2 Solar Activity: Concept and Observations

2.1 The concept of solar activity

The sun is known to be far from a static state, the so-called “quiet” sun described by simple stellar-evolution theories, but instead goes through various nonstationary active processes. Such nonstationary and nonequilibrium (often eruptive) processes can be broadly regarded as solar activity. Whereas the concept of solar activity is quite a common term nowadays, it is neither straightforwardly interpreted nor unambiguously defined. For instance, solar-surface magnetic variability, eruption phenomena, coronal activity, radiation of the sun as a star or even interplanetary transients and geomagnetic disturbances can be related to the concept of solar activity. A variety of indices quantifying solar activity have been proposed in order to represent different observables and caused effects. Most of the indices are highly correlated to each other due to the dominant 11-year cycle, but may differ in fine details and/or long-term trends. In addition to the solar indices, indirect proxy data is often used to quantify solar activity via its presumably known effect on the magnetosphere or heliosphere. The indices of solar activity that are often used for long-term studies are reviewed below.

2.2 Indices of solar activity

Solar (as well as other) indices can be divided into physical and synthetic according to the way they are obtained/calculated. Physical indices quantify the directly-measurable values of a real physical observable, such as, e.g., the radioflux, and thus have clear physical meaning as they quantify physical features of different aspects of solar activity and their effects. Synthetic indices (the most common being sunspot number) are calculated (or synthesized) using a special algorithm from observed (often not measurable in physical units) data or phenomena. Additionally, solar activity indices can be either direct (i.e., directly relating to the sun) or indirect (relating to indirect effects caused by solar activity), as discussed in subsequent Sections 2.2.1 and 2.2.2.

2.2.1 Direct solar indices

The most commonly used index of solar activity is based on sunspot number. Sunspots are dark areas on the solar disc (of size up to tens of thousands of km, lifetime up to half-a-year), characterized by a strong magnetic field, which leads to a lower temperature (about 4000 K compared to 5800 K in the photosphere) and observed as darkening.

Sunspot number is a synthetic, rather than a physical, index, but it has still become quite a useful parameter in quantifying the level of solar activity. This index presents the weighted number of individual sunspots and/or sunspot groups, calculated in a prescribed manner from simple visual solar observations. The use of the sunspot number makes it possible to combine together thousands and thousands of regular and fragmentary solar observations made by earlier professional and amateur astronomers. The technique, initially developed by Rudolf Wolf, yielded the longest series of directly and regularly-observed scientific quantities. Therefore, it is common to quantify solar magnetic activity via sunspot numbers. For details see the review on sunspot numbers and solar cycles (Hathaway and Wilson, 2004Jump To The Next Citation Point; Hathaway, 2010Jump To The Next Citation Point).

Wolf sunspot number (WSN) series

The concept of the sunspot number was developed by Rudolf Wolf of the Zürich observatory in the middle of the 19th century. The sunspot series, initiated by him, is called the Zürich or Wolf sunspot number (WSN) series. The relative sunspot number Rz is defined as

Rz = k(10 G + N ), (1 )
where G is the number of sunspot groups, N is the number of individual sunspots in all groups visible on the solar disc and k denotes the individual correction factor, which compensates for differences in observational techniques and instruments used by different observers, and is used to normalize different observations to each other.
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Figure 1: Sunspot numbers since 1610. a) Monthly (since 1749) and yearly (1700 – 1749) Wolf sunspot number series. b) Monthly group sunspot number series. The grey line presents the 11-year running mean after the Maunder minimum. Standard (Zürich) cycle numbering as well as the Maunder minimum (MM) and Dalton minimum (DM) are shown in the lower panel.

