aa r X i v : . [ a s t r o - ph . S R ] N ov Title of your IAU SymposiumProceedings IAU Symposium No. xxx, 2010A.C. Editor, B.D. Editor & C.E. Editor, eds. c (cid:13) Dynamo models of grand minima
Arnab Rai Choudhuri
Department of Physics, Indian Institute of Science, Bangalore-560012email: [email protected]
Abstract.
Since a universally accepted dynamo model of grand minima does not exist at the presenttime, we concentrate on the physical processes which may be behind the grand minima. Aftersummarizing the relevant observational data, we make the point that, while the usual sourcesof irregularities of solar cycles may be sufficient to cause a grand minimum, the solar dynamohas to operate somewhat differently from the normal to bring the Sun out of the grand mini-mum. We then consider three possible sources of irregularities in the solar dynamo: (i) nonlineareffects; (ii) fluctuations in the poloidal field generation process; (iii) fluctuations in the merid-ional circulation. We conclude that (i) is unlikely to be the cause behind grand minima, but acombination of (ii) and (iii) may cause them. If fluctuations make the poloidal field fall muchbelow the average or make the meridional circulation significantly weaker, then the Sun may bepushed into a grand minimum.
Keywords.
Sun: dynamo — Sun: activity — sunspots
1. Introduction
At the very outset, I would like to mention that the subject of this invited talk wasnot chosen by me. The organizers felt that this symposium should have an invited talkon dynamo models of grand minima and requested me to give it. Only after considerableinitial hesitation, I finally agreed. The reason behind my initial hesitation is that atpresent we have no dynamo model of grand minima which is completely satisfactory orwhich is generally accepted in the community. No two self-respecting dynamo theoristsseem to agree how grand minima are produced! To the best of my knowledge, this isthe first time an invited talk on this subject is being given in a major internationalconference.Given this situation, I have decided to adopt the following strategy. I shall mainlyfocus on the various bits of physics which go into making models of grand minima ratherthan discussing specific models of grand minima in detail. While many of the present-daymodels of grand minima may eventually fall by the wayside, I believe that the bits ofphysics that we consider relevant today will still remain relevant after 20 or 30 yearswhen there may be a better understanding of what occasionally pushes the Sun into thegrand minima.
2. Observational characteristics of grand minima
Before getting into the theoretical discussion, let us see what we can learn about thecharacteristics of grand minima from the very limited observational data available to us.Several authors have studied the archival records of sunspots during the Maunderminimum (Sokoloff & Nesme-Ribes 1994; Hoyt & Schatten 1996). The few sunspots seenduring the Maunder minimum mostly appeared in the southern hemisphere. Sokoloff& Nesme-Ribes (1994) have used the archival data to construct a butterfly diagram1 Choudhurifor a part of the Maunder minimum from 1670, showing a clear trend of hemisphericasymmetry. It is an open question whether hemispheric asymmetry played any crucial rolein creating the Maunder minimum (Charbonneau 2005). Usoskin, Mursula & Kovaltsov(2000) argued that the Maunder minimum started abruptly but ended in a gradualmanner, indicating that the strength of the dynamo must be building up as the Suncame out of the Maunder minimum. However, some recent evidence suggests that theonset of the Maunder minimum may not be as abrupt as believed earlier (Vaquero et al.2011).When solar activity is stronger, magnetic fields in the solar wind suppress the cosmicray flux, reducing the production of Be and C which can be used as proxies for solaractivity. From the analysis of Be abundance in a polar ice core, Beer, Tobias & Weiss(1998) concluded that the solar activity cycle continued during the Maunder minimum,although the overall level of the activity was lower than usual. Miyahara et al. (2004)drew the same conclusion from their analysis of C abundance in tree rings.This method of using various proxies (like the abundances of Be and C) for sunspotactivity can be extended to study even the earlier grand minima before the Maunderminimum. Usoskin, Solanki & Kovaltsov (2007) estimated that there have been about27 grand minima in the last 11,000 years. They also identified about 19 grand maxima,i.e. periods during which sunspot activity was unusually high, like what was seen duringmuch of the twentieth century.
