Light on sunspots

Solar dynamo models developed by scientists at the Indian Institute of Science, Bangalore, seem to be able to correctly reproduce many of the features of the sunspot cycle.

ABOUT two years ago Frontline (December 17, 2010) carried an article by Biman Nath, titled Vanishing sunspots, in the wake of the first signs of solar activity after about a two-year lull during which very few sunspots or solar flares had been seen on the suns surface. Towards the end of 2010, the sun seemed to be slowly coming out of that slumber.

The number of sunspots waxes and wanes in an approximate 11-year cycle (Figure 1). This cyclical behaviour was first discovered by the German astronomer Samuel Heinrich Schwabe in 1844. In 1849, the Swiss astronomer Rudolf Wolf reconstructed the cycles back up to 1745 and numbered them, starting with the 1755-1766 cycle as 1. In that scheme, we are now in the midst of Cycle 24. According to David Hathaway of the Marshall Space Flight Centre (MSFC) of the National Aeronautics and Space Administration (NASA), an expert in the field, Cycle 24 began in May 2008.

It is clear from Figure 1 that not all sunspot cycles are either of equal strength or of exactly equal duration. The average period of a solar cycle is actually 131 months with a statistical spread (standard deviation) of 14 months. Cycle 23 lasted 142 months, which is well within the standard deviation and thus not abnormal at all. The lull at the beginning of Cycle 24 was also well within historical norms for the solar cycle, says Hathaway. It was an extended solar minimum after Cycle 23. We are now well over four years into Cycle 24 and are currently close to its peaking phase. In fact, the cycle had a strong peaking in late 2011, with a smoothed sunspot number of around 90. (The SSN is the 12-month running average for each of the months specified.)

According to Hathaways predictions (revised in November), Cycle 24 will have an SSN of only 73 at its maximum, which it will be reached in the Fall of 2013 (Figure 2). Interestingly, at the beginning of Cycle 24, Hathaway had predicted an intense or strong Cycle 24, with a peak SSN in the 130-140 range. In January 2009, however, this was revised downwards to the 100-110 range, in May 2009 to 80-90 and in October 2010 down to 60-70. The latest prediction, November 2012, is about the same. Fixing [the cycle start] date and then finding the cycle amplitude that best fits the sunspot number data yields the current (revised) prediction, said the MSFCs November 2 posting on its website. The current predicted and observed size makes this the smallest sunspot cycle after Cycle 14, which had a maximum of 64.2 in February 1906.

Solar activity & space missions

Predictions based on empirical data, says Hathaway, become fairly reliable only once the cycle is well under way (about three years after the minimum in SSN occurs). But predicting the behaviour of a solar cycle well before the cycle sets in is important for the following reason. Matter in stars such as the sun is in the form of very high temperature plasma, a fluid-like state comprising ionised gases (atoms that have been stripped of their electrons) with the charged nuclei and electrons of atoms moving freely. When a cycle is at its peak, the consequent solar flares (gigantic explosions around the sunspot regions) and coronal mass ejections (huge blobs of plasma ejected from the sun at enormous speeds) can cause disturbances in the ionosphere. These can disrupt terrestrial radio communications, damage electronic equipment on satellites and even trip power supply grids as has happened in the past. Planning for satellite orbits and space missions often requires knowledge of solar activity levels years in advance.

So it is important to be able to theoretically predict well in advance the strength of the oncoming solar maximum. But theoretical models to predict the amplitude or the maximum of the cycle in advance were not developed until recently. Models that predict the amplitude of the sunspot cycle, when the sun is near or at the minimum of the previous activity cycle, only came into being about five to six years ago. Cycle 24 will be a test for these models as the cycle nears its peak in 2013.