The value of Rz (see Figure 1View Imagea) is calculated for each day using only one observation made by the “primary” observer (judged as the most reliable observer during a given time) for the day. The primary observers were Staudacher (1749 – 1787), Flaugergues (1788 – 1825), Schwabe (1826 – 1847), Wolf (1848 – 1893), Wolfer (1893 – 1928), Brunner (1929 – 1944), Waldmeier (1945 – 1980) and Koeckelenbergh (since 1980). If observations by the primary observer are not available for a certain day, the secondary, tertiary, etc. observers are used (see the hierarchy of observers in Waldmeier, 1961). The use of only one observer for each day aims to make Rz a homogeneous time series. As a drawback, such an approach ignores all other observations available for the day, which constitute a large fraction of the existing information. Moreover, possible errors of the primary observer cannot be caught or estimated. The observational uncertainties in the monthly Rz can be up to 25% (e.g., Vitinsky et al., 1986Jump To The Next Citation Point). The WSN series is based on observations performed at the Zürich Observatory during 1849 – 1981 using almost the same technique. This part of the series is fairly stable and homogeneous although an offset due to the change of the weighting procedure might have been introduced in 1945 – 1946 (Svalgaard, 2012Jump To The Next Citation Point). However, prior to that there have been many gaps in the data that were interpolated. If no sunspot observations are available for some period, the data gap is filled, without note in the final WSN series, using an interpolation between the available data and by employing some proxy data. In addition, earlier parts of the sunspot series were “corrected” by Wolf using geomagnetic observation (see details in Svalgaard, 2012Jump To The Next Citation Point), which makes the series less homogeneous. Therefore, the WSN series is a combination of direct observations and interpolations for the period before 1849, leading to possible errors and inhomogeneities as discussed, e.g, by Vitinsky et al. (1986Jump To The Next Citation Point); Wilson (1998); Letfus (1999Jump To The Next Citation Point); Svalgaard (2012). The quality of the Wolf series before 1749 is rather poor and hardly reliable (Hoyt et al., 1994; Hoyt and Schatten, 1998Jump To The Next Citation Point; Hathaway and Wilson, 2004Jump To The Next Citation Point).

Note that the sun has been routinely photographed since 1876 so that full information on daily sunspot activity is available (the Greenwich series) with observational uncertainties being negligible for the last 140 years.

The routine production of the WSN series was terminated in Zürich in 1982. Since then, the sunspot number series is routinely updated as the International sunspot number Ri, provided by the Solar Influences Data Analysis Center in Belgium (Clette et al., 2007). The international sunspot number series is computed using the same definition (Equation 1View Equation) as WSN but it has a significant distinction from the WSN; it is based not on a single primary solar observation for each day but instead uses a weighted average of more than 20 approved observers.

In addition to the standard sunspot number Ri, there is also a series of hemispheric sunspot numbers RN and RS, which account for spots only in the northern and southern solar hemispheres, respectively (note that Ri = RN + RS). These series are used to study the N-S asymmetry of solar activity (Temmer et al., 2002).

Group sunspot number (GSN) series

Since the WSN series is of lower quality before the 1850s and is hardly reliable before 1750, there was a need to re-evaluate early sunspot data. This tremendous work has been done by Hoyt and Schatten (1996Jump To The Next Citation Point, 1998Jump To The Next Citation Point), who performed an extensive archive search and nearly doubled the amount of original information compared to the Wolf series. They have produced a new series of sunspot activity called the group sunspot numbers (GSN – see Figure 1View Imageb), including all available archival records. The daily group sunspot number Rg is defined as follows:

12.08 ∑ Rg = ------ k′iGi, (2 ) n i
where Gi is the number of sunspot groups recorded by the i-th observer, ′ k is the observer’s individual correction factor, n is the number of observers for the particular day, and 12.08 is a normalization number scaling Rg to Rz values for the period of 1874 – 1976. Rg is more robust than Rz since it is based on more easily identified sunspot groups and does not include the number of individual spots. The GSN series includes not only one “primary” observation, but all available observations, and covers the period since 1610, being, thus, 140 years longer than the original WSN series. It is particularly interesting that the period of the Maunder minimum (1645 – 1715) was surprisingly well covered with daily observations (Ribes and Nesme-Ribes, 1993Jump To The Next Citation Point; Hoyt and Schatten, 1996Jump To The Next Citation Point) allowing for a detailed analysis of sunspot activity during this grand minimum (see also Section 4.2). Systematic uncertainties of the Rg values are estimated to be about 10% before 1640, less than 5% from 1640 – 1728 and from 1800 – 1849, 15 – 20% from 1728 – 1799, and about 1% since 1849 (Hoyt and Schatten, 1998Jump To The Next Citation Point). The GSN series is more reliable and homogeneous than the WSN series before 1849. The two series are nearly identical after the 1870s (Hoyt and Schatten, 1998Jump To The Next Citation Point; Letfus, 1999; Hathaway and Wilson, 2004). However, the GSN series still contains some lacunas, uncertainties and possible inhomogeneities (see, e.g., Letfus, 2000; Usoskin et al., 2003a; Vaquero et al., 2012).