3. Are grand minima merely extremes of cycle irregularities?
We know that the solar cycle is only approximately periodic. Both the strength and theperiod vary from one cycle to another. We begin our theoretical discussion by raising thequestion whether the grand minima are merely extreme examples of cycle irregularities.Are the theoretical ideas used to model irregularities of solar cycles adequate to explainthe occurrences of grand minima, or do we need to invoke some qualitatively differentideas? We do not yet have a definitive answer to this question. Any dynamo theorist isentitled to have his or her own personal opinion. Let me put forth my personal opinion.Our simulations (to be discussed later) seem to suggest that nothing very extraordi-nary may be needed to push the Sun into a grand minimum. If the fluctuations whichcause the usual cycle irregularities are sufficiently large, they may sometimes cause grandminima. However, even after the Sun is pushed into the grand minimum, a subdued cy-cle has to continue (as discussed in §
2) and eventually the Sun has to come out of thegrand minimum. These are more problematic to explain. There are certain mechanisms ofmagnetic field generation which crucially depend on the existence of sunspots. Certainlythose mechanisms cannot be operative during a grand minimum. So we need to invokealternative mechanisms.Let us look at the question how magnetic fields are generated in the dynamo process.The basic idea of solar dynamo is that the toroidal and poloidal components of the solarmagnetic field sustain each other through a feedback loop. It is fairly easy to generatethe toroidal field by the stretching of the poloidal field due to differential rotation. Sincehelioseismology has shown that the differential rotation is concentrated in the tachocline,the generation of toroidal field mainly takes place there. To complete the loop, we needto generate the poloidal field from the toroidal field. The historically important idea ofParker (1955) — which was further elaborated by Steenbeck, Krause & R¨adler (1966) — isthat the cyclonic turbulence in the convection zone twists the toroidal field to produce thepoloidal field. This mechanism is often called the α -effect because the crucial parameterdescribing this process is usually denoted by the symbol α . Within certain approximation ynamo models of grand minima α = − v . ( ∇ × v ) τ, (1)where v is the fluctuating part of the velocity field and τ is the correlation time (see,for example, Choudhuri 1998, § α -effect mechanism can be operative only ifthe toroidal field is not too strong such that the helical turbulence is able to twist it.However, the flux tube rise simulations by several authors (Choudhuri & Gilman 1987;Choudhuri 1989; D’Silva & Choudhuri 1993; Fan, Fisher & DeLuca 1993; Caligari et al.1995) indicated that the toroidal field at the base of the solar convection zone has to beas strong as 10 G. Such a strong field cannot be twisted by helical turbulence and wecannot invoke α -effect to generate the poloidal field from such a strong toroidal field. Analternative mechanism which has been widely used in many recent dynamo simulationsis due to Babcock (1961) and Leighton (1969). Bipolar sunspots on the solar surfacehave a tilt with respect to the solar equator and this tilt increases with latitude. Thiswas discovered by Joy in 1919 and is known as Joy’s law . D’Silva & Choudhuri (1993)provided the first theoretical explanation of Joy’s law by showing that the tilt is producedby the Coriolis force acting on the flux tubes rising through the convection zone due tomagnetic buoyancy. When a tilted bipolar sunspot decays, fluxes of opposite polaritiesdiffuse at slightly different latitudes, contributing to the poloidal field. According to thisBabcock–Leighton mechanism, a tilted bipolar sunspot pair is a conduit for convertingthe toroidal field to the poloidal field. The sunspot pair forms due to the buoyant rise ofthe toroidal field and we get the poloidal field after its decay.Since we see the Babcock–Leighton mechanism clearly operational at the solar surface,most of the recent flux transport dynamo models take this as the primary generationmechanism of the poloidal field. The α -effect cannot operate on the strong toroidal fieldat the base of the convection zone. But this strong toroidal field is expected to be highlyintermittent (Choudhuri 2003) and the α -effect is likely to be operative in those regionsof the convection zone where the toroidal field is weak, although the nature, the spatialdistribution and even the algebraic sign of the α parameter remain unclear at the presenttime. The Babcock–Leighton mechanism presumably cannot work during the grand min-imum when there are no sunspots. So we have to fall back upon the α -effect to continuethe cycles during the grand minimum and eventually to pull the Sun out of it. Our lackof knowledge about the α -process limits our understanding of these phenomena. In themean field dynamo equations, the term capturing the Babcock–Leighton process is for-mally very similar to the term capturing the α -effect — often even using the symbol α .Hence many dynamo models of the grand minima are worked out at the present time bysolving the same equations during and outside the grand minima. But it should be keptin mind that the physics behind the symbol α must be very different during and outsidethe grand minima.In summary, our view is that the usual sources of irregularities in solar cycles aresufficient for the onset of a grand minimum, but to pull the Sun out of a grand minimumwe need some physics different from the physics behind the usual solar cycles. Presumablythe situation is somewhat different for grand maxima. Not only are the usual irregularitiesexpected to cause a grand maximum, we also do not require anything unusual to takethe Sun out of the grand maximum. The usual poloidal field generation by the Babcock–Leighton mechanism continues during the grand maximum. It is intriguing that Usoskin,Solanki & Kovaltsov (2007) concluded that the lengths of grand maxima correspond toan exponential distribution, but the lengths of grand minima have a more complicatedbimodal distribution. Is this connected with the fact that grand maxima do not involve Choudhuriany physical processes different from the normal, but grand minima require the generationprocess of the poloidal field to be different from the normal situation?
