Solar Cycle Prediction

1.1 The sunspot number

In addition to introducing the relative sunspot number, Wolf (1861) also used earlier observational records available to him to reconstruct its monthly mean values since 1749. In this way, he reconstructed 11-year sunspot cycles back to that date, introducing their still universally used numbering. (In a later work he also determined annual mean values for each calendar year going back to 1700.)

In 1981, the observatory responsible for the official determination of the sunspot number changed from Zürich to the Royal Observatory of Belgium in Brussels. The website of the SIDC (originally Sunspot Index Data Center, recently renamed Solar Influences Data Analysis Center), http://sidc.oma.be, is now the most authoritative source of archive sunspot number data. But it has to be kept in mind that the sunspot number is also regularly determined by other institutions: these variants are informally known as the American sunspot number (collected by AAVSO and available from the National Geophysical Data Center, http://www.ngdc.noaa.gov/ngdc.html) and the Kislovodsk Sunspot Number (available from the web page of the Pulkovo Observatory, http://www.gao.spb.ru). Cycle amplitudes determined by these other centers may differ by up to 6 – 7% from the SIDC values, NOAA numbers being consistently lower, while Kislovodsk numbers show no such systematic trend.

These significant disagreements between determinations of R _W by various observatories and observers are even more pronounced in the case of historical data, especially prior to the mid-19th century. In particular, the controversial suggestion that a whole solar cycle may have been missed in the official sunspot number series at the end of the 18th century is taken by some as glaring evidence for the unreliability of early observations. Note, however, that independently of whether the claim for a missing cycle is well founded or not, there is clear evidence that this controversy is mostly due to the very atypical behaviour of the Sun itself in the given period of time, rather than to the low quality and coverage of contemporary observations. These issues will be discussed further in Section 3.2.2.

In what follows, R _max ⁽ⁿ⁾ and R _min ⁽ⁿ⁾ will refer to the maximum and minimum value of R in cycle n (the minimum being the one that starts the cycle). Similarly, t _max ⁽ⁿ⁾ and t _min ⁽ⁿ⁾ will denote the epochs when R takes these extrema.

1.2 Other indicators of solar activity

Reconstructions of R prior to the early 19th century are increasingly uncertain. In order to tackle problems related to sporadic and often unreliable observations, Hoyt and Schatten (1998) introduced the Group Sunspot Number (GSN) as an alternative indicator of solar activity. In contrast to R _W , the GSN only relies on counts of sunspot groups as a more robust indicator, disregarding the number of spots in each group. Furthermore, while R _W is determined for any given day from a single observer’s measurements (a hierarchy of secondary observers is defined for the case if data from the primary observer were unavailable), the GSN uses a weighted average of all observations available for a given day. The GSN series has been reproduced for the whole period 1611 – 1998 (Figure 1) and it is generally agreed that for the period 1611 – 1818 it is a more reliable reconstruction of solar activity than the relative sunspot number. Yet there have been relatively few attempts to date to use this data series for solar cycle prediction. One factor in this could be the lack of regular updates of the GSN series, i.e., the unavailability of precise GSN values for the past decade.

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Instead of the sunspot number, the total area of all spots observed on the solar disk might seem to be a less arbitrary measure of solar activity. However, these data have been available since 1874 only, covering a much shorter period of time than the sunspot number data. In addition, the determination of sunspot areas, especially farther from disk center, is not as trivial as it may seem, resulting in significant random and systematic errors in the area determinations. Area measurements performed in two different observatories often show discrepancies reaching ∼ 30% for smaller spots (cf. the figure and discussion in Appendix A of Petrovay et al., 1999).

A number of other direct indicators of solar activity have become available from the 20th century. These include, e.g., various plage indices or the 10.7 cm solar radio flux — the latter is considered a particularly good and simple to measure indicator of global activity (see Figure 2). As, however, these data sets only cover a few solar cycles, their impact on solar cycle prediction has been minimal.

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Of more importance are proxy indicators such as geomagnetic indices (the most widely used of which is the aa index), the occurrence frequency of aurorae or the abundances of cosmogenic radionuclides such as ¹⁴C and ¹⁰Be. For solar cycle prediction uses such data sets need to have a sufficiently high temporal resolution to reflect individual 11-year cycles. For the geomagnetic indices such data have been available since 1868, while an annual 10Be series covering 600 years has been published very recently by Berggren et al. (2009). Attempts have been made to reconstruct the epochs and even amplitudes of solar maxima during the past two millennia from oriental naked eye sunspot records and from auroral observations (Stephenson and Wolfendale, 1988; Nagovitsyn, 1997), but these reconstructions are currently subject to too many uncertainties to serve as a basis for predictions. Isotopic data with lower temporal resolution are now available for up to 50 000 years in the past; while such data do not show individual Schwabe cycles, they are still useful for the study of long term variations in cycle amplitude. Inferring solar activity parameters from such proxy data is generally not straightforward.

1.3 The solar cycle and its variation

The series of R values determined as described in Section 1.1 is plotted in Figure 3. It is evident that the sunspot cycle is rather irregular. The mean length of a cycle (defined as lasting from minimum to minimum) is 11.02 years (median 10.7 years), with a standard deviation of 1.2 years. The mean amplitude is 113 (median 115), with a standard deviation of 40. It is this wide variation that makes the prediction of the next cycle maximum such an interesting and vexing issue.

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It should be noted that the period covered by the relative sunspot number record includes an extended interval of atypically strong activity, the so called Modern Maximum (see below), cycles 17 – 23. Excluding these cycles from the averaging, the mean, and median values of the cycle amplitude are very close to 100, with a standard deviation of 35. The mean and median cycle length then become 11.1 and 11.2 years, respectively, with a standard deviation of 1.3 years.

1.4 Secular activity variations

The Gleissberg filtered sunspot number series is plotted in Figure 4. One long-term trend is an overall secular increase of solar activity, the last six or seven cycles being unusually strong. (Four of them are markedly stronger than average and none is weaker than average.) This period of elevated sunspot activity level from the mid-20th century is known as the “Modern Maximum”. On the other hand, cycles 5, 6, and 7 are unusually weak, forming the so-called “Dalton Minimum”. Finally, the rather long series of moderately weak cycles 12 – 16 is occasionally referred to as the “Gleissberg Minimum” — but note that most of these cycles are less than 1 σ below the long-term average.

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While the Dalton and Gleissberg minima are but local minima in the ever changing Gleissberg filtered SSN series, the conspicuous lack of sunspots in the period 1640 – 1705, known as the Maunder Minimum (Figure 1) quite obviously represents a qualitatively different state of solar activity. Such extended periods of high and low activity are usually referred to as grand maxima and grand minima. Clearly, in comparison with the Maunder Minimum, the Dalton Minimum could only be called a “semi-grand minimum”, while for the Gleissberg Minimum even that adjective is undeserved.

