Journal of Atmospheric and Solar-Terrestrial Physics 176 (2018) 5–9

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Journal of Atmospheric and Solar-Terrestrial Physics

journal homepage: www.elsevier.com/locate/jastp

Flux transport dynamo: From modelling irregularities to making predictions

Arnab Rai Choudhuri
Department of Physics, Indian Institute of Science, Bangalore, 560012, India

A R T I C L E I N F O A B S T R A C T

Keywords: The flux transport dynamo, in which the poloidal magnetic field is generated by the Babcock–Leighton mecha-
Dynamo nism and the meridional circulation plays a crucial role, has emerged as an attractive model for the solar cycle.
Solar cycle Based on theoretical calculations done with this model, we argue that the fluctuations in the Babcock–Leighton
mechanism and the fluctuations in the meridional circulation are the most likely causes of the irregularities of the
solar cycle. With our increased theoretical understanding of how these irregularities arise, it can be possible to
predict a future solar cycle by feeding the appropriate observational data in a theoretical dynamo model.

1. Introduction 2. The basic periodic model

The flux transport dynamo model, which started being developed One completely non-controversial aspect of solar dynamo models is
about a quarter century ago (Wang et al., 1991; Choudhuri et al., 1995; the generation of the toroidal field from the poloidal field by differential
Durney, 1995), has emerged as an attractive theoretical model for the rotation. Since differential rotation has now been mapped by helio-
solar cycle. There are several modern reviews (Choudhuri, 2011, 2014; seismology, this process can now be included in theoretical dynamo
Charbonneau, 2014; Karak et al., 2014a) surveying the current status of models quite realistically. The toroidal field is primarily produced in the
the field. The present paper is not a comprehensive review, but is based tachocline at the bottom of the convection zone and rises from there due
on a talk in a Workshop at the International Space Science Institute (ISSI) to magnetic buoyancy to create the sunspots. Although some authors
highlighting the works done by the author and his coworkers. Readers have argued that the near-surface shear layer discovered by helio-
are assumed to be familiar with the phenomenology of the solar cycle and seismology can also be important for the generation of the toroidal field
the basic concepts of MHD (such as flux freezing and magnetic buoy- (Brandenburg, 2005), the general view is that magnetic buoyancy would
ancy). Readers not having this background are advised to look at the limit the growth of magnetic field in this region of strong super-adiabatic
earlier reviews by the author (Choudhuri, 2011, 2014), which were gradient. To this generally accepted view that the toroidal field is pri-
written for wider readership. marily produced in the tachocline, the flux transport dynamo model adds
The initial effort in this field of flux transport dynamo was directed the following assumptions.
towards developing periodic models to explain the various periodic as-
pects of the solar cycle. Once sufficiently sophisticated periodic models  The generation of the poloidal field from the toroidal field takes place
were available, the next question was whether these theoretical models due to the Babcock–Leighton mechanism.
can be used to understand how the irregularities of the solar cycle arise.  The meridional circulation of the Sun plays a crucial role in the dy-
There is also a related question: if we understand what causes the ir- namo process.
regularities of the cycle, then will that enable us to predict future cycles?
We discuss the basic periodic model of the flux transport dynamo in We now comment on these two assumptions.
the next Section. Then x 3 discusses the possible causes of the irregu- Bipolar sunspots on the solar surface appear with a tilt statistically
larities of the solar cycle. Afterwards in x 4 we address the question increasing with latitude, in accordance with the so-called Joy's law. This
whether we are now in a position to predict future cycles. Finally, in x 5 tilt is produced by the Coriolis force acting on the rising flux tubes
we summarize the limitations of the 2D kinematic dynamo models and (D'Silva and Choudhuri, 1993). Babcock (1961) and Leighton (1964)
the recent efforts of going beyond such simple models. suggested that the poloidal field of 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 the poloidal field does get built up

E-mail address: arnab@physics.iisc.ernet.in.

http://dx.doi.org/10.1016/j.jastp.2017.08.002
Received 22 February 2017; Received in revised form 15 June 2017; Accepted 2 August 2017
Available online 5 August 2017
1364-6826/© 2017 Elsevier Ltd. All rights reserved.
A.R. Choudhuri Journal of Atmospheric and Solar-Terrestrial Physics 176 (2018) 5–9