Besides the cycle-to-cycle variability in solar activity, there is also a long-term variability in the form of grand minima, when almost no sunspots are seen. The last such grand minimum, called the Maunder Minimum, appeared between 1645 and 1715 and coincided with the coldest period of the 17th century Little Ice Age. Reliable sunspot data exist only from 1610, but terrestrial abundances of cosmogenic isotopes are good proxy data for solar activity. Analysis of such data shows that there have been about 27 grand minima in the last 11,000 years. That is, in 1,000 solar cycles, 2.7 per cent had conditions that drove the sun into prolonged minimum activity. It is also, therefore, important to understand the origin of solar grand minima.

Solar dynamo

On the basis of the splitting of spectral lines in the presence of a magnetic field (Zeeman splitting), George Hale discovered in 1908 that sunspots are actually dark regions of a concentrated magnetic field (about 3,000 gauss, or G, which is 10,000 times stronger than the earths magnetic field). It is thus apparent that the sunspot cycle is a consequence of the magnetic cycle of the sun. It is also now generally believed that the suns magnetic field is generated by a magnetic dynamo within the sun.

In the ordinary dynamo we are familiar with, a conducting coil rotates in a magnetic field. The cutting of the magnetic field by the rotating coil gives rise to an electric current across the coil in accordance with Faradays Law of electromagnetic induction. In stars such as the sun, the rotating masses of plasma, being highly charged, are good conductors of electricity. If there is a magnetic field, the dynamo principle will work here as well generating electric currents. The electric currents thus generated can in turn create magnetic fields, which, under appropriate conditions, reinforce the existing magnetic field.

Sunspots are the most striking manifestations of the solar magnetic field and, hence, the best indicators of its nature and behaviour. The fact that the suns magnetic field undergoes significant changes within just a few years and that it does so in a cyclical manner indicates that the magnetic field is continually being generated within the sun. So any theoretical model developed to explain the solar magnetic field must be able to explain this basic feature of sunspots. The basic solar dynamo problem that a successful theory must resolve can be stated as How can a magnetic field be maintained by the motion of an electrically conducting fluid despite the continual drain of magnetic energy by the resistance of fluid? Besides, the theory must also explain the other observed spatio-temporal features of solar activity. The current understanding of the solar dynamo is, however, far from complete.

The theoretical framework in which dynamo models are studied is magnetohydrodynamics (MHD). In MHD, the plasma is treated as an electrically conducting continuum fluid obeying equations of fluid dynamics. The coupled system of Maxwells electromagnetic equations and fluid dynamics equations must be solved in three dimensions in a self-consistent manner. However, these equations do not easily lend themselves to analytical solutions. Solar dynamo model investigations are, therefore, mostly done by numerical simulations and analysis using modern-day computers.

Like the earth, the sun rotates about its axis. But unlike the earth, which rotates as a single solid body, regions of the sun at different latitudes rotate with different speeds (differential rotation). While an equatorial region takes about 25 days to complete one rotation, at the poles it takes more than 30 days. Similarly, helioseismologythe study of the waves that propagate within the sun has been used since the 1990s to determine the radial differential rotation, when different layers of the suns interior rotate with different speeds. Differential rotation, as a function of latitude as well as of radius, is an important component of the solar dynamo process.

Individual sunspots have a life of a few days to a few weeks. One of the peculiar features of sunspots is that they appear at lower and lower latitudes as the solar cycle progresses. In the early phase of cycle they appear between the latitudes of 30 and 40. As the cycle progresses, they move towards the solar equator. Then a fresh half-cycle begins, with sunspots showing up again at higher latitudes. This equator-ward drift can be seen in the so-called butterfly diagram (Figure 3). Further, sunspots often appear in pairs with opposite polarities. Since hot plasma streams are seen to loop from one spot of a pair to the other, the sunspot pairs are like the poles of a horseshoe magnet (picture 1).