A number of possibilities have been proposed to explain the phenomenon of grand minima and maxima, including chaotic behaviour of the nonlinear solar dynamo (Weiss et al., 1984), stochastic fluctuations in dynamo parameters (Moss et al., 2008; Usoskin et al., 2009b) or a bimodal dynamo with stochastically induced alternation between two stationary states (Petrovay, 2007).

The analysis of long-term proxy data, extending over several millennia further showed that there exist systematic long-term statistical trends and periods such as the so called secular and supersecular cycles (see Section 3.2).

1.4.2 Waldmeier effect and amplitude-frequency correlation

“Greater activity on the Sun goes with shorter periods, and less with longer periods. I believe this law to be one of the most important relations among the Solar actions yet discovered.”

(Wolf, 1861)

It is apparent from Figure 3 that the profile of sunspot cycles is asymmetrical, the rise being steeper than the decay. Solar activity maxima occur 3 to 4 years after the minimum, while it takes another 7 – 8 years to reach the next minimum. It can also be noticed that the degree of this asymmetry correlates with the amplitude of the cycle: to be more specific, the length of the rise phase anticorrelates with the maximal value of R (Figure 5), while the length of the decay phase shows weak or no such correlation.

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Historically, the relation was first formulated by Waldmeier (1935) as an inverse correlation between the rise time and the cycle amplitude; however, as shown by Tritakis (1982), the total rise time is a weak (inverse logarithmic) function of the rise rate, so this representation makes the correlation appear less robust. (Indeed, when formulated with the rise time it is not even present in some activity indicators, such as sunspot areas - cf. Dikpati et al., 2008b.) As pointed out by Cameron and Schüssler (2008), the weak link between rise time and slope is due to the fact that in steeper rising cycles the minimum will occur earlier, thus partially compensating for the shortening due to a higher rise rate. The effect is indeed more clearly seen when the rate of the rise is used instead of the rise time (Lantos, 2000; Cameron and Schüssler, 2008). The observed correlation between rise rate and maximum cycle amplitude is approximately linear, good (correlation coefficient r ∼ 0:85), and quite robust, being present in various activity indices.

Nevertheless, when coupled with the nearly nonexistent correlation between the decay time and the cycle amplitude, even the weaker link between the rise time and the maximum amplitude is sufficient to forge a weak inverse correlation between the total cycle length and the cycle amplitude (Figure 5). This inverse relationship was first noticed by Wolf (1861).

A stronger inverse correlation was found between the cycle amplitude and the length of the previous cycle by Hathaway et al. (1994). This correlation is also readily explained as a consequence of the Waldmeier effect, as demonstrated in a simple model by Cameron and Schüssler (2007). Note that in a more detailed study Solanki et al. (2002) find that the correlation coefficient of this relationship has steadily decreased during the course of the historical sunspot number record, while the correlation between cycle amplitude and the length of the third preceding cycle has steadily increased. The physical significance (if any) of this latter result is unclear.

In what follows, the relationships found by Wolf (1861), Hathaway et al. (1994), and Solanki et al. (2002), discussed above, will be referred to as “R_max - t_cycle;n correlations” with n = 0, -1 or -3, respectively.

Modern time series analysis methods offer several ways to define an instantaneous frequency f in a quasiperiodic series. One simple approach was discussed in the context of Bracewell’s transform, Equation (3), above. Mininni et al. (2000) discuss several more sophisticated methods to do this, concluding that Gábor’s analytic signal approach yields the best performance. This technique was first applied to the sunspot record by Paluš and Novotná (1999), who found a significant long term correlation between the smoothed instantaneous frequency and amplitude of the signal. On time scales shorter than the cycle length, however, the frequency-amplitude correlation has not been convincingly proven, and the fact that the correlation coefficient is close to the one reported in the right hand panel of Figure 5 indicates that all the fashionable gadgetry of nonlinear dynamics could achieve was to recover the effect already known to Wolf. It is clear from this that the “frequency-amplitude correlation” is but a secondary consequence of the Waldmeier effect.

On the left hand panel of Figure 5, within the band of correlation the points seem to be sitting neatly on two parallel strings. Any number of faint hearted researchers would dismiss this as a coincidence or as another manifestation of the “Martian canal effect”. But Kuklin (1986) boldly speculated that the phenomenon may be real. Fair enough, cycles 22 and 23 dutifully took their place on the lower string even after the publication of Kuklin’s work. This speculation was supported with further evidence by Nagovitsyn (1997) who offered a physical explanation in terms of the amplitude-frequency diagram of a forced nonlinear oscillator (cf. Section 4.5).

Indeed, an anticorrelation between cycle length and amplitude is characteristic of a class of stochastically forced nonlinear oscillators and it may also be reproduced by introducing a stochastic forcing in dynamo models (Stix, 1972; Ossendrijver et al., 1996; Charbonneau and Dikpati, 2000). In some such models the characteristic asymmetric profile of the cycle is also well reproduced (Mininni et al., 2000, 2002). The predicted amplitude-frequency relation has the form

$$\log R_{\max }^{(n)} = C_1 + C_2 f.$$

(5)

Nonlinear dynamo models including some form of α-quenching also have the potential to reproduce the effects described by Wolf and Waldmeier without recourse to stochastic driving. In a dynamo with a Kleeorin-Ruzmaikin type feedback on α, Kitiashvili and Kosovichev (2009) are able to qualitatively reproduce the Waldmeier effect. Assuming that the sunspot number is related to the toroidal field strength according to the Bracewell transform, Equation (3), they find a strong link between rise time and amplitude, while the correlations with fall time and cycle length are much weaker, just as the observations suggest. They also find that the form of the growth time- amplitude relationship differs in the regular (multiperiodic) and chaotic regimes. In the regular regime the plotted relationship suggests

$$R_{\max }^{(n)} = C_1 - C_2 \left( {t_{\max }^{(n)} - t_{\min }^{(n)} } \right),$$

(6)

while in the chaotic case

$$R_{\max }^{(n)} \propto \left[ {1/\left( {t_{\max }^{(n)} - t_{\min }^{(n)} } \right)} \right].$$

(7)

Note that based on the actual sunspot number series Waldmeier originally proposed

$$\log R_{\max }^{(n)} = C_1 - C_2 \left( {t_{\max }^{(n)} - t_{\min }^{(n)} } \right),$$

(8)

while according to Dmitrieva et al. (2000) the relation takes the form

$$\log R_{\max }^{(n)} \propto \left[ {1/\left( {t_{\max }^{(n)} - t_{\min }^{(n)} } \right)} \right].$$

(9)

At first glance, these logarithmic empirical relationships seem to be more compatible with the relation (5) predicted by the stochastic models. These, on the other hand, do not actually reproduce the Waldmeier effect, just a general asymmetric profile and an amplitude-frequency correlation. At the same time, inspection of the the left hand panel in Figure 5 shows that the data is actually not incompatible with a linear or inverse rise time-amplitude relation, especially if the anomalous cycle 19 is ignored as an outlier. (Indeed, a logarithmic representation is found not to improve the correlation coefficient - its only advantage is that cycle 19 ceases to be an outlier.) All this indicates that nonlinear dynamo models may have the potential to provide a satisfactory quantitative explanation of the Waldmeier effect, but more extensive comparisons will need to be done, using various models and various representations of the relation.