4. The origin of irregularities in the flux transport dynamo
We now discuss the possible sources of irregularities in the flux transport dynamo— the most widely studied model of the solar cycle in recent years. Let us begin byrecapitulating some basic facts about the flux transport dynamo. The toroidal field isgenerated in the tachocline by the strong differential rotation and then rises to the surfacedue to magnetic buoyancy to form tilted bipolar sunspots. When these sunspots decay,we get the poloidal field by the Babcock–Leighton mechanism. The meridional circulationof the Sun, which is found to be poleward in the upper layers of the convection zone andmust have a hitherto unobserved equatorward branch at the bottom of the convectionzone in order to conserve mass, plays a very crucial role in the flux transport dynamo. Themeridional circulation causes the observed poleward transport of the poloidal field. Atthe base of the convection zone, it is responsible for making the dynamo wave propagateequatorward, such that sunspots are produced at lower and lower latitudes with theprogress of a cycle. In the absence of the meridional circulation, the dynamo wave at thebottom of the convection zone would propagate poleward in accordance with the Parker–Yoshimura sign rule (Parker 1955; Yoshimura 1975) contradicting observations. The fluxtransport dynamo could become a serious model of the solar cycle only after Choudhuri,Sch¨ussler & Dikpati (1995) demonstrated that a sufficiently strong meridional circulationcould overrule the Parker–Yoshimura sign rule and make the dynamo wave propagate inthe correct direction.The original flux transport dynamo model of Choudhuri, Sch¨ussler & Dikpati (1995)led to two offsprings: a high diffusivity model and a low diffusivity model. The diffusiontimes in these two models are of the order of 5 years and 200 years respectively. The highdiffusivity model has been developed by a group working in IISc Bangalore (Choudhuri,Nandy, Chatterjee, Jiang, Karak), whereas the low diffusivity model has been developedby a group working in HAO Boulder (Dikpati, Charbonneau, Gilman, de Toma). Thedifferences between these models have been systematically studied by Jiang, Chatterjee &Choudhuri (2007) and Yeates, Nandy & Mckay (2008). Both these models are capable ofgiving rise to oscillatory solutions resembling solar cycles. However, when we try to studythe irregularities of the cycles, the two models give completely different results. We needto introduce fluctuations to cause irregularities in the cycles. In the high diffusivity model,fluctuations spread all over the convection zone in about 5 years. On the other hand, in thelow diffusivity model, fluctuations essentially remain frozen during the cycle period. Thusthe behaviours of the two models are totally different on introducing fluctuations. Overthe last few years, several independent arguments have been advanced in support of thehigh diffusivity model (Chatterjee, Nandy & Choudhuri 2004; Chatterjee & Choudhuri2006; Jiang, Chatterjee & Choudhuri 2007; Goel & Choudhuri 2009; Hotta & Yokoyama2010). We adopt the point of view here the solar dynamo is most likely a high diffusivityflux transport dynamo.Three main sources of irregularities in dynamo models have been studied by differentauthors over the years: (i) chaotic behaviours introduced by nonlinearities of the dynamoprocess; (ii) fluctuations in the generation of the poloidal field; (iii) fluctuations in themeridional circulations. The three following sections will focus on these three sourcesof irregularities and discuss the question whether they can cause grand minima. Someof these sources of irregularities have been investigated even before the flux transportdynamo model became popular, by applying them to the earlier solar dynamo models. ynamo models of grand minima
5. Effects of nonlinearities