The search for other lost or missing records of past solar instrumental observations has not ended even since the extensive work by Hoyt and Schatten. Archival searches still give new interesting findings of forgotten sunspot observations, often outside major observatories – see a detailed review book by Vaquero and Vázquez (2009Jump To The Next Citation Point) and original papers by Casas et al. (2006); Vaquero et al. (2005, 2007); Arlt (2008Jump To The Next Citation Point, 2009Jump To The Next Citation Point). Interestingly, not only sunspot counts but also regular drawings, forgotten for centuries, are being restored nowadays in dusty archives. A very interesting work has been done by Rainer Arlt (Arlt, 2008Jump To The Next Citation Point, 2009Jump To The Next Citation Point; Arlt and Abdolvand, 2011Jump To The Next Citation Point; Arlt, 2013Jump To The Next Citation Point) on recovering, digitizing, and analyzing regular drawings by S.H. Schwabe of 1825 – 1867 and J.C. Staudacher of 1749 – 1796. This work led to the extension of the Maunder butterfly diagram for several solar cycles backwards (Arlt, 2009Jump To The Next Citation Point; Usoskin et al., 2009cJump To The Next Citation Point; Arlt and Abdolvand, 2011; Arlt, 2013Jump To The Next Citation Point) – see a newly built diagram for solar cycles Nos. 7 – 10 shown in Figure 2View Image. In particular, this data confirms that GSN series is more homogenous before 1874 that WSN. A recent finding of the lost data by G. Marcgraf and correcting some earlier uncertain data for the period 1636 – 1642 by Vaquero et al. (2011Jump To The Next Citation Point) made it possible to revise the pattern of the beginning of the Maunder minimum.

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Figure 2: Maunder butterfly diagram of sunspot occurrence reconstructed by Arlt (2013) for 1825 – 1867 using recovered drawing of S.H. Schwabe.

Other indices

An example of a synthetic index of solar activity is the flare index, representing solar flare activity (e.g., Özgüç et al., 2003Jump To The Next Citation Point; Kleczek, 1952). The flare index quantifies daily flare activity in the following manner; it is computed as a product of the flare’s relative importance I in the H α-range and duration t, Q = I t, thus being a rough measure of the total energy emitted by the flare. The daily flare index is produced by Bogazici University (Özgüç et al., 2003) and is available since 1936.

A traditional physical index of solar activity is related to the radioflux of the sun in the wavelength range of 10.7 cm and is called the F10.7 index (e.g., Tapping and Charrois, 1994). This index represents the flux (in solar flux units, 1 sfu = 10–22 Wm–2 Hz–1) of solar radio emission at a centimetric wavelength. There are at least two sources of 10.7 cm flux – free-free emission from hot coronal plasma and gyromagnetic emission from active regions (Tapping, 1987). It is a good quantitative measure of the level of solar activity, which is not directly related to sunspots. Close correlation between the F10.7 index and sunspot number indicates that the latter is a good index of general solar activity, including coronal activity. The solar F10.7 cm record has been measured continuously since 1947.

Another physical index is the coronal index (e.g., Rybanský et al., 2005), which is a measure of the irradiance of the sun as a star in the coronal green line. Computation of the coronal index is based on observations of green corona intensities (Fe XIV emission line at 530.3 nm wavelength) from coronal stations all over the world, the data being transformed to the Lomnický Štit photometric scale. This index is considered a basic optical index of solar activity. A synthesized homogeneous database of the Fe XIV 530.3 nm coronal-emission line intensities has existed since 1943 and covers seven solar cycles.