Fig. 1. A complicated meridional circulation used by Hazra et al. (2014a) in a dynamo calculation—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 drifting to lower latitudes
with time). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

in this way. presumably caused by the action of turbulence in the convection zone on
The meridional circulation is observed to be poleward at the solar the rising flux tubes (Longcope and Choudhuri, 2002). This scatter in the
surface and advects the poloidal field generated there, in conformity with tilt angles is expected to introduce fluctuations in the Babcock–Leighton
observational data of surface magnetic fields. The material which is mechanism (Choudhuri et al., 2007). By including this fluctuation in the
advected to the poles has to flow back equatorward through deeper dynamo models, it is possible to explain many aspects of the irregularities
layers within the solar convection zone. Since this circulation is driven by of the cycles including the grand minima (Choudhuri and Karak, 2009).
the turbulent stresses in the convection zone, we expect the meridional One other source of irregularities is the fluctuations in the meridional
circulation not to penetrate much below the bottom of the convection circulation. A faster meridional circulation will make the solar cycles
zone, although a slight penetration helps in explaining several aspects of shorter and vice versa. While we have actual data of meridional circu-
observational data (Nandy and Choudhuri, 2002; Chakraborty et al., lation variations for not more than a couple of decades, we have data for
2009). The early models of the flux transport dynamo assumed the return durations of solar cycles for more than a century, providing indications
flow of the meridional circulation to take place at the bottom of the that the meridional circulation had fluctuations in the past with corre-
convection zone, where the toroidal field generated by the differential lation times of the order of 30–40 years (Karak and Choudhuri, 2011).
rotation is advected equatorward with this flow, giving a natural expla- When the meridional circulation is slow and the cycles longer, diffusion
nation of the butterfly diagram representing the equatorward shift of the has more time to act on the magnetic fields, making the cycles weaker.
sunspot belt (Choudhuri et al., 1995). Such dynamo models have been On such ground, we expect longer cycles to be weaker and shorter cycles
remarkably successful in explaining many aspects of the observational to be stronger, leading to what is called the Waldmeier effect (Karak and
data (Chatterjee et al., 2004). Choudhuri, 2011). Also, when the meridional circulation is sufficiently
While we still do not have unambiguous measurements of the return weak, theoretical dynamo models show that even grand minima can be
flow of the meridional circulation, some groups claim to have found induced (Karak, 2010). To get these results, the correlation time of the
evidence for the return flow well above the bottom of the convection meridional circulation fluctuations was taken to be considerably longer
zone (Hathaway, 2012; Zhao et al., 2013; Schad et al., 2013). However, than the cycle period. If the correlation time is taken too short, then one
Rajaguru and Antia (2015) argue that the available helioseismology data may not get these results (Mu~ noz-Jaramillo et al., 2010). We also
still cannot rule out a one-cell meridional circulation spanning the whole emphasize that the effect of diffusion in making longer cycles weaker is
of the convection zone in each hemisphere. Hazra et al. (2014a) showed vital for getting these results. We need to take the value of diffusivity
that, even with a meridional circulation much more complicated than the sufficiently high such that the diffusion time scale is shorter than or of the
one-cell pattern assumed in the earlier flux transport dynamo papers, it is order of the cycle period. This is not the case in the model of Dikpati and
still possible to match the relevant observational data as long as there is Gilman (2006) in which diffusivity is very low. A longer cycle in such a
an equatorward flow at the bottom of the convection zone (see Fig. 1). low-diffusivity model tends to be stronger because differential rotation
has time to generate more toroidal field during a cycle, giving the
3. The origin of the irregularities of the solar cycle opposite of the Waldmeier effect. Differences between high- and
low-diffusivity dynamos were studied by Yeates et al. (2008). Clearly the
The earliest attempts of explaining irregularities of the solar cycle high-diffusivity model yields results more in conformity with observa-
were by regarding them as nonlinear chaos arising out of the non- tional data.
linearities of the dynamo equations (Weiss et al., 1984). Charbonneau By analyzing the contents of C-14 in old tree trunks and Be-10 in polar
et al. (2007) argued that the Gnevyshev-Ohl rule in solar cycles (i.e. the ice cores, it has now been possible to reconstruct the history of solar
tendency of alternate cycles to lie above and below the running mean of activity over a few millenia (Usoskin, 2013). It has been estimated that
cycle amplitudes) arises out of period doubling due to nonlinearities. there have been about 27 grand minima in the last 11,000 years (Usoskin
However, the simplest kinds of nonlinearities expected in dynamo et al., 2007). Since grand minima can be caused both by fluctuations in
equations tend to make the cycles more stable rather than producing the Babcock-Leighton mechanism and fluctuations in the meridional
irregularities and it has been suggested that stochastic fluctuations are circulation, a full theoretical model of grand minima should combine
more likely to be the primary reason behind producing irregularities both types of fluctuations. If, at the beginning of a cycle, the poloidal field
(Choudhuri, 1992). is too weak due to the fluctuations in the Babcock–Leighton mechanism
The Babcock–Leighton mechanism for the generation of the poloidal or the meridional circulation is too weak, then the Sun may be forced into
field depends on the tilts of bipolar sunspots. While the average tilt is a grand minimum. Assuming a Gaussian distribution for fluctuations in
given by Joy's law, we see considerable scatter around this average tilt, both the Babcock–Leighton mechanism and the meridional circulation,

