Flux Transport Dynamo: From Modelling Irregularities to Making Predictions
aa r X i v : . [ a s t r o - ph . S R ] A ug Flux Transport Dynamo: From Modelling Irregularitiesto Making Predictions
Arnab Rai Choudhuri a a Department of Physics, Indian Institute of Science,Bangalore 560012, India
Abstract
The flux transport dynamo, in which the poloidal magnetic field is gen-erated by the Babcock–Leighton mechanism and the meridional circulationplays a crucial role, has emerged as an attractive model for the solar cycle.Based on theoretical calculations done with this model, we argue that thefluctuations in the Babcock–Leighton mechanism and the fluctuations in themeridional circulation are the most likely causes of the irregularities of thesolar cycle. With our increased theoretical understanding of how these ir-regularities arise, it can be possible to predict a future solar cycle by feedingthe appropriate observational data in a theoretical dynamo model.
Keywords: dynamo, solar cycle
1. Introduction
The flux transport dynamo model, which started being developed abouta quarter century ago (Wang et al., 1991; Choudhuri et al., 1995; Durney,1995), has emerged as an attractive theoretical model for the solar cycle.There are several modern reviews (Choudhuri, 2011, 2014; Charbonneau,2014; Karak et al., 2014a) surveying the current status of the field. Thepresent paper is not a comprehensive review, but is based on a talk in aWorkshop at the International Space Science Institute (ISSI) highlightingthe works done by the author and his coworkers. Readers are assumed to befamiliar with the phenomenology of the solar cycle and the basic conceptsof MHD (such as flux freezing and magnetic buoyancy). Readers not having
Email address: [email protected] (Arnab Rai Choudhuri)
Preprint submitted to Elsevier November 5, 2018 his background are advised to look at the earlier reviews by the author(Choudhuri, 2011, 2014), which were written for wider readership.The initial effort in this field of flux transport dynamo was directed to-wards developing periodic models to explain the various periodic aspects ofthe solar cycle. Once sufficiently sophisticated periodic models were avail-able, the next question was whether these theoretical models can be usedto understand how the irregularities of the solar cycle arise. There is also arelated question: if we understand what causes the irregularities of the cycle,then will that enable us to predict future cycles?We discuss the basic periodic model of the flux transport dynamo in thenext Section. Then § § §
2. The Basic Periodic Model
One completely non-controversial aspect of solar dynamo models is thegeneration of the toroidal field from the poloidal field by differential rotation.Since differential rotation has now been mapped by helioseismology, thisprocess can now be included in theoretical dynamo models quite realistically.The toroidal field is primarily produced in the tachocline at the bottom of theconvection zone and rises from there due to magnetic buoyancy to create thesunspots. Although some authors have argued that the near-surface shearlayer discovered by helioseismology can also be important for the generationof the toroidal field (Brandenburg, 2005), the general view is that magneticbuoyancy would limit the growth of magnetic field in this region of strongsuper-adiabatic gradient. To this generally accepted view that the toroidalfield is primarily produced in the tachocline, the flux transport dynamo modeladds the following assumptions. • The generation of the poloidal field from the toroidal field takes placedue to the Babcock–Leighton mechanism. • The meridional circulation of the Sun plays a crucial role in the dynamoprocess.We now comment on these two assumptions.2ipolar sunspots on the solar surface appear with a tilt statistically in-creasing with latitude, in accordance with the so-called Joy’s law. This tilt isproduced by the Coriolis force acting on the rising flux tubes (D’Silva and Choudhuri,1993). Babcock (1961) and Leighton (1964) suggested that the poloidal fieldof the Sun is produced from the decay of such tilted bipolar sunspot pairs.There is now enough evidence from observations of the solar surface that thepoloidal field does get built up in this way.The meridional circulation is observed to be poleward at the solar surfaceand advects the poloidal field generated there, in conformity with obser-vational data of surface magnetic fields. The material which is advectedto the poles has to flow back equatorward through deeper layers withinthe solar convection zone. Since this circulation is driven by the turbu-lent stresses in the convection zone, we expect the meridional circulationnot to penetrate much below the bottom of the convection zone, although aslight penetration helps in explaining several aspects of observational data(Nandy and Choudhuri, 2002; Chakraborty et al., 2009). The early modelsof the flux transport dynamo assumed the return flow of the meridional circu-lation to take place at the bottom of the convection zone, where the toroidalfield generated by the differential rotation is advected equatorward with thisflow, giving a natural explanation of the butterfly diagram representing theequatorward shift of the sunspot belt (Choudhuri et al., 1995). Such dynamomodels have been remarkably successful in explaining many aspects of theobservational data (Chatterjee et al., 2004).While we still do not have unambiguous measurements of the return flowof the meridional circulation, some groups claim to have found evidence forthe return flow well above the bottom of the convection zone (Hathaway,2012; Zhao et al., 2013; Schad et al., 2013). However, Rajaguru and Antia(2015) argue that the available helioseismology data still cannot rule out aone-cell meridional circulation spanning the whole of the convection zone ineach hemisphere. Hazra et al. (2014a) showed that, even with a meridionalcirculation much more complicated than the one-cell pattern assumed in theearlier flux transport dynamo papers, it is still possible to match the relevantobservational data as long as there is an equatorward flow at the bottom ofthe convection zone (see Figure 1). 3 igure 1: A complicated meridional circulation used by Hazra et al. (2014a) in a dynamocalculation—red corresponding to streamlines of clockwise circulation and blue to anti-clockwise circulation. Note that the flow near the bottom at low latitudes is equatorward.The butterfly diagram obtained with this circulation is solar-like (sunspot activity driftingto lower latitudes with time).