Often sunspot area is considered as a physical index representing solar activity (e.g., Baranyi et al., 2001; Balmaceda et al., 2005). This index gives the total area of visible spots on the solar disc in units of millionths of the sun’s visible hemisphere, corrected for apparent distortion due to the curvature of the solar surface. The area of individual groups may vary between tens of millionths (for small groups) up to several thousands of millionths for huge groups. This index has a physical meaning related to the solar magnetic flux emerging at sunspots. Sunspot areas are available since 1874 in the Greenwich series obtained from daily photographic images of the sun. In addition, some fragmentary data of sunspot areas, obtained from solar drawings, are available for earlier periods (Vaquero et al., 2004; Arlt, 2008Jump To The Next Citation Point).

An important quantity is solar irradiance, total and spectral (Fröhlich, 2012). Irradiance variations are physically related to solar magnetic variability (e.g., Solanki et al., 2000Jump To The Next Citation Point), and are often considered manifestations of solar activity, which is of primary importance for solar-terrestrial relations.

Other physical indices include spectral sun-as-star observations, such as the Ca II-K index (e.g., Donnelly et al., 1994Jump To The Next Citation Point; Foukal, 1996), the space-based Mg II core-to-wing ratio as an index of solar UVI (e.g., Donnelly et al., 1994; Viereck and Puga, 1999; Snow et al., 2005) and many others.

All the above indices are closely correlated to sunspot numbers on the solar-cycle scale, but may depict quite different behavior on short or long timescales.

2.2.2 Indirect indices

Sometimes quantitative measures of solar-variability effects are also considered as indices of solar activity. These are related not to solar activity per se, but rather to its effect on different environments. Accordingly, such indices are called indirect, and can be roughly divided into terrestrial/geomagnetic and heliospheric/interplanetary.

Geomagnetic indices quantify different effects of geomagnetic activity ultimately caused by solar variability, mostly by variations of solar-wind properties and the interplanetary magnetic field. For example, the aa-index, which provides a global index of magnetic activity relative to a quiet-day curve for a pair of antipodal magnetic observatories (in England and Australia), is available from 1868 (Mayaud, 1972). An extension of the geomagnetic series is available from the 1840s using the Helsinki Ak(H) index (Nevanlinna, 2004a,b). Although the homogeneity of the geomagnetic series is compromised (e.g., Lukianova et al., 2009; Love, 2011), it still remains an important indirect index of solar activity. A review of the geomagnetic effects of solar activity can be found, e.g., in Pulkkinen (2007). It is noteworthy that geomagnetic indices, in particular low-latitude aurorae (Silverman, 2006), are associated with coronal/interplanetary activity (high-speed solar-wind streams, interplanetary transients, etc.) that may not be directly related to the sunspot-cycle phase and amplitude, and therefore serve only as an approximate index of solar activity. One of the earliest instrumental geomagnetic indices is related to the daily magnetic declination range, the range of diurnal variation of magnetic needle readings at a fixed location, and is available from the 1780s (Nevanlinna, 1995). However, this data exists as several fragmentary sets, which are difficult to combine into a homogeneous data series.

Heliospheric indices are related to features of the solar wind or the interplanetary magnetic field measured (or estimated) in the interplanetary space. For example, the time evolution of the total (or open) solar magnetic flux is extensively debated (e.g., Lockwood et al., 1999Jump To The Next Citation Point; Wang et al., 2005Jump To The Next Citation Point; Krivova et al., 2007Jump To The Next Citation Point).

A special case of heliospheric indices is related to the galactic cosmic-ray intensity recorded in natural terrestrial archives. Since this indirect proxy is based on data recorded naturally throughout the ages and revealed now, it makes possible the reconstruction of solar activity changes on long timescales, as discussed in Section 3.