2.1 Cycle parameters as precursors and the Waldmeier effect

The simplest weather forecast method is saying that “tomorrow the weather will be just like today” (works in about 2/3 of the cases). Similarly, a simple approach of sunspot cycle prediction is correlating the amplitudes of consecutive cycles. There is indeed a marginal correlation, but the correlation coefficient is quite low (0.35). The existence of the correlation is related to secular variations in solar activity, while its weakness is due to the significant cycle-to-cycle variations.

Cameron and Schüssler (2007) point out that the activity level three years before the minimum is an even better predictor of the next maximum. Indeed, playing with the value of time shift we find that the best correlation coefficient corresponds to a time shift of 2.5 years, as shown in the right hand panel of Figure 6 (but this may depend on the particular time period considered, so we will refer to this method in Table 1 as “minimax3” for brevity). The linear regression is

$$R_{\max } = 41.9 + 1.68R(t_{\min } - 2.5).$$

(11)

For cycle 24 the value of the predictor is 16.3, so this indicates an amplitude of 69, suggesting that the upcoming cycle may be comparable in strength to those during the Gleissberg minimum at the turn of the 19th and 20th centuries.

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As the epoch of the minimum of R cannot be known with certainty until about a year after the minimum, the practical use of these methods is rather limited: a prediction will only become available 2 – 3 years before the maximum, and even then with the rather low reliability reflected in the correlation coefficients quoted above. In addition, as convincingly demonstrated by Cameron and Schüssler (2007) in a Monte Carlo simulation, these methods do not constitute real cycle-to-cycle prediction in the physical sense: instead, they are due to a combination of the overlap of solar cycles with the Waldmeier effect. As stronger cycles are characterized by a steeper rise phase, the minimum before such cycles will take place earlier, when the activity from the previous cycle has not yet reached very low levels.

The same overlap readily explains the R_max - t_cycle;n correlations discussed in Section 1.3.3. These relationships may also be used for solar cycle prediction purposes (e.g., Kane, 2008) but they lack robustness. For cycle 24 the R_max – t_cycle;−1 correlation, as formulated by Hathaway (2010b) predicts R_max = 80 while the methods used by Solanki et al. (2002) yield values ranging from 86 to about 110, depending on the relative weights of t_cycle;−1 and t_cycle;−3. The forecast is not only sensitive to the value of n used but also to the data set (relative or group sunspot numbers) (Vaquero and Trigo, 2008).

2.2 Polar precursors

Direct measurements of the magnetic field in the polar areas of the Sun have been available from Wilcox Observatory since 1976 (Svalgaard et al., 1978; Hoeksema, 1995). Even before a significant amount of data had been available for statistical analysis, solely on the basis of the Babcock-Leighton scenario of the origin of the solar cycle, Schatten et al. (1978) suggested that the polar field measurements may be used to predict the amplitude of the next solar cycle. Data collected in the four subsequent solar cycles have indeed confirmed this suggestion. As it was originally motivated by theoretical considerations, this polar field precursor method might also be a considered a model-based prediction technique. As, however, no particular detailed mathematical model is underlying the method, numerical predictions must still be based on empirical correlations - hence our categorization of this technique as a precursor method.

The shortness of the available direct measurement series represents a difficulty when it comes to finding empirical correlations to solar activity. This problem can to some extent be circumvented by the use of proxy data. For instance, Obridko and Shelting (2008) use H_α synoptic maps to reconstruct the polar field strength at the source surface back to 1915. Spherical harmonic expansions of global photospheric magnetic measurements can also be used to deduce the field strength near the poles. The use of such proxy techniques permits a forecast with a sufficiently restricted error bar to be made, despite the shortness of the direct polar field data set.

The polar fields reach their maximal amplitude near minima of the sunspot cycle. In its most commonly used form, the polar field precursor method employs the value of the polar magnetic field strength (typically, the absolute value of the mean field strength poleward of 55° latitudes, averaged for the two hemispheres) at the time of sunspot minimum. It is indeed remarkable that despite the very limited available experience, forecasts using the polar field method have proven to be consistently in the right range for cycles 21, 22, and 23 (Schatten and Sofia, 1987; Schatten et al., 1996).

By virtue of the definition (2), the time of the minimum of R cannot be known earlier than 6 months after the minimum - indeed, to make sure that the perceived minimum was more than just a local dip in R, at least a year or so needs to elapse. This would suggest that the predictive value of polar field measurements is limited, the prediction becoming available 2 – 3 years before the upcoming maximum only.

To remedy this situation, Schatten and Pesnell (1993) introduced a new activity index, the “Solar Dynamo Amplitude” (SoDA) index, combining the polar field strength with a traditional activity indicator (the 10.7 cm radio flux F10.7). Around minimum, SoDA is basically proportional to the polar precursor and its value yields the prediction for F10.7 at the next maximum; however, it was constructed so that its 11-year modulation is minimized, so theoretically it should be rather stable, making predictions possible well before the minimum. That is the theory, anyway — in reality, SoDA based forecasts made more than 2 – 3 years before the minimum usually proved unreliable. It is then questionable to what extent SoDA improves the prediction skill of the polar precursor, to which it is more or less equivalent in those late phases of the solar cycle when forecasts start to become reliable.

Fortunately, however, the maxima of the polar field curves are often rather flat (see Figure 7), so approximate forecasts are feasible several years before the actual minimum. Using the current, rather flat and low maximum in polar field strength, Svalgaard et al. (2005) have been able to predict a relatively weak cycle 24 (peak R value 75 ± 8) as early as 4 years before the sunspot minimum took place in December 2008! Such an early prediction is not always possible: early polar field predictions of cycles 22 and 23 had to be corrected later and only forecasts made shortly before the actual minimum did finally converge. Nevertheless, even the moderate success rate of such early predictions seems to indicate that the suggested physical link between the precursor and the cycle amplitude is real.

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In addition to their above mentioned use in reconstructing the polar field strength, various proxies or alternative indices of the global solar magnetic field during the activity minimum may also be used directly as activity cycle precursors. H_α synoptic charts are now available from various observatories from as early as 1870. As H_α filaments lie on the magnetic neutral lines, these maps can be used to reconstruct the overall topology, if not the detailed map, of the large-scale solar magnetic field. Tlatov (2009) has shown that several indices of the polar magnetic field during the activity minimum, determined from these charts, correlate well with the amplitude of the incipient cycle.

