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.

Inspecting Figure 3 one can discern an obvious long term variation. For the study of such long term variations, the series of cycle parameters is often smoothed on time scales significantly longer than a solar cycle: this procedure is known as secular smoothing. One popular method is the so-called Gleissberg filter or 12221 filter (Gleissberg, 1967). For instance, the Gleissberg filtered amplitude of cycle n is given by

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 below the long-term average.

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).

Following customary usage, by “memory” we will refer to some physical (or, in the case of a model, mathematical) mechanism by which the state of a system at a given time will depend on its previous states. In any system there may be several different such mechanisms at work simultaneously – if this is so, again following common usage we will speak of different “types” of memory. A very mundane example are the RAM and the hard disk in a computer: devices that store information over very different time scales and the effect of which manifests itself differently in the functioning of the system.

There is no question that the solar dynamo (i.e., the mechanism that gives rise to the sunspot number series) does possess a memory that extends at least over the course of a single solar cycle. Obviously, during the rise phase solar activity “remembers” that it should keep growing, while in the decay phase it keeps decaying, even though exactly the same range of R values are observed in both phases. Furthermore, profiles of individual sunspot cycles may, in a first approximation, be considered a one-parameter ensemble (Hathaway et al., 1994). This obvious effect will be referred to here as intracycle memory.

As we will see, correlations between activity parameters in different cycles are generally much weaker than those within one cycle, which strongly suggests that the intracycle memory mechanism is different from longer term memory effects, if such are present at all. Referring back to our analogy, the intracycle memory may work like computer RAM, periodically erased at every reboot (i.e., at the start of a new cycle).

The interesting question is whether, in addition to the intracycle memory effect, any other type of memory is present in the solar dynamo or not. To what extent is the amplitude of a sunspot cycle determined by previous cycles? Are subsequent cycles essentially independent, randomly drawn from some stochastic distribution of cycle amplitudes around the long term average? Or, in the alternative case, for how many previous cycles do we need to consider solar activity for successful forecasts?

The existence of long lasting grand minima and maxima suggests that the sunspot number record must have a long-term memory extending over several consecutive cycles. Indeed, elementary combinatorical calculations show that the occurrence of phenomena like the Dalton minimum (3 of the 4 lowest maxima occurring in a row) or the Modern maximum (4 of the 5 highest maxima occurring within a series of 5 cycles) in a random series of 24 recorded solar maxima has a rather low probability (5 % and 3 %, respectively). This conclusion is corroborated by the analysis of long-term proxy data, extending over several millennia, which showed that the occurrence of grand minima and grand maxima is more common than what would follow from Gaussian statistics (Usoskin et al., 2007).

It could be objected that for sustained grand minima or maxima a memory extending only from one cycle to the next would suffice. In contrast to long-term (multidecadal or longer) memory, this would constitute another kind of short-term ( 10 years) memory: a cycle-to-cycle or intercycle memory effect. In our computer analogy, think of system files or memory cache written on the hard disk, often with the explicit goal of recalling the system status (e.g., desktop arrangement) after the next reboot. While these files survive the reboot, they are subject to erasing and rewriting in every session, so they have a much more temporary character than the generic data files stored on the disk.

The intercycle memory explanation of persistent secular activity minima and maxima, however, would imply a good correlation between the amplitudes of subsequent cycles, which is not the case (cf. Section 2.1 below). With the known poor cycle-to-cycle correlation, strong deviations from the long-term mean would be expected to be damped on time scales short compared to, e.g., the length of the Maunder minimum. This suggests that the persistent states of low or high activity are due to truly long term memory effects extending over several cycles.

Further evidence for a long-term memory in solar activity comes from the persistence analysis of activity indicators. The parameter determined in such studies is the Hurst exponent . Essentially, is the steepness of the growth of the total range of measured values plotted against the number n of data in a time series, on a logarithmic plot: . For a Markovian random process with no memory . Processes with are persistent (they tend to stay in a stronger-than-average or weaker-than-average state longer), while those with are anti-persistent (their fluctuations will change sign more often).

Hurst exponents for solar activity indices have been derived using rescaled range analysis by many authors (Mandelbrot and Wallis, 1969; Ruzmaikin et al., 1994; Komm, 1995; Oliver and Ballester, 1996; Kilcik et al., 2009). All studies coherently yield a value for time scales exceeding a year or so, and somewhat lower values () on shorter time scales. Some doubts regarding the significance of this result for a finite series have been raised by Oliver and Ballester (1998); however, Qian and Rasheed (2004) have shown using Monte Carlo experiments that for time series of a length comparable to the sunspot record, values exceeding 0.7 are statistically significant.

A complementary method, essentially equivalent to rescaled range analysis is detrended fluctuation analysis. Its application to solar data (Ogurtsov, 2004) has yielded results in accordance with the values quoted above.

The overwhelming evidence for the persistent character of solar activity and for the intermittent appearance of secular cyclicities, however, is not much help when it comes to cycle-to-cycle prediction. It is certainly reassuring to know that forecasting is not a completely idle enterprise (which would be the case for a purely Markovian process), and the long-term persistence and trends may make our predictions statistically somewhat different from just the long-term average. There are, however, large decadal scale fluctuations superposed on the long term trends, so the associated errors will still be so large as to make the forecast of little use for individual cycles.

“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)

(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.

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 ), 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. Update

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 “ correlations” with n = 0, –1 or –3, respectively.

Modern time series analysis methods offer several ways to define an instantaneous frequency 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

Nonlinear dynamo models including some form of -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 , 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

while in the chaotic caseNote that based on the actual sunspot number series Waldmeier originally proposed

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

Living Rev. Solar Phys. 7, (2010), 6
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