It is well known that nonlinear dynamical systems can show complicated chaotic be-haviours. Some of the earliest efforts of modelling solar cycle irregularities invoked theidea of nonlinear chaos. The full dynamo problem is certainly a nonlinear problem inwhich the magnetic fields produced by the fluid motions react back on the fluid motions.The simplest way of capturing the effect of this in a kinematic dynamo model (in whichthe fluid equations are not solved) is to consider a quenching of the α parameter asfollows: α = α | B/B | , (2)where B is the average of the magnetic field produced by the dynamo and B is thevalue of magnetic field beyond which nonlinear effects become important. There is a longhistory of dynamo models studied with such quenching (Stix 1972; Ivanova & Ruzmaikin1977; Yoshimura 1978; Brandenburg et al. 1989; Schmitt & Sch¨ussler 1989). In most ofthe nonlinear calculations, however, the dynamo eventually settles to a periodic modewith a given amplitude rather than showing sustained irregular behaviour. The reasonfor this is intuitively obvious. Since a sudden increase in the amplitude of the magneticfield would diminish the dynamo activity by reducing α given by (2) and thereby pulldown the amplitude again (a decrease in the amplitude would do the opposite), the α -quenching mechanism tends to lock the system to a stable mode once the system relaxesto it. In fact, Krause & Meinel (1988) and Brandenburg et al. (1989) argued that thenonlinear stability may determine the mode in which the dynamo is found. Yoshimura(1978) was able to reproduce some irregular features of the solar cycle by introducingan unrealistic delay time of 29 years between the magnetic field and its effect on the α -coefficient. In some highly truncated models with the suppression of differential rotation,one could find the evidence of chaos in limited parts of the parameter space (Weiss,Cattaneo & Jones 1984). K¨uker, Arlt & R¨udiger (1999) suggested that the quenching ofdifferential rotation might have caused the Maunder minimum. This seems unlikely nowon the ground that torsional oscillations — periodic modulations of differential rotationcaused by the dynamo-generated magnetic field — appear like small perturbations.It does not seem that the irregularities of solar cycles are primarily caused by nonlin-earities. But that does not mean that nonlinearities have no important consequences inthe currently favoured flux transport dynamo models. In order to explain the even-odd orthe Gnevyshev–Ohl effect of solar cycles, Charbonneau, St-Jean & Zacharias (2005) andCharbonneau, Beaubien & St-Jean (2007) made the highly provocative suggestion thatthe solar dynamo may be sitting in a region of period doubling just beyond the point ofnonlinear bifurcation. Recently the effects of nonlinearities introduced by the quenchingof turbulent diffusion (Guerrero, Dikpati & de Gouveia Dal Pino 2009) and meridionalcirculation (Karak & Choudhuri 2012) are being investigated.
6. Fluctuations in poloidal field generation
Since the mean field dynamo equations are derived by averaging over turbulence, weexpect fluctuations to be present around the mean. Choudhuri (1992) was the first tosuggest that these fluctuations will be particularly important in the poloidal field gen-eration. It is now difficult to believe that this was an unorthodox and radical idea in1992 when it was proposed, though this idea was explored further by Moss et al. (1992),Hoyng (1993) and Ossendrijver, Hoyng & Schmitt (1996). This idea was applied to theflux transport dynamo by Charbonneau & Dikpati (2000). Choudhuri
Year La t i t ude (a) Year S un s po t nu m be r (b) Figure 1.
The theoretical model of the Maunder minimum from Choudhuri & Karak (2009).This is based on a simulation in which the poloidal field was reduced to 0.0 and 0.4 of its averagevalue in the two hemispheres. The upper panel shows the butterfly diagram. The dotted anddashed lines in the lower panel are sunspot numbers in the northern and southern hemispheres,whereas the solid line is their sum.