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A.R. Choudhuri Journal of Atmospheric and Solar-Terrestrial Physics 176 (2018) 5–9

4. Predicting solar cycles

The first attempts of predicting solar cycles were based on using

observational precursors of solar cycles. There is considerable evidence
that the polar field at the end of a solar cycle is correlated with the next
cycle. Since the polar field at the end of cycle 23 was rather weak, several
authors (Svalgaard et al., 2005; Schatten, 2005) predicted that the next
cycle 24 would be weak.
The sunspot minimum between the cycles 23 and 24 (around
2005–2008) was the first sunspot minimum when sufficiently sophisti-
cated models of the flux transport dynamo were available. Whether these
models could be used to predict the next cycle became an important
question. When a kinematic mean field dynamo code is run without
introducing any fluctuations, one finds that the code settles down to a
periodic solution if the various dynamo parameters are in the correct
range. In order to model actual solar cycles, one has to feed some
observational data to the theoretical model in an appropriate manner 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 to feed into the
Fig. 2. According to the calculations of Choudhuri and Karak (2012), the poloidal field theoretical model and how. An understanding of what causes the irreg-
strength (γ is the value of the poloidal field compared to its average value) and the
ularities of the solar cycle is of utmost interest in deciding this. An
amplitude of the meridional circulation at the surface have to lie in the shaded region at
the beginning of a cycle in order to force the dynamo into a grand minimum. They esti- attempt by Dikpati and Gilman (2006) produced the prediction that the
mated the probability of this to be about 1.3%, corresponding to about 13 grand minima in cycle 24 would be very strong, in contradiction to what was predicted on
11,000 years. the basis of the weak polar field at the end of the cycle 23.
Assuming that the fluctuation in the Babcock–Leighton mechanism is
the main cause of irregularities in the solar cycle, Choudhuri et al. (2007)
Choudhuri and Karak (2012) developed a comprehensive theory of grand
devised a methodology of feeding observational data of the polar mag-
minima that agreed remarkably well with the statistical data of grand
netic field into the theoretical model to account for the random kick
minima (see Fig. 2 and its caption). However, if there are no sunspots
received by the dynamo due to fluctuations in the Babcock–Leighton
during grand minima, then the Babcock–Leighton mechanism which
mechanism. The dynamo model of Choudhuri et al. (2007) predicted that
depends on sunspots may not be operational and how the Sun comes out
the cycle 24 would be weak, in conformity with the weakness of the polar
of the grand minima is still rather poorly understood (Karak and
field at the end of cycle 23. Jiang et al. (2007) explained the physical
Choudhuri, 2013; Hazra et al., 2014b).
basis of what causes the correlation between the polar field at the end of a
While discussing irregularities of the solar cycle, it may be mentioned
cycle and the strength of the next cycle. Suppose the fluctuations in the
that these irregularities are correlated reasonably well in the two hemi-
Babcock–Leighton mechanism produced a poloidal field stronger than
spheres of the Sun. Strong cycles are usually strong in both the hemi-
the usual. This strong poloidal field will be advected to the poles to
spheres and weak cycles are weak in both. This requires a coupling
produce a strong polar field at the end of the cycle and, if the turbulent
between the two hemispheres, implying that the turbulent diffusion time
diffusion time across the convection zone is not more than a few years,
over the convection zone could not be more than a few years (Chatterjee
this poloidal field will also diffuse to the bottom of the convection zone to
and Choudhuri, 2006; Goel and Choudhuri, 2009).
provide a strong seed for the next cycle, making the next cycle strong. On
the other hand, if the poloidal field produced in a cycle is weaker than the
average, then we shall get a weak polar field at the end of the cycle and a
weak subsequent cycle. This will give rise to a correlation between the
polar field at the end of a cycle and the strength of the next cycle. If the
diffusion is assumed to be weak—as in the model of Dikpati and Gilman
(2006)—then different regions of the convection zone may not be able to
communicate through diffusion in a few years and we shall not get this
correlation. The prediction of Choudhuri et al. (2007) that the cycle 24
would be weak was a robust prediction in their model because the polar
field at the end of a cycle is correlated to the next cycle in their model and
they had fed the data of the weak polar field at the end of cycle 23 into
their theoretical model in order to generate their prediction. As can be
seen in Fig. 3, the actual amplitude of cycle 24 turned out to be very close
to what was predicted by Choudhuri et al. (2007), making this to be the
first successful prediction of a solar cycle from a theoretical dynamo
model in the history of our subject.
As we have pointed out in the previous Section, the fluctuations of the
meridional circulation also can cause irregularities in the solar cycle. This
was not realized when the various predictions for cycle 24 were made
during 2005–2007. It is observationally found that there is a correlation
between the decay 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, surely will make the decay rate faster and also the
Fig. 3. The sunspot number in the last few years. The upper star indicates the predicted
cycle stronger, as pointed out already (a slower meridional circulation
amplitude of cycle 24 according to Dikpati and Gilman (2006), while the lower star in-
dicates the predicted amplitude according to Choudhuri et al. (2007). The circle on the would do the opposite). If the effect of the fluctuating meridional cir-
horizontal axis indicates the time when these predictions were made. culation on the decay rate is immediate, but on the cycle strength is