An important feature of such sunspot pairs is that the lines joining the two spots are nearly parallel to the equator. There is a systematic tilt of these lines with respect to the equator, and this tilt increases with latitude (Joys Law). But there is also a large scatter in the tilt angles, indicating fluctuations in the underlying mechanism. The leading spots have opposite polarities in the two hemispheres, and there is also a reversal of polarities from one cycle to the next (Hales Law) (Figure 3). The reversal actually occurs during the solar maximum when the number of sunspots on the surface is the highest (Figure 4). Therefore, while sunspot numbers have a cycle of 11 years, the magnetic cycle has a period of 22 years.

Magnetic fields can be regarded as continuous loops of lines of force having both tension and pressure. Like rubber bands, magnetic fields can be strengthened by stretching them (which increases the magnetic flux density), twisting them and folding them back on themselves. The stretching, twisting and folding is done by the fluid flows in the sun. Nearly all of the suns fluid flows, both on its surface and in its interior, may contribute in one way or another to the production of its magnetic field. According to the MHD equations, the magnetic flux is nearly frozen in the plasma, and the flux transport is almost entirely because of the fluid flow.

The solar magnetic field is predominantly azimuthal or toroidal (east-west oriented), and this is indicated by the fact that the lines joining sunspot pairs are nearly parallel to the equator. But it can be seen from Figure 3 that besides the high concentrated fields at the sunspots there is also a much weaker diffused magnetic field present outside the sunspots in the poleward direction. A drift of large unipolar (positive and negative) swathes of this diffuse field towards the suns poles has also been observed.

This poleward migration of the magnetic field is now understood as being due to meridional circulation of the plasma on the solar surface. H.W. Babcock (1961) and R.B. Leighton (1969) showed how the surface differential rotation could convert the toroidal field of decaying sunspots to a poloidal (north-south oriented) field. This poloidal field is carried towards the poles by the fluid flow along the meridians on the surface. This N-S fluid flow velocity is about 20 metres/second, while the E-W flow velocity due to the suns rotation is about 2,000 m/s.

The observed polarity reversal probably results when sufficient field of opposite polarity accumulates at the poles. The average field at the poles is about 10 G, but since this field reverses its polarity during the solar maximum (Figure 4), it stands to reason that this process is somehow linked to the strong field at the sunspots and is part of the magnetic cycle responsible for the sunspots. Only in recent years has it been realised that this diffuse field is important for the solar magnetic cycle and needs to be taken into account in any solar dynamo model.

Two dynamo models

There is a class of solar dynamo models called Flux Transport Dynamo Models. A version of FTDMs developed by Arnab Rai Choudhuri and his associates at the Indian Institute of Science (IISc), Bangalore, have since the mid-1990s seemed to correctly reproduce many of the features of the sunspot cycle. In particular, as mentioned in Biman Naths 2007 article, Choudhuri and two of his research students correctly predicted, using their FTDM and data of an appropriate predictor from Cycle 23, that Cycle 24 would be a weak one with fewer SSNs on average and a smaller amplitude of the maximum. On the other hand, in 2005, Peter Gilman and Mausumi Dikpati at the High Altitude Observatory of the National Centre for Atmospheric Research (NCAR), United States, predicted, using a slightly different version, that Cycle 24 would be much stronger (with 150-180 amplitude) than Cycle 23 as indeed NASA/MSFC too had originally predicted.

These two papers based on FTDMs were the first ever to theoretically predict the oncoming solar cycle on the basis of data from the decaying one. The main difference between the two models, however, lay in the assumptions made in the models and the assimilated predictor data of the previous cycle. Given the current data on Cycle 24 (Figure 2), and the recent statement by NASA that it is likely to be weakest in a century, the IISc prediction is likely to be borne out. So what exactly is an FTDM and how does it differ from the earlier conventional solar dynamo models?