Let us consider the question how fluctuations in poloidal field generation arise in theflux transport dynamo. The Babcock–Leighton mechanism of poloidal field generationdepends the tilts of bipolar sunspot pairs. While the average tilts are given by Joy’s law,one finds a large scatter around this average. Longcope & Choudhuri (2002) provided atheoretical model of this scatter on the basis of the idea that the rising flux tubes arebuffeted by turbulence in the convection zone. This scatter around Joy’s law producesfluctuations in the poloidal field generation process and we identify this as a primarysource of irregularities in the solar cycle. It may be noted that Choudhuri, Chatterjee& Jiang (2007) and Jiang, Chatterjee & Choudhuri (2007) modelled the last few cyclesby assuming the fluctuations in poloidal field generation to be the main source of irreg-ularities in solar cycles and predicted that the forthcoming cycle 24 will be weak. Thisprediction was based on the high diffusivity model. There are enough indications by nowthat the upcoming cycle is going to be a weak one, providing further support to the highdiffusivity model.We now come to question whether fluctuations in the poloidal field generation can pro-duce grand minima. Several authors found that intermittencies resembling grand minimacan be obtained in simple dynamo models by introducing fluctuations (Schmitt, Sch¨ussler& Ferriz-Mas 1996; Mininni, Gomez & Mindlin 2001; Brandenburg & Spiegel 2008). Theeffect of such fluctuations on flux transport dynamo models has been investigated onlyrecently. Charbonneau, Blais-Laurier & St-Jean (2004) carried out a simulation by intro-ducing 100% fluctuations in α (the poloidal field generation parameter) in a flux transportdynamo with low diffusivity. They found intermittencies in their simulations resembling ynamo models of grand minima not be entering another grand minimum. On making the poloidalfield in the northern and southern hemispheres fall to respectively 0.0 and 0.4 of its av-erage value, Choudhuri & Karak (2009) found that many characteristics of the Maunderminimum were reproduced. Fig. 1 shows the theoretical plots of butterfly diagram andsunspot number, which compare favourably with the corresponding observational plotsgiven in Fig. 1(a) of Sokoloff & Nesme-Ribes (1994) and Fig. 1 of Usoskin, Mursula& Kovaltsov (2000). We find that the theoretical model reproduced the fact that theMaunder minimum started abruptly, but ended gradually. It is basically the growth timeof the dynamo which determines the duration of the grand minimum during which themagnetic field has to grow up again to return to normalcy. As pointed out in §
3, theoperation of the dynamo during the grand minimum presumably depends on the α -effectand the theoretical model shows an ongoing but subdued cycle of magnetic field in thesolar wind. We sum up the theoretical results in the following words. If the poloidal fieldat the end of a cycle turns out to be very weak due to fluctuations in its generationprocess, then that can push the dynamo into a grand minimum, from which it recoversgradually in the dynamo growth time.
7. Fluctuations in meridional circulation
It is well known that the period of the flux transport dynamo varies roughly as theinverse of the meridional circulation speed. The period of the dynamo is approximatelygiven by the time taken by meridional circulation at the bottom of the convection zoneto move from higher latitudes to lower latitudes. In other words, the period of a fluxtransport dynamo does not depend too much on the details of poloidal field genera-tion mechanism. Probably this is the reason why the period of the dynamo during agrand minimum like the Maunder minimum does not change drastically, even though thepoloidal field generation mechanism may be different from normal times as explained §
10 11 12 13 14 1500.10.20.30.40.50.60.70.80.91 v γ Longer thanMaunder minimum Shorter thanMaunder minimum
Figure 2.