2.3 Solar activity observations in the pre-telescopic epoch

Instrumental solar data is based on regular observation (drawings or counting of spots) of the sun using optical instruments, e.g., the telescope used by Galileo in the early 17th century. These observations have mostly been made by professional astronomers whose qualifications and scientific thoroughness were doubtless. They form the basis of the Group sunspot-number series (Hoyt and Schatten, 1998Jump To The Next Citation Point), which can be more-or-less reliably extended back to 1610 (see discussion in Section 2.2.1). However, some fragmentary records of qualitative solar and geomagnetic observations exist even for earlier times, as discussed below (Sections

2.3.1 Instrumental observations: Camera obscura

The invention of the telescope revolutionized astronomy. However, another solar astronomical instrument, the camera obscura, also made it possible to provide relatively good solar images and was still in use until the late 18th century. Camera obscuras were known from early times, and they have been used in major cathedrals to define the sun’s position (see the review by Vaquero, 2007Jump To The Next Citation Point; Vaquero and Vázquez, 2009). The earliest known drawing of the solar disc was made by Frisius, who observed the solar eclipse in 1544 using a camera obscura. That observation was performed during the Spörer minimum and no spots were observed on the sun. The first known observation of a sunspot using a camera obscura was done by Kepler in May 1607, who erroneously ascribed the spot on the sun to a transit of Mercury. Although such observations were sparse and related to other phenomena (solar eclipses or transits of planets), there were also regular solar observations by camera obscura. For example, about 300 pages of logs of solar observations made in the cathedral of San Petronio in Bologna from 1655 – 1736 were published by Eustachio Manfredi in 1736 (see the full story in Vaquero, 2007).

Therefore, observations and drawings made using camera obscura can be regarded as instrumental observations.

2.3.2 Naked-eye observations

Even before regular professional observations performed with the aid of specially-developed instruments (what we now regard as scientific observations) people were interested in unusual phenomena. Several historical records exist based on naked-eye observations of transient phenomena on the sun or in the sky.

From even before the telescopic era, a large amount of evidence of spots being observed on the solar disc can be traced back as far as to the middle of the 4th century BC (Theophrastus of Athens). The earliest known drawing of sunspots is dated to December 8, 1128 AD as published in “The Chronicle of John of Worcester” (Willis and Stephenson, 2001). However, such evidence from occidental and Moslem sources is scarce and mostly related to observations of transits of inner planets over the sun’s disc, probably because of the dominance of the dogma on the perfectness of the sun’s body, which dates back to Aristotle’s doctrine (Bray and Loughhead, 1964). Oriental sources are much richer for naked-eye sunspot records, but that data is also fragmentary and irregular (see, e.g., Clark and Stephenson, 1978Jump To The Next Citation Point; Wittmann and Xu, 1987; Yau and Stephenson, 1988). Spots on the sun are mentioned in official Chinese and Korean chronicles from 165 BC to 1918 AD. While these chronicles are fairly reliable, the data is not straightforward to interpret since it can be influenced by meteorological phenomena, e.g., dust loading in the atmosphere due to dust storms (Willis et al., 1980) or volcanic eruptions (Scuderi, 1990) can facilitate sunspots observations. Direct comparison of Oriental naked-eye sunspot observations and European telescopic data shows that naked-eye observations can serve only as a qualitative indicator of sunspot activity, but can hardly be quantitatively interpreted (see, e.g., Willis et al., 1996, and references therein). Moreover, as a modern experiment of naked-eye observations (Mossman, 1989) shows, Oriental chronicles contain only a tiny (1 1 ∕200– ∕1000) fraction of the number of sunspots potentially visible with the naked eye (Eddy et al., 1989). This indicates that records of sunspot observations in the official chronicles were highly irregular (Eddy, 1983Jump To The Next Citation Point) and probably dependent on dominating traditions during specific historical periods (Clark and Stephenson, 1978). Although naked-eye observations tend to qualitatively follow the general trend in solar activity according to a posteriori information (e.g., Vaquero et al., 2002), extraction of any independent quantitative information from these records seems impossible.

