2.4 The solar cycle and its variations

2.4.1 Quasi-periodicities

The main feature of solar activity is its pronounced quasi-periodicity with a period of about 11 years, known as the Schwabe cycle. However, the cycle varies in both amplitude and duration. The first observation of a possible regular variability in sunspot numbers was made by the Danish astronomer Christian Horrebow in the 1770s on the basis of his sunspot observations from 1761 – 1769 (see details in Gleissberg, 1952Vitinsky, 1965Jump To The Next Citation Point), but the results were forgotten. It took over 70 years before the amateur astronomer Schwabe announced in 1844 that sunspot activity varies cyclically with a period of about 10 years. This cycle, called the 11-year or Schwabe cycle, is the most prominent variability in the sunspot-number series. It is recognized now as a fundamental feature of solar activity originating from the solar-dynamo process. This 11-year cyclicity is prominent in many other parameters including solar, heliospheric, geomagnetic, space weather, climate and others. The background for the 11-year Schwabe cycle is the 22-year Hale magnetic polarity cycle. Hale found that the polarity of sunspot magnetic fields changes in both hemispheres when a new 11-year cycle starts (Hale et al., 1919). This relates to the reversal of the global magnetic field of the sun with the period of 22 years. It is often considered that the 11-year Schwabe cycle is the modulo of the sign-alternating Hale cycle (e.g., Sonett, 1983Jump To The Next Citation PointBracewell, 1986Kurths and Ruzmaikin, 1990Jump To The Next Citation Pointde Meyer, 1998Jump To The Next Citation PointMininni et al., 2001Jump To The Next Citation Point). A detailed review of solar-cyclic variability can be found in (Hathaway, 2010). UpdateJump To The Next Update Information

Sometimes the regular time evolution of solar activity is broken up by periods of greatly depressed activity called grand minima. The last grand minimum (and the only one covered by direct solar observations) was the famous Maunder minimum from 1645 – 1715 (Eddy, 1976Jump To The Next Citation Point1983). Other grand minima in the past, known from cosmogenic isotope data, include, e.g., the Spörer minimum around 1450 – 1550 and the Wolf minimum around the 14th century (see the detailed discussion in Section 4.3). Sometimes the Dalton minimum (ca. 1790 – 1820) is also considered to be a grand minimum. However, sunspot activity was not completely suppressed and still showed Schwabe cyclicity during the Dalton minimum. As suggested by Schüssler et al. (1997), this can be a separate, intermediate state of the dynamo between the grand minimum and normal activity, or an unsuccessful attempt of the sun to switch to the grand minimum state (Frick et al., 1997Jump To The Next Citation PointSokoloff, 2004Jump To The Next Citation Point). This is observed as the phase catastrophe of solar-activity evolution (e.g., Vitinsky et al., 1986Jump To The Next Citation PointKremliovsky, 1994Jump To The Next Citation Point). A peculiarity in the phase evolution of sunspot activity around 1800 was also noted by Sonett (1983Jump To The Next Citation Point) who ascribed it to a possible error in Wolf sunspot data and by Wilson (1988a), who reported on a possible misplacement of sunspot minima for cycles 4 – 6 in the WSN series. It has been also suggested that the phase catastrophe can be related to a tiny cycle, which might have been lost at the end of the 18th century because of very sparse observations (Usoskin et al., 2001b2002a2003bZolotova and Ponyavin, 2007). We note that a new independent evidence proving the existence of the lost cycle has been found recently in the reconstructed sunspot butterfly diagram for that period (Usoskin et al., 2009cJump To The Next Citation Point). UpdateJump To The Next Update Information Another period associated with a phase catastrophe is the period of sunspot-activity recovery after the Maunder minimum. The recovery of the 11-year cycle passed through the time of distorted phase evolution (Usoskin et al., 2001aJump To The Next Citation Point) with the concurrent loss of the 88-year cycle phase (Feynman and Gabriel, 1990Jump To The Next Citation Point).

The long-term change (trend) in the Schwabe cycle amplitude is known as the secular Gleissberg cycle (Gleissberg, 1939). However, the Gleissberg cycle is not a cycle in the strict periodic sense but rather a modulation of the cycle envelope with a varying timescale of 60 – 120 years (e.g., Gleissberg, 1971Kuklin, 1976Ogurtsov et al., 2002Jump To The Next Citation Point). This secular cycle has also been reported, using a spectral analysis of radiocarbon data as a proxy for solar activity (see Section 3), to exist on long timescales (Feynman and Gabriel, 1990Jump To The Next Citation PointPeristykh and Damon, 2003Jump To The Next Citation Point), but the question of its phase locking and persistency/intermittency still remains open.