The parameter space indicating the reduction factor γ of the poloidal field and theamplitude of the meridional circulation needed to produce Maunder-like grand minima. dynamo-generated magnetic field slows down the meridional circulation at the time of thesunspot maximum. Karak & Choudhuri (2012) found that this quenching of meridionalcirculation by the Lorentz force does not produce irregularities in the cycle, provided thediffusivity is high as we believe. We disagree with the model of Nandy, Mu˜noz-Jaramillo& Martens (2011) which assumes that the meridional circulation changes abruptly ateach sunspot maximum. Our point of view is that the periodic variation of meridionalcirculation due to the Lorentz force cannot be responsible for solar cycle irregularitiesand we need to consider other kinds of fluctuations in meridional circulation.We have reliable observational data on the variation of meridional circulation only fora little more than a decade. To draw any conclusions about the variation of meridionalcirculation at earlier times, we have to rely on indirect arguments. If we assume the cycleperiod to go inversely as meridional circulation, then we can use periods of differentpast solar cycles to infer how meridional circulation has varied with time in the last fewcenturies. On the basis of such considerations, it appears that the meridional circulationhad random fluctuations in the last few centuries with correlation time of the order of30–40 years (Karak & Choudhuri 2011). We now come to question what effect theserandom fluctuations of meridional circulation may have on the dynamo. Based on theanalysis of Yeates, Nandy & Mckay (2008), we can easily see that dynamos with highand low diffusivity will be affected very differently. Suppose the meridional circulationhas suddenly fallen to a low value. This will increase the period of the dynamo and leadto two opposing effects. On the one hand, the differential rotation will have more time togenerate the toroidal field and will try to make the cycles stronger. On the other hand,diffusion will also have more time to act on the magnetic fields and will try to make thecycles weaker. Which of these two competing effects wins over will depend on the valueof diffusivity. If the diffusivity is high, then the action of diffusivity is more importantand the cycles become weaker when the meridional circulation is slower. The oppositehappens if the diffusivity is low.As we pointed out in § §
6, there are enough indications that the diffusivity ofthe solar dynamo is high. If that is the case, then a slowing of the meridional circulationwould make the cycles weaker. Karak (2010) found that the flux transport dynamo canbe pushed into a grand minimum if the meridional circulation drops to 0.4 of its normalvalue. This is another possible mechanism for producing a grand minimum. ynamo models of grand minima
8. Concluding remarks
Although there are many uncertainties in our theoretical understanding of grand min-ima, it appears that fluctuations in poloidal field generation and fluctuations in merid-ional circulation are the main causes of irregularities in solar cycles and can also producegrand minima. We believe the solar dynamo to be a high diffusivity dynamo in whicha fall in the meridional circulation makes cycles weaker. If fluctuations in poloidal fieldgeneration alone and fluctuations in meridional circulation alone are to produce grandminima, then the poloidal field at the end of a cycle or the meridional circulation has tofall to rather low values to cause grand minima. The situation is a little less constrainedif we consider simultaneous fluctuations in both. Fig. 2 taken from Karak (2010) showsthe values to which the poloidal field at the end of a cycle and the meridional circulationhave to fall if a grand minimum is to be caused by their simultaneously falling to lowvalues. This seems to be the most likely scenario we have at the present time for explain-ing grand minima. One important question is whether we can estimate how often thisis likely to happen. Can we explain why there were 27 grand minima in the last 11,000years? We are looking at this question right now.I end by summarizing what appears to me to be the most plausible theoretical scenariofor grand minima based on the flux transport dynamo. Due to fluctuations in poloidalfield generation and meridional circulation, if both of them simultaneously happen tobecome sufficiently weak, that may push the Sun into a grand minimum. Within thegrand minimum, the dynamo keeps operating on the basis of the α -effect and ultimatelybounces out of the grand minimum in the dynamo growth time. Acknowledgments
My participation in IAU Symposium 286 was made possible by a JC Bose Fellowshipawarded by Department of Science and Technology, Government of India.
