A history of solar activity over millennia

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 2004; Hathaway 2015).

2.2.1.1 Wolf (WSN) and International (ISN) sunspot number series

The concept of the sunspot number was developed by Rudolf Wolf of the Zürich observatory in the middle of the nineteenth century. The sunspot series, initiated by him, is called the Zürich or Wolf sunspot number (WSN) series. The relative sunspot number \(R_{z}\) is defined as

$$\begin{aligned} R_z = k\, (10\, G + N) , \end{aligned}$$

(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.

Open image in new window

The value of \(R_{z}\) (see Fig. 1a) 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 \(R_{z}\) 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 \(R_{z}\) can be up to 25% (e.g., Vitinsky et al. 1986). 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 (Clette et al. 2014) but the correction for this effect is not clear and leads to uncertainties (Lockwood et al. 2014a; Friedli 2016). 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 2012), 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. (1986), Wilson (1998), Letfus (1999), Svalgaard (2012), Clette et al. (2014). The quality of the Wolf series before 1749 is rather poor and hardly reliable (Hoyt et al. 1994; Hoyt and Schatten 1998; Hathaway and Wilson 2004).

The main problem of the WSN was a lack of documentation so that only the final product was available without information of the raw data, that made a full revision of the series hardly possible. Although this information does exist, it was hidden in hand-written notes of Rudolf Wolf and his successors. The situation is being improved now with an effort of the Rudolf Wolf Gesellschaft (http://www.wolfinstitute.ch) to scan and digitize the original Wolf’s notes (Friedli 2016).

Note that the sun has been routinely photographed since 1876 so that full information on daily sunspot activity is available (the Greenwich series) 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 (ISN) \(R_{i}\), 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 (Eq. 1) 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. The ISN (see SILSO, http://sidc.be/silso/datafiles) has been recently updated to version 2 with corrections to some known inhomogeneities (Clette et al. 2014). A potential user should know that the ISN (v.2) is calibrated to Wolfer, in contrast to earlier WSN and ISN (v.1) calibrated to Wolf. As a result, a constant scaling factor 0.6 should be applied to compare ISN (v.2) to ISN (v.1). The two versions are shown in Fig. 1a. One can see that the ISN v.2 (scaled by the 0.6 factor) is somewhat smaller than v.1 after the 1940s, because of the correction for the Waldmeier discontinuity (see below), but they are nearly identical before that.

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

2.2.1.3 Updates of the series

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 (2009) and original papers by Casas et al. (2006), Vaquero et al. (2005, 2007), Arlt (2008), Arlt (2009). 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 2008, 2009; Arlt and Abdolvand 2011; Arlt et al. 2013) 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 2009; Usoskin et al. 2009c; Arlt and Abdolvand 2011; Arlt et al. 2013)—see a newly built butterfly diagram for solar cycles Nos. 7–10 shown in Fig. 2. 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. (2011) made it possible to revise the pattern of the beginning of the Maunder minimum. Recent corrections to the group number database have been collected (http://haso.unex.es/?q=content/data) by Vaquero et al. (2016) who updated the database of Hoyt and Schatten (1998) by correcting some errors and inexactitudes.

Open image in new window

Several inconsistencies and discontinuities have been found in the existing sunspot series. For instance, Leussu et al. (2013) have shown that the values of WSN before 1848 (when Wolf had started his own observations) were overestimated by \({\approx }20\%\) because of the incorrect k-factor ascribed by Wolf to Schwabe. This error, called the “Wolf discontinuity”, erroneously alters the WSN/ISN series but does not affect the GSN series. Another reported error is the so-called “Waldmeier discontinuity” around 1947 (Clette et al. 2014), related to the fact that Waldmeier had modified the procedure of counting spots, including ‘weighting’ sunspot number, without a proper noticing which led to a greater sunspot number compared to the standard technique. This suggests that the WSN/ISN may be overestimated by 10–20% past 1947 (Clette et al. 2014; Lockwood et al. 2014b), but this does not directly affect the GSN series. As studied by Friedli (2016), this weighting might have been introduced intermittently already in the early twentieth century.