Visual observations of aurorae borealis at middle latitudes form another proxy for solar activity (e.g., Siscoe, 1980; Schove, 1983; Křivský, 1984; Silverman, 1992Jump To The Next Citation Point; Schröder, 1992; Lee et al., 2004; Basurah, 2004; Vázquez and Vaquero, 2010). Fragmentary records of aurorae can be found in both occidental and oriental sources since antiquity. The first known dated notation of an aurora is from March 12, 567 BC from Babylon (Stephenson et al., 2004). Aurorae may appear at middle latitudes as a result of enhanced geomagnetic activity due to transient interplanetary phenomena. Although auroral activity reflects coronal and interplanetary features rather than magnetic fields on the solar surface, there is a strong correlation between long-term variations of sunspot numbers and the frequency of aurora occurrences. Because of the phenomenon’s short duration and low brightness, the probability of seeing aurora is severely affected by other factors such as the weather (sky overcast, heat lightnings), the Moon’s phase, season, etc. The fact that these observations were not systematic in early times (before the beginning of the 18th century) makes it difficult to produce a homogeneous data set. Moreover, the geomagnetic latitude of the same geographical location may change quite dramatically over centuries, due to the migration of the geomagnetic axis, which also affects the probability of watching aurorae (Siscoe and Verosub, 1983; Oguti and Egeland, 1995). For example, the geomagnetic latitude of Seoul (37.5° N 127° E), which is currently less than 30°, was about 40° a millennium ago (Kovaltsov and Usoskin, 2007). This dramatic change alone can explain the enhanced frequency of aurorae observations recorded in oriental chronicles.

2.3.3 Mathematical/statistical extrapolations

Due to the lack of reliable information regarding solar activity in the pre-instrumental era, it seems natural to try to extend the sunspot series back in time, before 1610 AD, by means of extrapolating its statistical properties. Indeed, numerous attempts of this kind have been made even recently (e.g., Nagovitsyn, 1997; de Meyer, 1998Jump To The Next Citation Point; Rigozo et al., 2001). Such models aim to find the main feature of the actually-observed sunspot series, e.g., a modulated carrier frequency or a multi-harmonic representation, which is then extrapolated backwards in time. The main disadvantage of this approach is that it is not a reconstruction based upon measured or observed quantities, but rather a “post-diction” based on extrapolation. This method is often used for short-term predictions, but it can hardly be used for the reliable long-term reconstruction of solar activity. In particular, it assumes that the sunspot time series is stationary, i.e., a limited time realization contains full information on its future and past. Clearly such models cannot include periods exceeding the time span of observations upon which the extrapolation is based. Hence, the pre- or post-diction becomes increasingly unreliable with growing extrapolation time and its accuracy is hard to estimate.

Sometimes a combination of the above approaches is used, i.e., a fit of the mathematical model to indirect qualitative proxy data. In such models a mathematical extrapolation of the sunspot series is slightly tuned and fitted to some proxy data for earlier times. For example, Schove (1955Jump To The Next Citation Point, 1979) fitted the slightly variable but phase-locked carrier frequency (about 11 years) to fragmentary data from naked-eye sunspot observations and auroral sightings. The phase locking is achieved by assuming exactly nine solar cycles per calendar century. This series, known as Schove series, reflects qualitative long-term variations of the solar activity, including some grand minima, but cannot pretend to be a quantitative representation in solar activity level. The Schove series played an important historical role in the 1960s. In particular, a comparison of the Δ14C data with this series succeeded in convincing the scientific community that secular variations of 14C in tree rings have solar and not climatic origins (Stuiver, 1961). This formed a cornerstone of the precise method of solar-activity reconstruction, which uses cosmogenic isotopes from terrestrial archives. However, attempts to reconstruct the phase and amplitude of the 11-year cycle, using this method, were unsuccessful. For example, Schove (1955) made predictions of forthcoming solar cycles up to 2005, which failed. We note that all these works are not able to reproduce, for example, the Maunder minimum (which cannot be represented as a result of the superposition of different harmonic oscillations), yielding too high sunspot activity compared to that observed. From the modern point of view, the Schove series can be regarded as archaic, but it is still in use in some studies.

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, 1952; Vitinsky, 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 Point; Bracewell, 1986; Kurths and Ruzmaikin, 1990Jump To The Next Citation Point; de Meyer, 1998Jump To The Next Citation Point; Mininni et al., 2001Jump To The Next Citation Point), but this is only a mathematical representation. A detailed review of solar cyclic variability can be found in (Hathaway, 2010).

