out of this difficulty. Suppose the meridional circulation penetrates a little bit below the bottom of the SCZ into the region of convective overshooting. Then the strong toroidal field produced at high latitudes within the tachocline will be pushed into the stable layers underneath the SCZ by the meridional circulation and will not be able to rise up to form active regions at high latitudes. This toroidal field will then be advected to lower latitudes through the stable layers by the meridional circulation, which will ultimately bring the toroidal field within the SCZ at low latitudes where active regions will form. On using meridional circulation confined within the SCZ, Nandy and Choudhuri (2002) found a butterfly diagram with activity concentrated at high latitudes (Fig. 2A in their paper). On making the meridional circulation penetrate below the bottom of SCZ, however, the butterfly diagram looked completely different and appeared very solar-like (Nandy and Choudhuri, 2002, Fig. 2B). It should be kept in mind that,
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unless there is some amount of overshooting turbulence below the bottom of SCZ, it is physically not possible for the meridional circulation to penetrate much (Gilman and Miesch, 2004). Dikpati et al. (2005) criticized this model on the ground that it is unphysical to assume that the meridional circulation penetrates below the bottom of SCZ. Choudhuri et al. (2005) respond that the bottom of SCZ is the least understood region of the Sun and one should be open-minded. They also point out that Dikpati and her colleagues themselves have been using deeply penetrating meridional circulation in their dynamo models without clearly mentioning it (Choudhuri et al., 2005, Fig. 2). Dikpati et al. (2004) is the first paper which claims to produce solar-like butterfly diagrams with non-penetrating meridional circulation. Surprisingly, there is no discussion in this paper how the authors solve the problem of magnetic buoyancy of toroidal field in high latitudes. Moreover, we and other groups (Jie Jiang, private communication) who tried to reproduce this model with other codes have not been able to reproduce the results of Dikpati et al. (2004). A second important uncertainty is regarding the distribution of a. If a is concentrated near the surface as expected for the Babcock–Leighton mechanism, Dikpati and Gilman (2001) found that the dynamo solution tends to be quadrupolar, whereas solar magnetic fields appear dipolar. Only on including an a at the bottom of SCZ, they were able to get dipolar solutions. This result was confirmed by Bonanno et al. (2002). Based on this result, Dikpati and Gilman (2001) argued that poloidal field generation should take place at the bottom of the SCZ. In a dipolar solution, poloidal field lines connect across the equator. Chatterjee et al. (2004) found that it is possible for this to happen and give dipolar solutions if the diffusivity of the poloidal field is sufficiently high, even when a is concentrated near the surface. Chatterjee et al. (2004) do not claim that a has to be strictly zero well below the surface. However, the argument that we must have a non-zero a at the base of SCZ to make the solutions dipolar is clearly incorrect. Finally we point out that there are two popular ways of handling magnetic buoyancy in two-dimensional dynamo models. 1. If B inside the SCZ is larger than a critical value, we move a part of it to the surface. Our group in Bangalore is following this recipe. 2. In the poloidal field source term aB in (2), we multiply a at the surface by the value of B at the bottom of SCZ rather than at the surface. This recipe was proposed by Choudhuri and Dikpati (1999) and is used in the dynamo models of the group in Boulder. Unfortunately, these two recipes can give solutions of totally different nature. Fig. 3 shows solutions obtained by treating magnetic buoyancy by these two different recipes in otherwise identical dynamo models. After Dikpati
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Fig. 3. Plots of the contours of toroidal field and the poloidal field lines. (a) and (b) correspond to dynamo solutions in which magnetic buoyancy is treated by Recipe 1 and Recipe 2, respectively.
et al. (2005) pointed out that they cannot reproduce the results of Chatterjee et al. (2004), we realized that they are getting different results merely because they treat magnetic buoyancy differently (Choudhuri et al., 2005). Certainly more careful studies are needed to ascertain the best way of treating magnetic buoyancy in two-dimensional dynamo models. 5. Conclusion After Parker (1955) wrote his fundamental paper on the solar dynamo, many theorists in 1960s and 1970s attempted to build detailed models of the solar dynamo. Not much was known about the interior of the SCZ at that time and theorists were making assumptions which seemed ‘reasonable’ at that time, but now appear like wild guesses completely off the mark. The new era of solar dynamo research dawned after (i) helioseismology mapped out the angular velocity distribution in the interior of the Sun; (ii) detailed flux tube simulations made it clear that the toroidal field at the base of the SCZ should be as strong as 105 G; and (iii) it became increasingly clear that meridional circulation must play an important role. The solar dynamo models constructed in the last few years appear much more realistic than the earlier models which, however, played historically important roles and provided us with insight that was crucial in building the recent models – a fact which should not be forgotten. We now have a broad understanding of how the solar dynamo operates. I have called it the standard cartoon. However, the mathematical formulation of this cartoon requires specification of several parameters, some of which seem to have large uncertainties at the present time. It is possible to construct theoretical models which explain various aspects of observational data reasonably well. But such theoretical models may not be unique, as the models developed in Bangalore and Boulder use very different combinations of parameters. We urgently need the induction of new groups in this field who may be able to bring
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