High resolution Hinode observations have now demonstrated that the polar magnetic field has a strongly intermittent structure, being concentrated in intense unipolar tubes that coincide with polar faculae (Tsuneta et al., 2008). The number of polar faculae should then also be a plausible proxy of the polar magnetic field strength and a good precursor of the incipient solar cycle around the minimum. This conclusion was indeed confirmed by Li et al. (2002) and, more recently, by Tlatov (2009).

These methods offer a prediction over a time span of 3 – 4 years, comparable to the rise time of the next cycle. A significantly earlier prediction possibility was, however, suggested by Makarov et al. (1989) and Makarov and Makarova (1996) based on the number of polar faculae observed at Kislovodsk, which was found to predict the next sunspot cycle with a time lag of 5 – 6 years; even short term annual variations or “surges” of sunspot activity were claimed to be discernible in the polar facular record. This surprising result may be partly due to the fact that Makarov et al. considered all faculae poleward of 50° latitude. Bona fide polar faculae, seen on Hinode images to be knots of the unipolar field around the poles, are limited to higher latitudes, so the wider sample may consist of a mix of such “real” polar faculae and small bipolar ephemeral active regions. These latter are known to obey an extended butterfly diagram, as recently confirmed by Tlatov et al. (2010): the first bipoles of the new cycle appear at higher latitudes about 4 years after the activity maximum. It is not impossible that these early ephemeral active regions may be used for prediction purposes (cf. also Badalyan et al., 2001); but whether or not the result of Makarov et al. (1989) may be attributed to this is doubtful, as Li et al. (2002) find that even using all polar faculae poleward of 50° from the Mitaka data base, the best autocorrelation still results with a time shift of about 4 years only.

2.3 Geomagnetic and interplanetary precursors

The component (a) of the geomagnetic variations actually follows sunspot activity with a variable time delay. Thus a geomagnetic precursor based on features of the cycle dominated by this component has relatively little practical utility. This would seem to be the case, e.g., with the forecast method first proposed by Ohl (1966), who noticed that the minimum amplitudes of the smoothed geomagnetic aa index are correlated to the amplitude of the next sunspot cycle (see also Du et al., 2009).

An indication that the total geomagnetic activity, resulting from both mechanisms does contain useful information on the expected amplitude of the next solar cycle was given by Thompson (1993), who found that the total number of disturbed days in the geomagnetic field in cycle n is related to the sum of the amplitudes of cycles n and n+1 (see also Dabas et al., 2008).

A method for separating component (b) was proposed by Feynman (1982) who correlated the annual aa index with the annual mean sunspot number and found a linear relationship between R and the minimal value of aa for years with such R values. She interpreted this linear relationship as representing the component (a) discussed above, while the amount by which aa in a given year actually exceeds the value predicted by the linear relation would be the contribution of type (b) (the “interplanetary component”). The interplanetary component usually peaks well ahead of the sunspot minimum and the amplitude of the peak seemed to be a good predictor of the next sunspot maximum. However, it is to be noted that the assumption that the “surplus” contribution to aa originates from the interplanetary component only is likely to be erroneous, especially for stronger cycles. It is known that the number of large solar eruptions shows no unique relation to R: in particular, for R > 100 their frequency may vary by a factor of 3 (see Figure 15 in Hathaway, 2010b), so in some years they may well yield a contribution to aa that greatly exceeds the minimum contribution. A case in point was the “Halloween events” of 2003, that very likely resulted in a large false contribution to the derived “interplanetary” aa index (Hathaway, 2010a). As a result, the geomagnetic precursor method based on the separation of the interplanetary component predicts an unusually strong cycle 24 (R _m ∼ 150), in contrast to most other methods, including Ohl’s method and the polar field precursor, which suggest a weaker than average cycle (R _m ∼ 80 – 90).

The actual run of cycle 24 will be certainly most revealing from the point of view of these complex interrelationships.

The open magnetic flux can also be derived from the extrapolation of solar magnetograms using a potential field source surface model. The magnetograms applied for this purpose may be actual observations or the output from surface flux transport models, using the sunspot distribution (butterfly diagram) and the meridional flow as input. Such models indicate that the observed latitude independence of the interplanetary field strength (“split monopole” structure) is only reproduced if the source surface is far enough (> 10R_⊙) and the potential field model is modified to take into account the heliospheric current sheet (current sheet source surface model, Schüssler and Baumann, 2006; Jiang et al., 2010a). The extrapolations are generally found to agree well with in situ measurements where these are available.

2.4 Flows in the photosphere

In the currently widely popular flux transport dynamo models the strong polar fields prevalent around sunspot minimum are formed by the advection of following polarity flux from active regions by the poleward meridional flow. Changes in this flow may thus influence the would-be polar fields and thereby may serve as precursors of the upcoming cycle.

Such changes, on the other hand, are also associated with the normal course of the solar activity cycle, the overall flow at mid-latitudes being slower before and during maxima and faster during the decay phase. Therefore, it is just the cycle-to-cycle variation in this normal pattern that may be associated with the activity variations between cycles. In this respect it is of interest to note that the poleward flow in the late phases of cycle 23 seems to have had an excess speed relative to the previous cycle (Hathaway and Rightmire, 2010). If this were a latitude-independent amplitude modulation of the flow, then most flux transport dynamo models would predict a stronger than average polar field at the minimum, contrary to observations. On the other hand, in the surface flux transport model of Wang et al. (2009) an increased poleward flow actually results in weaker polar fields, as it lets less leading polarity flux to diffuse across the equator and cancel there. As the recent analysis by Muñoz-Jaramillo et al. (2010) has shown, the discrepancy resulted from the neglect of leading polarity flux in the Babcock-Leighton source term in flux transport dynamo models, and it can be remedied by substituting a pair of opposite polarity flux rings as source term instead of the α-term. With this correction, 2D flux transport and surface flux transport models agree in predicting a weaker polar field for faster meridional flow.

It is known from helioseismology that meridional flow speed fluctuations follow a characteristic latitudinal pattern associated with torsional oscillations and the butterfly diagram, consisting of a pair of axisymmetric bands of latitudinal flows converging towards the activity belts, migrating towards the equator, and accompanied by similar high-latitude poleward branches. This suggests interpreting the unusual meridional flow speeds observed during cycle 23 as an increased amplitude of this migrating modulation, rather than a change in the large-scale flow speed (Cameron and Schüssler, 2010). In this case, the flows converging on the activity belts tend to inhibit the transport of following polarities to the poles, again resulting in a lower than usual polar field, as observed (Jiang et al., 2010b; note, however, that Švanda et al., 2007 find no change in the flux transport in areas with increased flows). It is interesting to note that the torsional oscillation pattern, and thus presumably the associated meridional flow modulation pattern, was shown to be fairly well reproduced by a microquenching mechanism due to magnetic flux emerging in the active belts (Petrovay and Forgács-Dajka, 2002). Observational support for this notion has been provided by the seismic detection of locally increased flow modulation near active regions (Švanda et al., 2007). This suggests that stronger cycles may be associated with a stronger modulation pattern, introducing a nonlinearity into the flux transport dynamo model, as suggested by Jiang et al. (2010b).