3. The Origin of the Irregularities of the Solar Cycle
The earliest attempts of explaining irregularities of the solar cycle were byregarding them as nonlinear chaos arising out of the nonlinearities of the dy-namo equations (Weiss et al., 1984). Charbonneau et al. (2007) argued thatthe Gnevyshev-Ohl rule in solar cycles (i.e. the tendency of alternate cyclesto lie above and below the running mean of cycle amplitudes) arises out ofperiod doubling due to nonlinearities. However, the simplest kinds of nonlin-earities expected in dynamo equations tend to make the cycles more stablerather than producing irregularities and it has been suggested that stochas-tic fluctuations are more likely to be the primary reason behind producingirregularities (Choudhuri, 1992).The Babcock–Leighton mechanism for the generation of the poloidal fielddepends on the tilts of bipolar sunspots. While the average tilt is givenby Joy’s law, we see considerable scatter around this average tilt, presum-ably caused by the action of turbulence in the convection zone on the risingflux tubes (Longcope and Choudhuri, 2002). This scatter in the tilt anglesis expected to introduce fluctuations in the Babcock–Leighton mechanism(Choudhuri et al., 2007). By including this fluctuation in the dynamo mod-4ls, it is possible to explain many aspects of the irregularities of the cyclesincluding the grand minima (Choudhuri and Karak, 2009).One other source of irregularities is the fluctuations in the meridionalcirculation. A faster meridional circulation will make the solar cycles shorterand vice versa. While we have actual data of meridional circulation varia-tions for not more than a couple of decades, we have data for durations ofsolar cycles for more than a century, providing indications that the merid-ional circulation had fluctuations in the past with correlation times of theorder of 30–40 years (Karak and Choudhuri, 2011). When the meridionalcirculation is slow and the cycles longer, diffusion has more time to act onthe magnetic fields, making the cycles weaker. On such ground, we expectlonger cycles to be weaker and shorter cycles to be stronger, leading to whatis called the Waldmeier effect (Karak and Choudhuri, 2011). Also, when themeridional circulation is sufficiently weak, theoretical dynamo models showthat even grand minima can be induced (Karak, 2010). To get these results,the correlation time of the meridional circulation fluctuations was taken tobe considerably longer than the cycle period. If the correlation time is takentoo short, then one may not get these results (Mu˜noz-Jaramillo et al., 2010).We also emphasize that the effect of diffusion in making longer cycles weakeris vital for getting these results. We need to take the value of diffusivity suf-ficiently high such that the diffusion time scale is shorter than or of the orderof the cycle period. This is not the case in the model of Dikpati and Gilman(2006) in which diffusivity is very low. A longer cycle in such a low-diffusivitymodel tends to be stronger because differential rotation has time to generatemore toroidal field during a cycle, giving the opposite of the Waldmeier ef-fect. Differences between high- and low-diffusivity dynamos were studied byYeates et al. (2008). Clearly the high-diffusivity model yields results more inconformity with observational data.By analyzing the contents of C-14 in old tree trunks and Be-10 in polar icecores, it has now been possible to reconstruct the history of solar activity overa few millenia (Usoskin, 2013). It has been estimated that there have beenabout 27 grand minima in the last 11,000 years (Usoskin et al., 2007). Sincegrand minima can be caused both by fluctuations in the Babcock-Leightonmechanism and fluctuations in the meridional circulation, a full theoreticalmodel of grand minima should combine both types of fluctuations. If, at thebeginning of a cycle, the poloidal field is too weak due to the fluctuations inthe Babcock–Leighton mechanism or the meridional circulation is too weak,then the Sun may be forced into a grand minimum. Assuming a Gaussian5 igure 2: According to the calculations of Choudhuri and Karak (2012), the poloidalfield strength ( γ is the value of the poloidal field compared to its average value) and theamplitude of the meridional circulation at the surface have to lie in the shaded region at thebeginning of a cycle in order to force the dynamo into a grand minimum. They estimatedthe probability of this to be about 1.3%, corresponding to about 13 grand minima in11,000 years. distribution for fluctuations in both the Babcock–Leighton mechanism andthe meridional circulation, Choudhuri and Karak (2012) developed a com-prehensive theory of grand minima that agreed remarkably well with thestatistical data of grand minima (see Figure 2 and its caption). However,if there are no sunspots during grand minima, then the Babcock–Leightonmechanism which depends on sunspots may not be operational and howthe Sun comes out of the grand minima is still rather poorly understood(Karak and Choudhuri, 2013; Hazra et al., 2014b).While discussing irregularities of the solar cycle, it may be mentioned thatthese irregularities are correlated reasonably well in the two hemispheresof the Sun. Strong cycles are usually strong in both the hemispheres andweak cycles are weak in both. This requires a coupling between the twohemispheres, implying that the turbulent diffusion time over the convectionzone could not be more than a few years (Chatterjee and Choudhuri, 2006;Goel and Choudhuri, 2009). 