Clette et al. (2014) and Cliver and Ling (2016) proposed, based on the ratio of number of groups reported by Wolfer to that based on the Royal Greenwich Observatory (RGO) data, that there might be a transition in the calibration of GSN around the turn of the nineteenth to twentieth century (or even until 1915) related to inhomogeneous quality of the RGO data used to build the GSN at that period. On the other hand, independent observations of David Hadden from Iowa or Madrid Observatory (Carrasco et al. 2013; Aparicio et al. 2014) reveal that the problem with RGO data is essential only before the 1880s but not after that. Willis et al. (2016) studied RGO data for the years 1874–1885 and found that the database of Hoyt and Schatten (1998) might have slightly underestimated the RGO number of groups during that time. Sarychev and Roshchina (2009) suggested that the RGO data are erroneous for the period 1874–1880 but quite homogeneous after that. Vaquero et al. (2014) reported that ISN values for the period prior to 1850 are discordant with the number of spotless days, and concluded that the problem could be related to the calibration constants by Wolf (as found by Leussu et al. 2013) and to the non-linearity of ISN for low values. Most of these errors affect the WSN/ISN series, while GSN is more robust.

Thus, such inconsistencies should be investigated, and new series, with corrections of the known problems, need to be produced. Several such efforts have been made recently leading to inconsistent solar activity reconstructions. One of the new reconstructions was made by Clette et al. (2014) who introduced a revised version of the ISN (v.2 – see Fig. 1a), correcting the two apparent discontinuities, Wolf and Waldmeier, as described above. In addition, the entire series was rescaled to the reference level of Wolfer, while the ‘classic’ WSN/ISN series was scaled to Wolf. This leads to a constant scaling with the factor \(1.667 = 1/6\) of the ISN_v.2 series with respect to other series. Keeping this scaling in mind, the ISN_v2 series is systematically different from the earlier one after the 1940s, and for a few decades in the mid-nineteenth century. This was not a fundamental revision by a scaling correction for a couple of errors.

A full revision of the GSN series was performed by Svalgaard and Schatten (2016) who used the number of sunspot group by Hoyt and Schatten (1998) but applied different method to construct the new GSN series. They also used a daisy-chain linear regression to calibrate different observers but doing it in several steps. A few key observers, called the ‘backbones’, were selected, and other observers were re-normalized to the ‘backbones’ using linear regressions. Then the ‘backbones’ were calibrated to each other, again using the linear scaling. Before 1800, when the daisy-chain calibration cannot be directly applied, two other methods, the ‘high-low’ (observers reporting larger number of groups were favored over those reporting smaller number of groups) and ‘brightest star’ (only the highest daily number of sunspot groups per year was considered) methods. This GSN series, called S16, is shown in Fig. 1b as the blue dotted curve. It suggests a much higher, than usually thought, level of solar activity in the 19th and especially 18th centuries, comparable to that during the mid-twentieth century. As a result of the ‘brightest star’ method, it yields moderate values during the Maunder minimum in contrast to the present paradigm of virtually no sunspots (Eddy 1976; Ribes and Nesme-Ribes 1993; Hoyt and Schatten 1996; Usoskin et al. 2015).

The use of the traditional k-factor method of linear scaling for inter-calibration of solar observers has been found invalid (Dudok et al. 2016; Lockwood et al. 2016c, b; Usoskin et al. 2016b), and a need for a modern non-parametric method has emerged. This is illustrated by Fig. 3 which shows the ratio of the sunspot group number reported by Wolfer, \(G_\mathrm{Wolfer}\), to that by Wolf, \(G_\mathrm{Wolf}\), for days when both reports are available. This ratio obviously depends on the level of activity, being about two for low-activity days \(G_\mathrm{Wolfer}=1\) and only \({\approx } 1.2\) for high-activity days. The ratio is strongly non-linear due to the fact that large sunspot groups dominate during periods of high activity. The horizontal dash-dotted line denotes the constant scaling k-factor of 1.667 used earlier (Clette et al. 2014) between Wolf and Wolfer. One can see that the use of the k-factor leads to a significant, by \({\approx } 40\%\), over-correction of the numbers from Wolf when scaling them to Wolfer.