The solar interior can be separated into four regions (Figure 5): the core where energy is generated, which forms the innermost quarter. This energy is transported mostly by radiation (X-rays and gamma rays) through the radiative zone (RZ), which extends up to 70 per cent of the solar radius. Beyond this, radiation is unable to carry the energy flux towards the surface when convection (energy transport by fluid motion) sets in. The convective zone (CZ) constitutes nearly the outermost 30 per cent of the interior (~200,000 km), extending up to the visible solar surface. The solar magnetic field is thought to be generated at the thin (~2 per cent of the solar radius: ~14,000 kilometres) interface layer between the RZ and the CZ called the tachocline, where there is maximum radial differential rotation and hence maximum shear. Also, helioseismological studies have shown that, while there is latitudinal differential rotation, there is not much radial differential motion within the CZ.

Differential rotation can stretch the magnetic fields out within the sun and wrap them up around it. This is called the Omega Effect. The latitudinal differential rotation can take a poloidal field and wrap it once around the sun in about eight months. That is, poloidal flux can be turned into toroidal flux by differential rotation (Figure 6a & b). Toroidal flux can also be twisted by the effects of the suns rotationthe Alpha Effect (Figure 6c)and, aided by turbulence in the CZ, turned into poloidal field. The problem for dynamo models is sustaining both these components of the solar magnetic field by the fluid flow and reproducing the other features of sunspots and their cycles.

Scientists believe that sunspots are the surface manifestations of a strong toroidal magnetic field generated within the solar interior, which is the strongest at the tachocline. If this field becomes unstable, flux line bundles can buckle and rise through the CZ (just as bubbles in water). Such instabilities can arise because of what is called magnetic buoyancy. Since a magnetic field exerts pressure, plasma in regions inside the magnetic field can expand and become lighter than the outside so that it is buoyantly lifted by gravity. In the CZ, convective instability and magnetic buoyancy can reinforce each other for the sub-surface toroidal flux bundles to occasionally break through the solar surface. The two sites where the field breaks through are seen as sunspots (Figure 6d-f).

It is also clear from Figure 6b that the toroidal field inside the sun, formed by the Omega Effect, will have opposite polarities in the two hemispheres. When the toroidal field breaks through the surface, then the bipolar sunspot pairs in the two hemispheres will have opposite polarities in accordance with Hales Law. Sunspots appear dark because the pressure and tension of the high magnetic field inside them suppress energy transport into it by convection.

Besides the necessary high shear at the tachocline to generate a strong toroidal field, the other reason why the tachocline is believed to be the source of high fields is that it retains the strength for a longer period than if the high field was generated in the CZ itself. From the CZ, the flux would rise to the surface too quickly owing to magnetic buoyancyin about a monthnot giving differential rotation enough time to act on it for the field to amplify by stretching. Only a strong field from the tachocline will not only be sufficiently buoyant to rise to surface and form sunspots but also allow enough time for the field amplification due to differential rotation to occur.

In 1993, studying the evolution of magnetic buoyancy on the flux tubes starting from the bottom of the CZ, Choudhuri and A. DSilva found that to reproduce the correct features of sunspotstheir emergence at the correct latitudes on the surface and observed tilts of the bipolar sunspots (Joys Law)a high initial toroidal magnetic field of at least 100,000 G was needed, as against the conventionally assumed value of 10,000 G. Indeed, in the IISc model only fields above 100,000 G at the tachocline have been considered. The observed scatter of tilts around the average given by Joys Law was attributed to the vigorous turbulence on the top layers of the CZ, which exerts a random force on the emerging flux tubes.

To complete the magnetic cycle, the dynamo model must also provide a mechanism for the toroidal field to be reconverted back into poloidal form in the sun. That is, to sustain the solar magnetic cycle, there has to be a continual conversion between poloidal and toroidal fields. Earlier, we saw how the decay of the tilted bipolar sunspots provided a mechanism for recreating a poloidal field from a toroidal field. The problem is to transport the poloidal field generated on the solar surface deep down to regenerate a strong toroidal field and maintain the cycle. Solar dynamo models differ in the manner in which this source poloidal field arises inside the sun and is driven to the tachocline or the bottom of the CZ for its conversion into toroidal field and thus maintain the magnetic cycle.