In addition to a variation in the amplitude of migrating flow modulations, their migration speed may also influence the cycle. Howe et al. (2009) point out that in the current minimum the equatorward drift of the torsional oscillation shear belt corresponding to the active latitude of the cycle has been slower than in the previous minimum. They suggest that this slowing may explain the belated start of cycle 24.

3.1 Linear regression

Linear regression techniques have been widely used for solar activity prediction during the course of an ongoing cycle. Their application for cycle-to-cycle prediction has been less common and successful (Lomb and Andersen, 1980; Box et al., 2008; Wei, 2005).

Brajša et al. (2009) applied an ARMA model to the series of annual values of R. A successful fit was found for p = 6, q = 6. Using this fit, the next solar maximum was predicted to take place around 2012.0 with an amplitude 90 ± 27, and the following minimum occurring in 2017.

Instead of applying an autoregression model directly to SSN data, Hiremath (2008) applied it to a forced and damped harmonic oscillator model claimed to well represent the SSN series. This resulted in a predicted amplitude of 110 ± 10 for solar cycle 24, with the cycle starting in mid-2008 and lasting 9.34 years.

3.2 Spectral methods

Spectral analysis of the sunspot number record is used for prediction under the assumption that the main reason of variability in the solar cycle is a long-term modulation due to one or more periods.

The usual approach to the problem is the purely formal one of representing the sunspot record with the superposition of eigenfunctions forming an orthogonal basis. From a technical point of view, spectral methods are a complicated form of linear regression. The analysis can be performed by any of the widely used means of harmonic analysis:

(1) Least squares (LS) frequency analysis (sometimes called “Lomb-Scargle periodogram”) consists in finding by trial and error the best fitting sine curve to the data using the least squares method, subtracting it (“prewhitening”), then repeating the procedure until the residuals become indistinguishable from white noise. The first serious attempt at sunspot cycle prediction, due to Kimura (1913), belonged to this group. The analysis resulted in a large number of peaks with dubious physical significance. The prediction given for the upcoming cycle 15 failed, the forecasted amplitude being ∼ 60 while the cycle actually peaked at 105. However, it is interesting to note that Kimura correctly predicted the long term strengthening of solar activity during the first half of the 20th century! LS frequency analysis on sunspot data was also performed by Lomb and Andersen (1980), with similar results for the spectrum.

(2) Fourier analysis is probably the most commonly used method of spectral decomposition in science. It has been applied to sunspot data from the beginning of the 20th century (Turner, 1913a,b; Michelson, 1913). Vitinsky (1973) judges Fourier-based forecasts even less reliable than LS periodogram methods. Indeed, for instance Cole (1973) predicted cycle 21 to have a peak amplitude of 60, while the real value proved to be nearly twice that.

(3) The maximum entropy method (MEM) relies on the Wiener-Khinchin theorem that the power spectrum is the Fourier transform of the autocorrelation function. Calculating the autocorrelation of a time series for M ≪ N points and extrapolating it further in time in a particular way to ensure maximal entropy can yield a spectrum that extends to arbitrarily low frequencies despite the shortness of the data segment considered, and also has the property of being able to reproduce sharp spectral features (if such are present in the data in the first place). A good description of the method is given by Ables (1974), accompanied with some propaganda for it - see Press et al. (1992) for a more balanced account of its pros and cons. The use of MEM for sunspot number prediction was pioneered by Currie (1973). Using maximum entropy method combined with multiple regression analysis (MRA) to estimate the amplitudes and phases, Kane (2007) arrived at a prediction of 80 to 101 for the maximum amplitude of cycle 24. It should be noted that the same method yielded a prediction (Kane, 1999) for cycle 23 that was far off the mark.

(4) Singular spectrum analysis (SSA) is a relatively novel method for the orthogonal decomposition of a time series. While in the methods discussed above the base was fixed (the trigonometric functions), SSA allows for the identification of a set of othogonal eigenfunctions that are most suitable for the problem. This is done by a principal component analysis of the covariance matrix r _ik = 〈R _i R_i+k〉. SSA was first applied to the sunspot record by Rangarajan (1998) who only used this method for pre-filtering before the application of MEM. Loskutov et al. (2001) who also give a good description of the method, already made a prediction for cycle 24: a peak amplitude of 117. More recently, the forecast has been corrected slightly downwards to 106 (Kuzanyan et al., 2008).

In practice, all three effects suggested above may play some part. The first mentioned effect, time dependence, may in fact be studied within the framework of spectral analysis. MEM and SSA are intrinsically capable of detecting or representing time dependence in the spectrum, while LS and Fourier analysis can study time dependence by sliding an appropriate data window across the period covered by observations. If the window is Gaussian with a width proportional to the frequency we arrive at the popular wavelet analysis. This method was applied to the sunspot number series by Ochadlick Jr et al. (1993), Vigouroux and Delachie (1994), Frick et al. (1997), Fligge et al. (1999), and Li et al. (2005) who could confirm the existence and slight variation of the 11-year cycle and the Gleissberg-cycle. Recently, Kolláth and Oláh (2009) called attention to a variety of other generalized time dependent spectral analysis methods, of which the pseudo-Wigner transform yields especially clear details (see Figure 9). The time varying character of the basic periods makes it difficult to use these results for prediction purposes but they are able to shed some light on the variation as well as the presistent or intermittent nature of the periods determining solar activity.

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In summary, it is fair to say that forecasts based on harmonic analysis are notoriously unreliable. The secular variation of the basic periods, obeying as yet unknown rules, would render harmonic analysis practically useless for the prediction of solar cycles even if solar activity could indeed be described by a superposition of periodic functions. Although they may be potentially useful for very long term prediction (on centennial scales), when it comes to cycle-to-cycle forecasts the best we can hope from spectral studies is apparently an indirect contribution, by constraining dynamo models with the inambiguously detected periodicities.

In what remains from this subsection, we briefly review what these apparently physically real periods are and what impact they may have on solar cycle prediction.

3.2.1 The 11-year cycle and its harmonics

As an example of the period spectrum obtained by these methods, in Figure 8 we present the FFT based power spectrum estimate of the smoothed sunspot number record. Three main features are immediately noticed:

• The dominant 11-year peak, with its sidelobes and its 5.5-year harmonic.

• The 22-year subharmonic, representing the even-odd rule.

• The significant power present at periods longer than 50 years, associated with the Gleissberg cycle.

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The dominant peak in the power spectrum is at ∼ 11 years. Significant power is also present at the first harmonic of this period, at 5.5 years. This is hardly surprising as the sunspot number cycles, as presented in Figure 3, have a markedly asymmetrical profile. It is a characteristic of Fourier decomposition that in any periodic series of cycles where the profiles of individual cycles are non-sinusoidal, all harmonics of the base period will appear in the spectrum.