6 . Predicting solar cycles The first attempts of predicting solar cycles were based on using ob-servational precursors of solar cycles. There is considerable evidence thatthe polar field at the end of a solar cycle is correlated with the next cy-cle. Since the polar field at the end of cycle 23 was rather weak, severalauthors (Svalgaard et al., 2005; Schatten, 2005) predicted that the next cy-cle 24 would be weak.The sunspot minimum between the cycles 23 and 24 (around 2005–2008)was the first sunspot minimum when sufficiently sophisticated models of theflux transport dynamo were available. Whether these models could be usedto predict the next cycle became an important question. When a kinematicmean field dynamo code is run without introducing any fluctuations, onefinds that the code settles down to a periodic solution if the various dynamoparameters are in the correct range. In order to model actual solar cycles, onehas to feed some observational data to the theoretical model in an appropriatemanner and then run the code for a future cycle to generate a prediction.The crucial issue here is to figure out what kind of observational data tofeed into the theoretical model and how. An understanding of what causesthe irregularities of the solar cycle is of utmost interest in deciding this.An attempt by Dikpati and Gilman (2006) produced the prediction that thecycle 24 would be very strong, in contradiction to what was predicted on thebasis of the weak polar field at the end of the cycle 23.Assuming that the fluctuation in the Babcock–Leighton mechanism isthe main cause of irregularities in the solar cycle, Choudhuri et al. (2007)devised a methodology of feeding observational data of the polar magneticfield into the theoretical model to account for the random kick received bythe dynamo due to fluctuations in the Babcock–Leighton mechanism. Thedynamo model of Choudhuri et al. (2007) predicted that the cycle 24 wouldbe weak, in conformity with the weakness of the polar field at the end ofcycle 23. Jiang et al. (2007) explained the physical basis of what causes thecorrelation between the polar field at the end of a cycle and the strength ofthe next cycle. Suppose the fluctuations in the Babcock–Leighton mechanismproduced a poloidal field stronger than the usual. This strong poloidal fieldwill be advected to the poles to produce a strong polar field at the end ofthe cycle and, if the turbulent diffusion time across the convection zone isnot more than a few years, this poloidal field will also diffuse to the bottomof the convection zone to provide a strong seed for the next cycle, making7
985 1990 1995 2000 2005 2010 2015050100150200250
Cycle 24 Sunspot Number Prediction22 23 24
Figure 3: The sunspot number in the last few years. The upper star indicates the pre-dicted amplitude of cycle 24 according to Dikpati and Gilman (2006), while the lower starindicates the predicted amplitude according to Choudhuri et al. (2007). The circle on thehorizontal axis indicates the time when these predictions were made. the next cycle strong. On the other hand, if the poloidal field produced ina cycle is weaker than the average, then we shall get a weak polar field atthe end of the cycle and a weak subsequent cycle. This will give rise to acorrelation between the polar field at the end of a cycle and the strengthof the next cycle. If the diffusion is assumed to be weak—as in the modelof Dikpati and Gilman (2006)—then different regions of the convection zonemay not be able to communicate through diffusion in a few years and weshall not get this correlation. The prediction of Choudhuri et al. (2007) thatthe cycle 24 would be weak was a robust prediction in their model becausethe polar field at the end of a cycle is correlated to the next cycle in theirmodel and they had fed the data of the weak polar field at the end of cycle 23into their theoretical model in order to generate their prediction. As can beseen in Figure 3, the actual amplitude of cycle 24 turned out to be very closeto what was predicted by Choudhuri et al. (2007), making this to be the firstsuccessful prediction of a solar cycle from a theoretical dynamo model in thehistory of our subject.As we have pointed out in the previous Section, the fluctuations of themeridional circulation also can cause irregularities in the solar cycle. This8as not realized when the various predictions for cycle 24 were made during2005–2007. It is observationally found that there is a correlation between thedecay rate of a cycle and the strength of the next cycle (Hazra et al., 2015).Now, a faster meridional circulation, which would make a cycle shorter, surelywill make the decay rate faster and also the cycle stronger, as pointed outalready (a slower meridional circulation would do the opposite). If the effectof the fluctuating meridional circulation on the decay rate is immediate, buton the cycle strength is delayed by a few years, then we can explain theobserved correlation. This is confirmed by theoretical dynamo calculations(Hazra et al., 2015). This shows that it may be possible to use the decayrate at the end of a cycle to predict the effect of the fluctuating meridionalcirculation on the next cycle. This issue needs to be looked at carefully.