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 Point, 1983). 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 4.2). 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 Point; Sokoloff, 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 Point; Kremliovsky, 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., 2001b, 2002a, 2003b; Zolotova 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).

The long-term change (trend) in the Schwabe cycle amplitude is known as the secular Gleissberg cycle (Gleissberg, 1939) with the mean period of about 90 years. 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, 1971; Kuklin, 1976; Ogurtsov et al., 2002Jump To The Next Citation Point).

Longer (super-secular) cycles cannot be studied using direct solar observations, but only indicatively by means of indirect proxies such as cosmogenic isotopes discussed in Section 3. Analysis of the proxy data also yields the Gleissberg secular cycle (Feynman and Gabriel, 1990Jump To The Next Citation Point; Peristykh and Damon, 2003Jump To The Next Citation Point), but the question of its phase locking and persistency/intermittency still remains open. Several longer cycles have been found in the 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 various cosmogenic data (e.g., Suess, 1980; Sonett and Finney, 1990Jump To The Next Citation Point; Zhentao, 1990; Usoskin 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 Point; Sonett and Finney, 1990; Vasiliev and Dergachev, 2002Jump To The Next Citation Point; Steinhilber et al., 2012Jump To The Next Citation Point; Abreu et al., 2012Jump 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 Point; Damon and Sonett, 1991Jump To The Next Citation Point; Vasiliev and Dergachev, 2002Jump To The Next Citation Point). However, the non-solar 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, 1990; Oliver and Ballester, 1996; Frick 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 Point; Sonett, 1983; Vitinsky 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, 1994; Weiss and Tobias, 2000Jump To The Next Citation Point; Charbonneau, 2001Jump To The Next Citation Point; Mininni 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, 1994; Sello, 2000Jump To The Next Citation Point; Polygiannakis 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 cycle formation (e.g., Moss et al., 2008Jump To The Next Citation Point; Käpylä et al., 2012). An old idea of the possible planetary influence on the dynamo has received a new pulse recently with some unspecified torque effect on the assumed quasi-rigid non-axisymmetric tahocline (Abreu et al., 2012). If confirmed this idea would imply a significant multi-harmonic driver of the solar activity, but the question is still open. Different numeric tests, such as an analysis of the Lyapunov exponents (Ostriakov and Usoskin, 1990Jump To The Next Citation Point; Mundt et al., 1991Jump To The Next Citation Point; Kremliovsky, 1995Jump To The Next Citation Point; Sello, 2000Jump To The Next Citation Point), Kolmogorov entropy (Carbonell et al., 1994Jump To The Next Citation Point; Sello, 2000) and Hurst exponent (Ruzmaikin et al., 1994; Oliver 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 Point; Ostriakov and Usoskin, 1990; Morfill et al., 1991; Mundt et al., 1991; Rozelot, 1995; Salakhutdinova, 1999; Serre and Nesme-Ribes, 2000; Hanslmeier et al., 2013). 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., 1993, 1994), and the results were strongly dependent on the choice of filtering methods (Price et al., 1992). Developing this approach, Mininni et al. (2000, 2001Jump 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, 2003; Charbonneau, 2010Jump To The Next Citation Point), which include stochastic processes (e.g., Weiss et al., 1984; Feynman and Gabriel, 1990Jump To The Next Citation Point; Schmalz and Stix, 1991; Moss et al., 1992; Hoyng, 1993Jump To The Next Citation Point; Brooke and Moss, 1994; Lawrence et al., 1995; Schmitt et al., 1996Jump To The Next Citation Point; Charbonneau and Dikpati, 2000Jump To The Next Citation Point; Brandenburg 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 Point; Ossendrijver et al., 1996Jump To The Next Citation Point; Brandenburg and Spiegel, 2008Jump To The Next Citation Point; Moss et al., 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.

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, 1927; Newton, 1928; Gleissberg, 1948; Vitinsky, 1965) and has greatly intensified recently with a hundred of journal articles being published to predict the solar cycle No. 24 maximum (see, e.g., the review by Pesnell, 2012Jump To The Next Citation Point), following the boost of space-technology development and increasing debates on solar-terrestrial relations. In fact, the situation has not been improved since the previous cycle, No. 23. The predictions for the peak sunspot number of solar cycle No. 24 range by a factor of 5, between 40 and 200, reflecting the lack of a reliable consensus method (Tobias et al., 2006Jump To The Next Citation Point). Detailed review of the solar activity prediction methods and results have been recently provided by (Hathaway, 2009Jump To The Next Citation Point; Petrovay, 2010; Pesnell, 2012Jump To The Next Citation Point).