Indeed, were it not for the 13-month smoothing, higher harmonics could also be expected to appear in the power spectrum. It has been proposed (Krivova and Solanki, 2002) that these harmonics are detected in the sunspot record and that they may be related to the periodicities of ∼ 1.3 years intermittently observed in solar wind speed (Richardson et al., 1994; Paularena et al., 1995; Szabo et al., 1995; Mursula and Zieger, 2000; Lockwood, 2001) and in the internal rotation velocity of the Sun (Howe, 2009, Sect. 10.1). An analoguous intermittent 2.5 year variation in the solar neutrino flux (Shirai, 2004) may also belong to this group of phenomena. It may be worth noting that, from the other end of the period spectrum, the 154-day Rieger period in solar flare occurrence (Rieger et al., 1984; Bai and Cliver, 1990) has also been tentatively linked to the 1.3-year periodicity. Unusually strong excitation of such high harmonics of the Schwabe cycle may possibly be explained by excitation due to unstable Rossby waves in the tachocline (Zaqarashvili et al., 2010).

The 11-year peak in the power spectrum has substantial width, related to the rather wide variation in cycle lengths in the range 9 – 13 years. Yet Figure 8 seems to suggest the presence of a well detached second peak in the spectrum at a period of ∼ 14 years. The presence of a distinct peak at the first harmonic and even at the subharmonic of this period seems to support its reality. Indeed, peaks at around 14 and 7 years were already found by other researchers (e.g., Kimura, 1913; Currie, 1973) who suggested that these may be real secondary periods of sunspot activity.

The situation is, however, more prosaic. Constraining the time interval considered to data more recent than 1850, from which time the sunspot number series is considered to be more reliable, the 14.5-year secondary peak and its harmonics completely disappear. On the other hand, the power spectrum for the years 1783 – 1835 indicates that the appearance of the 14.5-year secondary peak in the complete series is almost entirely due to the strong predominance of this period (and its harmonic) in that interval. This interval covers the unusually long cycle 4 and the Dalton minimum, consisting of three consecutive unusually weak cycles, when the “normal” 11-year mode of operation was completely suppressed.

As pointed out by Petrovay (2010), this probably does not imply that the Sun was operating in a different mode during the Dalton minimum, the cycle length being 14.5 years instead of the usual 11 years. Instead, the effect may be explained by the well known inverse correlation between cycle length and amplitude, which in turn is the consequence of the strong inverse correlation between rise rate and cycle amplitude (Waldmeier effect), combined with a much weaker or nonexistent correlation between decay rate and amplitude (see Section 1.3.3). The cycles around the Dalton minimum, then, seem to lie at the low amplitude (or long period) end of a continuum representing the well known cycle length-amplitude relation, ultimately explained by the Waldmeier effect.

A major consequence of this is that the detailed distribution of peaks varies significantly depending on the interval of time considered. Indeed, Kolláth and Oláh (2009) recently applied time dependent harmonic analysis to the sunspot number series and found that the dominant periods have shown systematic secular changes during the past 300 years (Figure 9). For instance, the basic period seems to have shortened from 11 years to 10 years between 1850 and 1950, with some moderate increase in the last 50 years. (This is consistent with the known anticorrelation between cycle length and amplitude, cf. Section 1.3.3.)

3.3 Nonlinear methods

The nonlinearities in the dynamo equations readily give rise to chaotic behaviour of the solutions. The long term behaviour of solar activity, with phenomena like grand minima and grand maxima, is also suggestive of a chaotic system. While chaotic systems are inherently unpredictable on long enough time scales, their deterministic nature does admit forecast within a limited range. It is therefore natural to explore this possibility from the point of view of solar cycle prediction.

4.1 The solar dynamo: a brief summary of current models

Extensive summaries of the current standing of solar dynamo theory are given in the reviews by Petrovay (2000), Ossendrijver (2003), Charbonneau (2010), and Solanki et al. (2006). As explained in detail in those reviews, all current models are based on the mean-field theory approach wherein a coupled system of partial differential equations governs the evolution of the toroidal and poloidal components of the large-scale magnetic field. The large-scale field is assumed to be axially symmetric in practically all current models. In some nonlinear models the averaged equation of motion, governing large-scale flows is also coupled into the system.

In the more mainstream solar dynamo models the strong toroidal field is now generally thought to reside near the bottom of the solar convective zone. Indeed, it is known that a variety of flux transport mechanisms such as pumping (Petrovay, 1994) remove magnetic flux from the solar convective zone on a timescale short compared to the solar cycle. Following earlier simpler numerical experiments, recent MHD numerical simulations have indeed demonstrated this pumping of large scale magnetic flux from the convective zone into the tachocline below, where it forms strong coherent toroidal fields (Browning et al., 2006). As this layer is also where rotational shear is maximal, it is plausible that the strong toroidal fields are not just stored but also generated here, by the winding up of poloidal field. The two main groups of dynamo models, interface dynamos and flux transport dynamos, differ mainly in their assumptions about the site and mechanism of the α-effect responsible for the generation of a new poloidal field from the toroidal field.

In interface dynamos α is assumed to be concentrated near the bottom of the convective zone, in a region adjacent to the tachocline, so that the dynamo operates as a wave propagating along the interface between these two layers. While these models may be roughly consistent and convincing from the physical point of view, they have only had limited success in reproducing the observed characteristics of the solar cycle, such as the butterfly diagram.

Flux transport dynamos, in contrast, rely on the Babcock-Leighton mechanism for α, arising due to the action of the Coriolis force on emerging flux loops, and they assume that the corresponding α-effect is concentrated near the surface. They keep this surface layer incommunicado with the tachocline by introducing some arbitrary unphysical assumptions (such as very low diffusivities in the bulk of the convective zone). The poloidal fields generated by this surface α-effect are then advected to the poles and there down to the tachocline by the meridional circulation - which, accordingly, has key importance in these models. The equatorward deep return flow of the meridional circulation is assumed to have a significant overlap with the tachocline (another controversial point), and it keeps transporting the toroidal field generated by the rotational shear towards the equator. By the time it reaches lower latitudes, it is amplified sufficiently for the flux emergence process to start, resulting in the formation of active regions and, as a result of the Babcock-Leighton mechanism, in the reconstruction of a poloidal field near the surface with a polarity opposed to that in the previous 11-year cycle. While flux transport models may be questionable from the point of view of their physical consistency, they can be readily fine-tuned to reproduce the observed butterfly diagram quite well.

It should be noted that while the terms “interface dynamo” and “flux transport dynamo” are now very widely used to describe the two main approaches, the more generic terms “advectiondominated” and “diffusion-dominated” would be preferable in several respects. This classification allows for a continuous spectrum of models depending on the numerical ratio of advective and diffusive timescales (for communication between surface and tachocline). In addition, even at the two extremes, classic interface dynamos and circulation-driven dynamos are just particular examples of advection or diffusion dominated systems with different geometrical structures.