In non-flux transport dynamo models, including the early ones, the source poloidal field for generating the toroidal field is generated within the CZ by the Alpha Effect in a mechanism first described by Eugene Parker in 1955 on the basis of turbulence within the CZ. According to Parkers picture, helical turbulent motions caused by the Coriolis force in the rising plasma (just like cyclones in the earths atmosphere) would twist the toroidal field in the plasma to give rise to large-scale poloidal field (Figure 7a-c). This poloidal field is carried back by plumes of convection that overshoot the base of the CZ into the tachocline to complete the cycle.

Only if the initial toroidal field is strong enough to withstand this convective helical twisting and shredding in the CZ, will it rise through the CZ owing to magnetic buoyancy and emerge from the surface to form a sunspot pair as described earlier. However, Choudhuri and associates argued that for high toroidal fields (100,000 G and above as required to explain sunspot features), Parkers mechanism cannot work. In their FTDM model, the Babcock-Leighton mechanism on the surface is the only way to generate the source poloidal field.

Since there is no accumulation of matter at the poles following the poleward meridional drift of fluid, and the advected magnetic field as a result of the Babcock-Leighton mechanism, mass conservation would require a return meridional flow of the material from the poles towards the equator through the solar interior. Thus, like a conveyor belt, meridional flow is believed to transport the poloidal magnetic field first from the sunspot towards the poles and then from the poles towards the equator in the interior, but in the opposite direction.

As this sub-surface meridional flow cannot be easily observed, different models vary in the assumptions made regarding the meridional flow and how the meridional flow cells close. FTDMs, in particular, assume that this return flow and the cell closure occur deep down at the bottom of the CZ or at the tachocline thus providing the source poloidal field for the magnetic cycle (Figure 6g-i). The models also assume a vertical diffusion of the plasma and the magnetic field from the surface down through the CZ.

The FTDMs of both Gilman and Dikpati and the IISc group can reproduce the general features of solar cycles. With a maximum meridional flow speed of 20-24 m/s near the surface, which decreases with the depth of the CZ to about 1-2 m/s at the tachocline, the advection by meridional circulation in both the models takes 15 to 20 years. The solar cycle period in FTDMs is inversely related to the flow speed, and using this, the models are also able to correctly reproduce the observed value of 11 years.

However, the models differ in the assumed diffusivity of the surface poloidal field through the CZ. The IISc model uses 50 times higher diffusivity than the other model. The diffusion times in the two models are five years and 200 years respectively. This factor becomes critical when the two models are used to study solar cycle variabilities and predict future cycles on the basis of past data. As the IISc authors say, while in the high-diffusivity model, fluctuations spread all over the CZ in about five years, in the low-diffusivity case the fluctuations essentially remain frozen during the cycle period.

As mentioned earlier, not all solar cycles are identical in strength and period. Cycle variabilities can arise in models only by introducing fluctuations in the input data. The two models differ in this respect as well. In the IISc model, the scatter in the tilt angles caused by the random convective stresses on the rising flux tubes introduces an inherent randomness in the generated surface poloidal field. This has been assumed to be the major source of fluctuations in the dynamo process. Further, since diffusion takes only five years, the strength of the poloidal field at the end of the previous cycle will be correlated to the strength of the toroidal field at the tachocline in the next cycle (Figure 4).

Choudhuri and co., therefore, used the measured strength of polar magnetic field at the end of a cycle as the predictor variable for the cycle for their prediction of Cycle 24s amplitude of 80. Gilman and Dikpati, on the other hand, used the measured sunspot areas, which are related to the poloidal field strength, as the predictor variable and predicted the amplitude to be 150-180. As mentioned earlier, the huge difference in the diffusivity assumed was the determining factor. But these models have been criticised in general for the use of polar field strengths for prediction. Magnetic diffusion is very poorly understood, pointed out Steve Tobias, an expert in the field at Leeds University, U.K. We really dont know if high diffusivity is possible, he said.