Longer (super-secular) cycles cannot be studied using direct solar observations, but several such cycles have been found in cosmogenic isotope data. A cycle with a period of 205 – 210 years, called the de Vries or Suess cycle in different sources, is a prominent feature, observed in varying cosmogenic data (e.g., Suess, 1980Sonett and Finney, 1990Jump To The Next Citation PointZhentao, 1990Usoskin et al., 2004Jump To The Next Citation Point). Sometimes variations with a characteristic time of 600 – 700 years or 1000 – 1200 years are discussed (e.g., Vitinsky et al., 1986Jump To The Next Citation PointSonett and Finney, 1990Vasiliev and Dergachev, 2002Jump To The Next Citation Point), but they are intermittent and can hardly be regarded as a typical feature of solar activity. A 2000 – 2400-year cycle is also noticeable in radiocarbon data series (see, e.g., Vitinsky et al., 1986Jump To The Next Citation PointDamon and Sonett, 1991Jump To The Next Citation PointVasiliev and Dergachev, 2002Jump To The Next Citation Point). However, the nonsolar origin of these super-secular cycles (e.g., geomagnetic or climatic variability) cannot be excluded.

2.4.2 Randomness vs. regularity

The short-term (days - months) variability of sunspot numbers is greater than the observational uncertainties indicating the presence of random fluctuations (noise). As typical for most real signals, this noise is not uniform (white), but rather red or correlated noise (e.g., Ostryakov and Usoskin, 1990Oliver and Ballester, 1996Frick et al., 1997Jump To The Next Citation Point), namely, its variance depends on the level of the signal. While the existence of regularity and randomness in sunspot series is apparent, their relationship is not clear (e.g., Wilson, 1994) – are they mutually independent or intrinsically tied together? Moreover, the question of whether randomness in sunspot data is due to chaotic or stochastic processes is still open.

Earlier it was common to describe sunspot activity as a multi-harmonic process with several basic harmonics (e.g., Vitinsky, 1965Jump To The Next Citation PointSonett, 1983Vitinsky et al., 1986Jump To The Next Citation Point) with an addition of random noise, which plays no role in the solar-cycle evolution. However, it has been shown (e.g., Rozelot, 1994Weiss and Tobias, 2000Jump To The Next Citation PointCharbonneau, 2001Jump To The Next Citation PointMininni et al., 2002) that such an oversimplified approach depends on the chosen reference time interval and does not adequately describe the long-term evolution of solar activity. A multi-harmonic representation is based on an assumption of the stationarity of the benchmark series, but this assumption is broadly invalid for solar activity (e.g., Kremliovsky, 1994Sello, 2000Jump To The Next Citation PointPolygiannakis et al., 2003). Moreover, a multi-harmonic representation cannot, for an apparent reason, be extrapolated to a timescale larger than that covered by the benchmark series. The fact that purely mathematical/statistical models cannot give good predictions of solar activity (as will be discussed later) implies that the nature of the solar cycle is not a multi-periodic or other purely deterministic process, but random (chaotic or stochastic) processes play an essential role in sunspot formation. Different numeric tests, such as an analysis of the Lyapunov exponents (Ostriakov and Usoskin, 1990Jump To The Next Citation PointMundt et al., 1991Jump To The Next Citation PointKremliovsky, 1995Jump To The Next Citation PointSello, 2000Jump To The Next Citation Point), Kolmogorov entropy (Carbonell et al., 1994Jump To The Next Citation PointSello, 2000) and Hurst exponent (Ruzmaikin et al., 1994Oliver and Ballester, 1998), confirm the chaotic/stochastic nature of the solar-activity time evolution (see, e.g., the recent review by Panchev and Tsekov, 2007).

It was suggested quite a while ago that the variability of the solar cycle may be a temporal realization of a low-dimensional chaotic system (e.g., Ruzmaikin, 1981). This concept became popular in the early 1990s, when many authors considered solar activity as an example of low-dimensional deterministic chaos, described by the strange attractor (e.g., Kurths and Ruzmaikin, 1990Jump To The Next Citation PointOstriakov and Usoskin, 1990Morfill et al., 1991Mundt et al., 1991Rozelot, 1995Salakhutdinova, 1999Serre and Nesme-Ribes, 2000). Such a process naturally contains randomness, which is an intrinsic feature of the system rather than an independent additive or multiplicative noise. However, although this approach easily produces features seemingly similar to those of solar activity, quantitative parameters of the low-dimensional attractor have varied greatly as obtained by different authors. Later it was realized that the analyzed data set was too short (Carbonell et al., 19931994), and the results were strongly dependent on the choice of filtering methods (Price et al., 1992). Developing this approach, Mininni et al. (20002001Jump To The Next Citation Point) suggest that one consider sunspot activity as an example of a 2D Van der Pol relaxation oscillator with an intrinsic stochastic component.