4.2 Is model-based cycle prediction feasible?

As it can be seen even from the very brief and sketchy presentation given above, all current solar dynamo models are based on a number of quite arbitrary assumptions and depend on a number of free parameters, the functional form and amplitude of which is far from being well constrained. For this reason, Bushby and Tobias (2007) rightfully say that all current solar dynamo models are only of “an illustrative nature”. This would suggest that as far as solar cycle prediction is concerned, the best we should expect from dynamo models is also an “illustrative” reproduction of a series of solar cycles with the same kind of long-term variations (qualitatively and, in the statistical sense, quantitatively) as seen in solar data. Indeed, Bushby and Tobias (2007) demonstrated that even a minuscule stochastic variation in the parameters of a particular flux transport model can lead to large, unpredictable variations in the cycle amplitudes. And even in the absence of stochastic effects, the chaotic nature of nonlinear dynamo solutions seriously limits the possibilities of prediction, as the authors find in a particular interface dynamo model: even if the very same model is used to reproduce the results of one particular run, the impossibility of setting initial conditions exactly representing the system implies that predictions are impossible even for the next cycle. Somewhat better results are achieved by an alternative method, based on the phase space reconstruction of the attractor of the nonlinear system - this is, however, a purely empirical time series analysis technique for which no knowledge of the detailed underlying physics is needed. (Cf. Section 3.3 above.)

Despite these very legitimate doubts regarding the feasibility of model-based prediction of solar cycles, in recent years several groups have claimed to be able to predict solar cycle 24 on the basis of dynamo models with a high confidence. So let us consider these claims.

4.3 Explicit models

The current buzz in the field of model-based solar cycle prediction was started by the work of the solar dynamo group in Boulder (Dikpati et al., 2006; Dikpati and Gilman, 2006). Their model is a flux transport dynamo, advection-dominated to the extreme. The strong suppression of diffusive effects is assured by the very low value (less than 20 km² s^-1) assumed for the turbulent magnetic diffusivity in the bulk of the convective zone. As a result, the poloidal fields generated near the surface by the Babcock-Leighton mechanism are only transported to the tachocline on the very long, decadal time scale of meridional circulation. The strong toroidal flux residing in the low-latitude tachocline, producing solar activity in a given cycle is thus the product of the shear amplification of poloidal fields formed near the surface about 2 – 3 solar cycles earlier, i.e., the model has a “memory” extending to several cycles. The mechanism responsible for cycle-to-cycle variation is assumed to be the stochastic nature of the flux emergence process. In order to represent this variability realistically, the model drops the surface α-term completely (a separate, smaller α term is retained in the tachocline); instead, the generation of poloidal field near the surface is represented by a source term, the amplitude of which is based on the sunspot record, while its detailed functional form remains fixed.

Dikpati and Gilman (2006) find that, starting off their calculation by fixing the source term amplitudes of sunspot cycles 12 to 15, they can predict the amplitudes of each subsequent cycle with a reasonable accuracy, provided that the relation between the relative sunspot numbers and the toroidal flux in the tachocline is linear, and that the observed amplitudes of all previous cycles are incorporated in the source term for the prediction of any given cycle. For the upcoming cycle 24 the model predicts peak smoothed annual relative sunspot numbers of 150 or more. Elaborating on their model, they proceeded to apply it separately to the northern and southern hemispheres, to find that the model can also be used to correctly forecast the hemispheric asymmetry of solar activity (Dikpati et al., 2007).

The extraordinary claims of this pioneering research have prompted a hot debate in the dynamo community. Besides the more general, fundamental doubt regarding the feasibility of model-based predictions (see Section 3.2 above), more technical concerns arose, to be discussed below.

Another flux transport dynamo code, the Surya code, originally developed by A. Choudhuri and coworkers in Bangalore, has also been utilized for prediction purposes. The crucial difference between the two models is in the value of the turbulent diffusivity assumed in the convective zone: in the Bangalore model this value is 240 km² s^-1, 1 – 2 orders of magnitude higher than in the Boulder model, and within the physically plausible range (Chatterjee et al., 2004). As a result of the shorter diffusive timescale, the model has a shorter memory, not exceeding one solar cycle. As a consequence of this relatively rapid diffusive communication between surface and tachocline, the poloidal fields forming near the surface at low latitudes due to the Babcock-Leighton mechanism diffuse down to the tachocline in about the same time as they reach the poles due to the advection by the meridional circulation. In these models, then, polar magnetic fields are not a true physical precursor of the low-latitude toroidal flux, and their correlation is just due to their common source. In the version of the code adapted for cycle prediction (Choudhuri et al., 2007; Jiang et al., 2007), the “surface” poloidal field (i.e., the poloidal field throughout the outer half of the convection zone) is rescaled at each minimum by a factor reflecting the observed amplitude of the Sun’s dipole field. The model shows reasonable predictive skill for the last three cycles for which data are available, and can even tackle hemispheric asymmetry (Goel and Choudhuri, 2009). For cycle 24, the predicted amplitude is 30 – 35% lower than cycle 23.

4.4 Truncated models

The “illustrative” nature of solar dynamo models is nowhere more clearly on display than in truncated or reduced models where some or all of the detailed spatial structure of the system is completely disregarded, and only temporal variations are explicitly considered. This is sometimes rationalised as a truncation or spatial integration of the equations of a more realistic inhomogenous system; in other cases, no such rationalisation is provided, representing the solar dynamo by an infinite, homogeneous or periodic turbulent medium where the amplitude of the periodic large-scale magnetic field varies with time only.

In the present subsection we deal with models that do keep one spatial variable (typically, the latitude), so growing wave solutions are still possible - these models, then, are still dynamos even though their spatial structure is not in a good correspondence with that of the solar dynamo.

This approach in fact goes back to the classic migratory dynamo model of Parker (1955) who radially truncated (i.e., integrated) his equations to simplify the problem. Parker seems to have been the first to employ a heuristic relaxation term of the form -B _r /τ _d in the poloidal field equation to represent the effect of radial diffusion; here, τ _d = d²/gh _T is the diffusive timescale across the thickness d of the convective zone. His model was recently generalized by Moss et al. (2008) and Usoskin et al. (2009b) to the case when the α-effect includes an additive stochastic noise, and nonlinear saturation of the dynamo is achieved by α-quenching. These authors do not make an attempt to predict solar activity with their model but they can reasonably well reproduce some features of the very long term solar activity record, as seen from cosmogenic isotope studies.

Another radially truncated model, this time formulated in a Cartesian system, is that of Kitiashvili and Kosovichev (2009). In this model stochastic effects are not considered and, in addition to using an α-quenching recipe, further nonlinearity is introduced by coupling in the Kleeorin-Ruzmaikin equation (Zel’dovich et al., 1983) governing the evolution of magnetic helicity, which in the hydromagnetic case contributes to α. Converting the toroidal field strength to relative sunspot number using the Bracewell transform, Equation (3), the solutions reproduce the asymmetric profile of the sunspot number cycle. For sufficiently high dynamo numbers the solutions become chaotic, cycle amplitudes show an irregular variation. Cycle amplitudes and minimum-maximum time delays are found to be related in a way reminiscent of the Waldmeier relation.