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A.R. Choudhuri Journal of Atmospheric and Solar-Terrestrial Physics 176 (2018) 5–9

Fig. 4. A study of magnetic field evolution on the solar surface from the 3D kinematic dynamo model of Hazra et al. (2017), showing how the polar field builds up from a single tilted
bipolar sunspot pair due to the Babcock–Leighton mechanism. The different panels show the distribution of magnetic field at the following epochs after the emergence of the bipolar
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.

delayed by a few years, then we can explain the observed correlation. in a more realistic fashion. A proper inclusion of flux tubes in a dynamo
This is confirmed by theoretical dynamo calculations (Hazra et al., 2015). model is essential for explaining such interesting observations as the
This shows that it may be possible to use the decay rate at the end of a predominance of negative helicity in the norther hemisphere (Pevtsov
cycle to predict the effect of the fluctuating meridional circulation on the et al., 1995), which is presumably caused by the wrapping of the poloidal
next cycle. This issue needs to be looked at carefully. field around the rising flux tubes (Choudhuri, 2003; Chatterjee et al.,
2006; Hotta and Yokoyama, 2012). This process can be modelled in 2D
5. Conclusion mean field dynamo models only through drastic simplifications
(Choudhuri et al., 2004). It should be possible to model this better
We have pointed out that over the years we have acquired an un- through 3D kinematic dynamo models. In other words, after the
derstanding of how the irregularities of the solar cycle arise and that this tremendous advances made by the 2D kinematic flux transport model
understanding helps us in predicting future solar activity. Our point of during the last quarter century, it appears that that 3D kinematic dynamo
view is that the fluctuations in the Babcock–Leighton mechanism and the models are likely to occupy the centre stage in the coming years.
fluctuations in the meridional circulation are the two primary sources of
irregularities in the solar cycles. These fluctuations have to be modelled Acknowledgements
realistically and fed into a theoretical dynamo model to generate
predictions. This work is partly supported by DST through a J.C. Bose Fellowship. I
It may be noted that we now have a huge amount of data on the thank VarSITI for travel support for attending the workshop at ISSI and
magnetic activity of solar-like stars (Choudhuri, 2017). Some solar-like thank ISSI for local hospitality during the workshop.
stars display grand minima and we have evidence for the Waldmeier
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