5. Conclusion
We have pointed out that over the years we have acquired an understand-ing of how the irregularities of the solar cycle arise and that this understand-ing helps us in predicting future solar activity. Our point of view is thatthe fluctuations in the Babcock–Leighton mechanism and the fluctuations inthe meridional circulation are the two primary sources of irregularities in thesolar cycles. These fluctuations have to be modelled realistically and fed intoa theoretical dynamo model to generate predictions.It may be noted that we now have a huge amount of data on the magneticactivity of solar-like stars (Choudhuri, 2017). Some solar-like stars displaygrand minima and we have evidence for the Waldmeier effect in some ofthem—see the concluding paragraph of Karak et al. (2014b). This suggeststhat dynamos similar to the solar dynamo may be operational in the interiorsof solar-type stars and the irregularities of their cycles also may be producedthe same way as the irregularities of solar cycles. Work on constructing fluxtransport dynamo models for solar-like stars has just begun (Karak et al.,2014b). Our ability to model stellar dynamos may throw more light on howthe solar dynamo works.All the theoretical results we discussed are based on axisymmetric 2Dkinematic dynamo models. One inherent limitation of such models is thatthe rise of a magnetic loop due to magnetic buoyancy and the Babcock–Leighton process of generating poloidal flux from it are intrinsically 3D pro-cesses and can be included in 2D models only through very crude averag-ing procedures (Nandy and Choudhuri, 2001; Mu˜noz-Jaramillo et al., 2010;9 igure 4: A study of magnetic field evolution on the solar surface from the 3D kinematicdynamo model of Hazra et al. (2017), showing how the polar field builds up from a singletilted bipolar sunspot pair due to the Babcock–Leighton mechanism. The different panelsshow the distribution of magnetic field at the following epochs after the emergence of thebipolar sunspots: (a) 0.025 yr, (b) 0.25 yr, (c) 1.02 yr, (d) 2.03 yr, (e) 3.05 yr, (f) 4.06 yr.
Choudhuri and Hazra, 2016). As we have discussed, the fluctuations in theBabcock–Leighton process play a crucial role in producing the irregulari-ties of the solar cycle. In order to model these fluctuations realistically,it is essential to treat the Babcock–Leighton process itself more realisti-cally than what is possible in 2D models. The next step should be theconstruction of 3D kinematic dynamo models for which efforts have begun(Yeates and Mu˜noz-Jaramillo, 2013; Miesch and Dikpati, 2014; Miesch and Teweldebirhan,2015; Hazra et al., 2017). Such 3D kinematic dynamo models can treat theBabcock–Leighton mechanism more realistically (see Figure 4) and shouldprovide a better understanding of how fluctuations in the Babcock–Leightonmechanism cause irregularities in the dynamo. The magnetic field presum-ably exists in the form of flux tubes throughout the convection zone and onelimitation of a mean field model is that flux tubes are not handled properly(Choudhuri, 2003). A 3D kinematic model allows one to model flux tubes ina more realistic fashion. A proper inclusion of flux tubes in a dynamo modelis essential for explaining such interesting observations as the predominanceof negative helicity in the norther hemisphere (Pevtsov et al., 1995), whichis presumably caused by the wrapping of the poloidal field around the risingflux tubes (Choudhuri, 2003; Chatterjee et al., 2006; Hotta and Yokoyama,2012). This process can be modelled in 2D mean field dynamo models onlythrough drastic simplifications (Choudhuri et al., 2004). It should be pos-sible to model this better through 3D kinematic dynamo models. In other10ords, after the tremendous advances made by the 2D kinematic flux trans-port model during the last quarter century, it appears that that 3D kinematicdynamo models are likely to occupy the centre stage in the coming years.
Acknowledgements
This work is partly supported by DST through a J.C. Bose Fellowship.I thank VarSITI for travel support for attending the workshop at ISSI andthank ISSI for local hospitality during the workshop.
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