Building on these results, Kitiashvili and Kosovichev (2008) attempt to predict solar cycles using a data assimilation method. The approach used is the so-called Ensemble Kalman Filter method. Applying the model for a “postdiction” of the last 8 solar cycles yielded astonishingly good results, considering the truncated and arbitrary nature of the model and the fundamental obstacles in the way of reliable prediction discussed above. While the presently available brief preliminary publication leaves several details of the method unclear, the question may arise whether the actual physics of the model considered has any significant role in this prediction, or we are dealing with something like the phase space reconstruction approach discussed in Section 3.3 above where basically any model with an attractor that looks reasonably similar to that of the actual solar dynamo would do. Either way, the method is remarkable, and the prediction for cycle 24 of a maximal smoothed annual sunspot number of 80, to be reached in 2013, will be worth comparing to the actual value.

In order to understand the origin of the predictive skill of the Boulder model, Cameron and Schüssler (2007) studied a radially truncated version of the model, wherein only the equation for the radial field component is solved as a function of time and latitude. The equation includes a source term similar to the one used in the Boulder model. As the toroidal flux does not figure in this simple model, the authors use the transequatorial flux Φ as a proxy, arguing that this may be more closely linked to the amplitude of the toroidal field in the upcoming cycle than the polar field. They find that Φ indeed correlates quite well (correlation coefficients r ∼ 0.8 – 0:9, depending on model details) with the amplitude of the next cycle, as long as the form of the latitude dependence of the source term is prescribed and only its amplitude is modulated with the observed sunspot number series (“idealized model”). But surprisingly, the predictive skill of the model is completely lost if the prescribed form of the source function is dropped and the actually observed latitude distribution of sunspots is used instead (“realistic model”). Cameron and Schüssler (2007) interpret this by pointing out that Φ is mainly determined by the amount of very low latitude flux emergence, which in turn occurs mainly in the last few years of the cycle in the idealized model, while it has a wider temporal distribution in the realistic model. The conclusion is that the root of the apparently good predictive skill of the truncated model (and, by inference, of the Boulder model it is purported to represent) is actually just the good empirical correlation between latephase activity and the amplitude of the next cycle, discussed in Section 2.1 above. This correlation is implicitly “imported” into the idealized flux transport model by assuming that the late-phase activity is concentrated at low latitudes, and therefore gives rise to cross-equatorial flux which then serves as a seed for the toroidal field in the next cycle. So if Cameron and Schüssler (2007) are correct, the predictive skill of the Boulder model is due to an empirical precursor and is thus ultimately explained by the good old Waldmeier effect (cf. Section 1.3.3)

The fact that the truncated model of Cameron and Schüssler (2007) is not identical to the Boulder model obviously leaves room for doubt regarding this conclusion. In particular, the effective diffusivity represented by the sink term in the truncated model is ∼ km² s^-1, significantly higher than in the Boulder model; consequently, the truncated model will have a more limited memory, cf. Yeates et al. (2008). The argument that the cross-equatorial flux is a valid proxy of the amplitude of the next cycle may be correct in such a short-memory model with no radial structure, but it is dubious whether it remains valid for flux transport models in general. In an attempt to appreciate the importance of the cross-equatorial flux in their model, Dikpati et al. (2008a) find that while this flux does indeed correlate fairly well (r = 0:76) with the next cycle amplitude, the toroidal flux is a much better predictor (r = 0:96). At first sight this seems to make it unlikely that the former can explain the latter; however, part of the difference in the predictive skill may be due to the fact that Φ shows much more short-term variability than the toroidal flux.

In any case, the obvious way to address the concerns raised by Cameron and Schüssler (2007) and further by Schüssler (2007) in relation to the Boulder model would be to run that model with a modified source function incorporating the realistic latitudinal distribution of sunspots in each cycle. The results of such a test are not yet available at the time of writing this review.

4.5 The Sun as an oscillator

(1) The most straightforward approach is to add a forcing term + sin(ω₀t) to the r.h.s. of Equations (18). Damping would cause the system to relax to the driving period 2π/gw₀ if there were no stochastic disturbances to this equilibrium. Hiremath (2006) fitted the parameters of the forced and damped oscillator model to each observed solar cycle individually; then in a later work (Hiremath, 2008) he applied linear regression to the resulting series to provide a forecast (see Section 3.1 above).

(2) Another trick is to account for the π/2 phase difference between poloidal and toroidal field components in a dynamo wave by introducing a phase factor i into the first term on the r.h.s. of Equation (17). This can also be given a more formal derivation as equations of this form result from the substitution of solutions of the form A ∝ e ^ikx , B ∝ e ^i(kx+π=2) into the 1D dynamo equations. This route, combined with a nonlinearity due to magnetic modulation of differential rotation described by a coupled third equation, was taken by Weiss et al. (1984). Their model displayed chaotic behaviour with intermittent episodes of low activity similar to grand minima.

(3) Wilmot-Smith et al. (2006) showed that another case where dynamo-like behaviour can be found in an equation like (18) is if the missing effects of finite communication time between parts of a spatially extended system are reintroduced by using a time delay δt, evaluating the first term on the r.h.s. at time t — Δt to get the value for the l.h.s. at time t.

These simple nonlinear oscillators were among the first physical systems where chaotic behaviour was detected (when a periodic forcing was added). Yet, curiously, they first emerged in the solar context precisely as an alternative to chaotic behaviour. Considering the mapping of the solar cycle in the differential phase space {B; dB/dt}, Mininni et al. (2000) got the impression that, rather than showing signs of a strange attractor. The SSN series is adequately modelled by a van der Pol oscillator with stochastic fluctuations. This concept was further developed by Lopes and Passos (2009) who fitted the parameters of the oscillator to each individual sunspot cycle. The parameter μ is related to the meridional flow speed and the fit indicates that a slower meridional flow may have been responsible for the Dalton minimum. This was also corroborated in an explicit dynamo model (the Surya code) — however, as we discussed in Section 2.4, this result of flux transport dynamo models is spurious and the actual effect of a slower meridional flow is likely to be opposite to that suggested by the van der Pol oscillator model.

In an alternative approach to the problem, Nagovitsyn (1997) attempted to constrain the properties of the solar oscillator from its amplitude-frequency diagram, suggesting a Duffing oscillator driven at two secular periods. While his empirical reconstruction of the amplitude-frequency plot may be subject to many uncertainties, the basic idea is certainly noteworthy.

In summary: despite its simplicity, the oscillator representation of the solar cycle is a relatively new development in dynamo theory, and its obvious potential for forecasting purposes has barely been exploited.