References
Babcock, H. W. 1961,
ApJ , 133, 572Basu, S., & Antia, H. M. 2010,
ApJ , 717, 488Beer, J., Tobias, S., & Weiss, N. 1998,
Solar Phys. , 181, 237Brandenburg, A., Krause, F., Meinel, R., Moss, D., & Tuominen I. 1989,
A&A , 213, 411Brandenburg, A., & Spiegel, E. A. 2008, AN , 329, 351Charbonneau, P. 2005, Solar Phys. , 229, 345Charbonneau, P., Beaubien, G., & St-Jean, C. 2007,
ApJ , 658, 657Charbonneau, P., Blais-Laurier, G., & St-Jean, C. 2004,
ApJ , 616, L183Charbonneau, P., & Dikpati, M. 2000,
ApJ , 543, 1027Charbonneau, P., St-Jean, C., & Zacharias, P. 2005,
ApJ , 619, 613Chatterjee, P., & Choudhuri, A. R. 2006,
Solar Phys. , 239, 29
Chatterjee, P., Nandy, D., & Choudhuri, A. R. 2004,
A&A , 427, 1019Choudhuri, A. R. 1989,
Solar Phys. , 123, 217Choudhuri, A. R. 1992,
A&A , 253, 277Choudhuri, A. R. 1998,
The Physics of Fluids and Plasmas: An Introduction for Astrophysicists (Cambridge University Press, Cambridge)Choudhuri, A. R. 2003,
Solar Phys. , 215, 31Choudhuri, A. R., Chatterjee, P., & Jiang, J. 2007,
Phys. Rev. Lett. , 98, 131103Choudhuri A. R., & Gilman P.A. 1987,
ApJ , 316, 788Choudhuri, A. R., & Karak, B. B. 2009,
RAA , 9, 953Choudhuri, A. R., Sch¨ussler, M., & Dikpati, M. 1995,
A&A , 303, L29D’Silva, S., & Choudhuri, A. R. 1993,
A&A , 272, 621Fan, Y., Fisher, G. H., & DeLuca, E. E. 1993,
ApJ , 405, 390Goel, A., & Choudhuri, A. R. 2009,
RAA , 9, 115Guerrero, G., Dikpati, M., & de Gouveia Dal Pino, E. M. 2009
ApJ , 701, 725Hathaway, D. H., & Rightmire, L. 2010,
Science , 327, 1350Hotta, H., & Yokoyama, T. 2010,
ApJ , 714, L308Hoyng, P. 1993,
A&A , 272, 321Hoyt, D. V., & Schatten, K. H. 1996,
Solar Phys. , 165, 181Ivanova, T. S., & Ruzmaikin, A. A. 1977,
SvA , 21, 479Jiang, J., Chatterjee, P., & Choudhuri, A. R. 2007,
MNRAS , 381, 1527Karak, B. B. 2010,
ApJ , 724, 1021Karak, B. B., & Choudhuri, A. R. 2011,
MNRAS , 410, 1503Karak, B. B., & Choudhuri, A. R. 2012,
Solar Phys. , submitted (arXiv:1111.1540)Krause, F., & Meinel, R. 1988,
GAFD , 43, 95K¨uker, M., Arlt, R., & R¨udiger, G. 1999,
A&A
ApJ , 156, 1Longcope, D. W., & Choudhuri, A. R. 2002,
Solar Phys. , 205, 63Mininni, P. D., Gomez, D. O., & Mindlin, G. B. 2001,
Solar Phys. , 201, 203Miyahara, H., Masuda, K., Muraki, Y., Furuzawa, H., Menjo, H., & Nakamura, T. 2004,
SolarPhys. , 224, 317Moss, D., Brandenburg, A., Tavakol, R., & Tuominen, I. 1992,
A&A , 265, 843Nandy, D., Mu˜noz-Jaramillo, A., & Martens, P. C. H. 2011
Nature
A&A , 313, 938Parker, E. N. 1955,
ApJ , 122, 293Schmitt, D., & Sch¨ussler, M. 1989,
A&A , 223, 343Schmitt, D., Sch¨ussler, M., & Ferriz-Mas, A. 1996,
A&A , 311, L1Sokoloff, D., & Nesme-Ribes, E. 1994,
A&A , 288, 293Steenbeck, M., Krause, F., & R¨adler, K. H. 1966,
Z. Naturforsch. , 21, 369Stix, M. 1972,
A&A , 20, 9Usoskin, I. G., Mursula, K., & Kovaltsov, G. A. 2000,
A&A , 354, L33Usoskin, I. G., Solanki, S. K., & Kovaltsov, G. A. 2007,
A&A , 471, 301Vaquero, J. M., Gallego, M. C., Usoskin, I. G., & Kovaltsov, G. A. 2011,
ApJ , 731, L24Weiss, N. O., Cattaaneo, F., & Jones, C. A. 1984,
GAFD , 30, 305Yeates, A. R., Nandy, D., & Mackay, D. H. 2008,
ApJ , 673, 544Yoshimura, H. 1975,
ApJ , 201, 740Yoshimura, H. 1978,