A detailed classification of the prediction methods is given by Pesnell (2012) who separates climatology, precursor, theoretical (dynamo model), spectral, neural network, and stock market prediction methods. All prediction methods can be generically divided into precursor and statistical (including the majority of the above classifications) techniques or their combinations (Hathaway et al., 1999Jump To The Next Citation Point). The fact that the prediction of solar cycle is not improved with adding more data (the new solar cycle) suggests that such methods are not able to give reliable prognoses.

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, 2009), 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, 1998; Hathaway et al., 1999; Li et al., 2001; Usoskin and Mursula, 2003Jump To The Next Citation Point; Kane, 2007) because of chaotic/stochastic behavior (see Section 2.4.2).

A new method based on sophisticated dynamo numerical simulations emerges (e.g., Dikpati and Gilman, 2006; Dikpati et al., 2008; Choudhuri et al., 2007; Jiang et al., 2007), but the results are contradictory with each other. Prospectives of this approach are also not clear because of the stochastic component, which drives the dynamo out of the deterministic regime, and uncertainties in the input parameters (Tobias et al., 2006Jump To The Next Citation Point; Bushby and Tobias, 2007; Karak and Nandy, 2012).

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; Tobias et al., 2006). 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.

Note that several “predictions” of the general decline of the coming solar activity have been made recently (Solanki et al., 2004Jump To The Next Citation Point; Abreu et al., 2008Jump To The Next Citation Point; Lockwood et al., 2011Jump To The Next Citation Point), however, these are not really true predictions but rather the acknowledge of the fact that the Modern Grand maximum (Usoskin et al., 2003cJump To The Next Citation Point; Solanki et al., 2004Jump To The Next Citation Point) must cease. Similar caution can be made about predictions of a Grand minimum (e.g., Lockwood et al., 2011Jump To The Next Citation Point; Miyahara et al., 2010) – a grand minimum should appear soon or later, but presently we are hardly able to predict its occurrence.

2.5 Summary

In this section, the concept of solar activity and quantifying indices is discussed, as well as the main features of solar-activity temporal behavior.

The concept of solar activity is quite broad and covers non-stationary and non-equilibrium (often eruptive) processes, in contrast to the “quiet” sun concept, and their effects upon the terrestrial and heliospheric environment. Many indices are used to quantify different aspects of variable solar activity. Quantitative indices include direct (i.e., related directly to solar variability) and indirect (i.e., related to terrestrial and interplanetary effects caused by solar activity), they can be physical or synthetic. While all indices depict the dominant 11-year cyclic variability, their relationships on other timescales (short scale or long-term trends) may vary to a great extent.

The most common and the longest available index of solar activity is the sunspot number, which is a synthetic index and is very useful for the quantitative representation of overall solar activity outside the grand minimum. During the grand Maunder minimum, however, it may give only a clue about solar activity whose level may drop below the sunspot formation threshold. The sunspot number series is available for the period from 1610 AD, after the invention of the telescope, and covers, in particular, the Maunder minimum in the late 17th century. Fragmentary non-instrumental observations of the sun before 1610, while giving a possible hint of relative changes in solar activity, cannot be interpreted in a quantitative manner.

Solar activity in all its manifestations is dominated by the 11-year Schwabe cycle, which has, in fact, a variable length of 9 – 14 years for individual cycles. The amplitude of the Schwabe cycle varies greatly – from the almost spotless Maunder minimum to the very high cycle 19, possibly in relation to the Gleissberg or secular cycle. Longer super-secular characteristic times can also be found in various proxies of solar activity, as discussed in Section 4.

Solar activity contains essential chaotic/stochastic components, that lead to irregular variations and make the prediction of solar activity for a timescale exceeding one solar cycle impossible.

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