However, in a recent work published in Physical Review Letters, Choudhuri and his student B.B. Karak have been able to explain the origin of the solar grand minima using their high-diffusivity model. That comes as an added boost to their model. For this, besides fluctuations arising from the irregularity in the poloidal field, they introduce another source of irregularity through fluctuations in the meridional circulation flow rate. Since in FTDMs flow speed is roughly inversely proportional to the cycle period, they use past data on the variations in the periods to obtain fluctuations in meridional circulation.

According to them, the solar dynamo can be driven into a grand minimum if the poloidal field at the end of a cycle and the meridional flow speed simultaneously fall to sufficiently low values.

Assuming a normal (bell curve) distribution for the two fluctuations, with their parameters fixed by past data of cycle periods and strengths, they estimate the probability for combined values of poloidal field and meridional circulation forcing the dynamo into grand minima. They find that over a period of 11,000 years, the number of grand minima would be around 24-30, very close to the figure obtained from isotope proxy data. Similarly, they find that the sun spends 10-15 per cent of the time in the grand minimum state, again close to the 17 per cent determined observationally.

The twin major successescorrect prediction of Cycle 24 strength and the explanation of the origin of solar grand minimawould suggest that the assumptions made in the IISc model are correct and that, in general, FTDM is the appropriate paradigm for solar dynamo theory. However, since a lot is unknown about the solar dynamo process, and many questions about assumptions made in FTDMs remain, Tobias cautions against such an inference.

I would warn you against confusing a successful prediction with a correct model, he says. These models are fraught with difficulties and have a number of ad hoc assumptions that cannot be justified, he adds. Most important among these is the assumption of deep meridional circulation for which there is no evidence yet. In fact, in a recent publication, Hathaway has concluded from observational data that the meridional circulation is rather shallow. The largest meridional circulation cell extends down only to a maximum depth of 35,000 km compared with the CZs depth of 200,000 km. If Hathaways inferences are correct, this spells the end for all FTDMs, says Tobias.

We do not take this result very seriously because of limited observations used, says Karak, Choudhuris co-author. But the authors do recognise that the most important issue to be resolved in the FTDMs is meridional circulation. This is a very important ingredient in this model but very poorly understood. We only have surface observations for about the last 20 years, and there is no data in the deeper parts of the CZ. Moreover, the theoretical understanding of meridional flow is very primitive because of the complex nature of the turbulent CZ. In fact, I am interested whether shallow meridional circulation can recover a regular solar cycle in our model, adds Karak.

The other important assumption is that the source poloidal field is generated only at the surface, with the whole of the CZ playing little role at all. This seems very unlikely to me, says Tobias. Simulations of turbulent convection indicate that only the very strongest fields make it to the solar surface. What happens to the rest of the [toroidal] flux? It has to be reprocessed and brought back to the tachocline by convective motions.

The mechanism of stochastic modulation proposed for grand minima requires rather special circumstances to occur, points out Tobias. The other mechanism, very strangely not discussed by the authors, is that the sun is in a non-linear state and it is the non-linear interactions that lead to modulation. This seems natural to me.

These FTDMs are the so-called kinematic dynamo models, where the rotation and meridional flow is put by hand and back-reaction of the magnetic field on these flows is ignored, points out H.M. Antia of the Tata Institute of Fundamental Research. Back-reaction is a dynamic effect, which is the force of the generated magnetic field on the current flows, the Lorentz force. This varies as the square of the magnetic field, and for high fields, as is the case here, this non-linear back-reaction cannot be ignored. Indeed, it is known that flows driven by the non-linear Lorentz force lead to torsional oscillations, which modulate the differential rotation. Detailed rotation profile in the interior provided by helioseismology has also not been used by them. They use some approximation, Antia adds. Dynamo theory is a major unsolved problem, and I dont think we are anywhere close to a breakthrough in this area, says Antia, notwithstanding the seemingly remarkable results of the IISc solar dynamo model.