Such phenomenological or basic principles models, while succeeding in reproducing (to some extent) the observed features of solar-activity variability, do not provide insight into the nature of regular and random components of solar variability. In this sense efforts to understand the nature of randomness in sunspot activity in the framework of dynamo theory are more advanced. Corresponding theoretical dynamo models have been developed (see reviews by Ossendrijver, 2003Charbonneau, 2005Jump To The Next Citation Point), which include stochastic processes (e.g., Weiss et al., 1984Feynman and Gabriel, 1990Jump To The Next Citation PointSchmalz and Stix, 1991Moss et al., 1992Hoyng, 1993Jump To The Next Citation PointBrooke and Moss, 1994Lawrence et al., 1995Schmitt et al., 1996Jump To The Next Citation PointCharbonneau and Dikpati, 2000Jump To The Next Citation PointBrandenburg and Sokoloff, 2002). For example, Feynman and Gabriel (1990) suggest that the transition from a regular to a chaotic dynamo passes through bifurcation. Charbonneau and Dikpati (2000) studied stochastic fluctuations in a Babcock–Leighton dynamo model and succeeded in the qualitative reproduction of the anti-correlation between cycle amplitude and length (Waldmeier rule). Their model also predicts a phase-lock of the Schwabe cycle, i.e., that the 11-year cycle is an internal “clock” of the sun. Most often the idea of fluctuations is related to the α-effect, which is the result of the electromotive force averaged over turbulent vortices, and thus can contain a fluctuating contribution (e.g., Hoyng, 1993Jump To The Next Citation PointOssendrijver et al., 1996Jump To The Next Citation PointBrandenburg and Spiegel, 2008Jump To The Next Citation Point). Note that a significant fluctuating component (with the amplitude more than 100% of the regular component) is essential in all these model. UpdateJump To The Next Update Information

2.4.3 A note on solar activity predictions

Randomness (see Section 2.4.2) in the SN series is directly related to the predictability of solar activity. Forecasting solar activity has been a subject of intense study for many years (e.g., Yule, 1927Newton, 1928Gleissberg, 1948Vitinsky, 1965) and has greatly intensified recently with a hundred journal articles being published before 2008 (see, e.g., the review by Kane, 2007Jump To The Next Citation Point) following the boosting of space-technology development and increasing debates on solar-terrestrial relations.

All prediction methods can be generically divided into precursor and statistical techniques or their combinations (Hathaway et al., 1999Jump To The Next Citation Point). The precursor methods are usually based on phenomenological, but sometimes physical, links between the poloidal solar-magnetic field, estimated, e.g., from geomagnetic activity in the declining phase of the preceding cycle or in the minimum time (e.g., Hathaway et al., 1999Jump To The Next Citation Point), with the toroidal field responsible for sunspot formation. These methods usually yield better short-term predictions of a forthcoming cycle maximum than the statistical methods, but cannot be applied to timescales longer than one solar cycle. Statistical methods, including a low-dimensional solar-attractor representation (Kurths and Ruzmaikin, 1990), are based solely on the statistical properties of sunspot activity and may give a reasonable result for short-term forecasting, but yield very poor results for long-term predictions (see reviews by e.g., Conway, 1998Hathaway et al., 1999Li et al., 2001Usoskin and Mursula, 2003Jump To The Next Citation PointKane, 2007) because of chaotic/stochastic behavior (see Section 2.4.2).

Some models, mostly based on precursor method, succeed in reasonable predictions of a forthcoming solar cycle (i.e., several years ahead), but they do not pretend to extend further in time. On the other hand, many claims of the solar activity forecast for 40 – 50 years ahead and even beyond have been made recently, often without sensible argumentation. However, so far there is no evidence of any method giving a reasonable prediction of solar activity beyond the solar-cycle scale (see, e.g., Section 2.3.3), probably because of the intrinsic limit of solar-activity predictability due to its stochastic/chaotic nature (Kremliovsky, 1995). Accordingly, such attempts can be regarded as speculative, unless they are verified by the actual behavior of solar activity. Note that even an exact prediction of the amplitude of one solar cycle can be just a random coincidence and cannot serve as a proof of the method’s veracity. Only a sequence of successful predictions can form a basis for confidence, which requires several decades.

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