Several new methods, free of the linear assumption, have been proposed recently. One, called the ADF-method (Usoskin et al. 2016b), is based on a comparison of statistics of the active-day fraction (ADF) in the sunspot (group) records of an observer with that of the reference data-set (the RGO record of sunspot groups for the period 1900–1976). By comparing them, the observational acuity threshold \(A_\mathrm{th}\) can be found defined so that the observer is supposed to report all the sunspot groups with area greater than the threshold and to miss smaller groups. This threshold characterizes the quality of the observer and is further used to calibrate his/her records. The values of the defined thresholds for some principal observers of the eighteenth to nineteenth century are given in Table 1. Based on the defined observational acuity thresholds for each observer, a new GSN series was constructed, called U16, is depicted in Fig. 1b as the red curve. It lies lower than GSN S16 around solar maxima but slightly higher than HS98, in the eighteenth–nineteenth centuries.

Open image in new window

Table 1

Values of the observational acuity threshold for the key observers of the 18–19th centuries (modified after Usoskin et al. 2016b)

Observer	\(T_\mathrm{obs}\)	N	\(SA_\mathrm{th}\)
Quimby	1889–1921	10830	22 \(\left( ^{28}_{16}\right) \)
Wolfer	1876–1928	7165	6 \(\left( ^{12}_{\, 0}\right) \)
Winkler\(^a\)	1882–1910	4812	53 \(\left( ^{66}_{45}\right) \)
Tacchini	1871–1900	6235	10 \(\left( ^{14}_{\, 7}\right) \)
Leppig	1867–1881	2463	45 \(\left( ^{33}_{55}\right) \)
Spörer	1861–1893	5386	3 \(\left( ^{5}_{0}\right) \)
Weber	1859–1883	6981	22 \(\left( ^{28}_{16}\right) \)
Wolf	1848–1893	8102	45 \(\left( ^{53}_{36}\right) \)
Shea	1847–1866	5538	25 \(\left( ^{33}_{18}\right) \)
Schmidt	1841–1883	6887	10 \(\left( ^{15}_{\,\,6}\right) \)
Schwabe	1825–1867	8570	13 \(\left( ^{18}_{\,\,8}\right) \)
Pastorff	1819–1833	1451	5 \(\left( ^{10}_{0}\right) \)
Stark\(^a\)	1813–1836	2406	60 \(\left( ^{70}_{50}\right) \)
Derfflinger\(^a\)	1802–1824	346	50 \(\left( ^{80}_{40}\right) \)
Herschel\(^a\)	1794–1818	344	23 \(\left( ^{35}_{10}\right) \)
Horrebow	1761–1776	1365	75 \(\left( ^{95}_{60}\right) \)
Schubert	1754–1758	404	10 \(\left( ^{16}_{\,5}\right) \)

\(^\mathrm{a}\) The observational threshold is likely overestimated

Columns are: Name of the observer; Period of observation \(T_\mathrm{obs}\); Number of observational days N; The threshold area \(A_\mathrm{th}\) in uncorrected msd; values in parentheses denote the upper and lower 1\(\sigma \) bound

Another new method was proposed by Friedli (2016) who revised the WSN series basing on recently digitized Wolf’s original notes and using the relation between numbers of groups and individual spots. This series appears consistent with the GSN Us16 series and close to the classical ISN series but is lower than the GSN S16 series.

Several attempts to test/validate different sunspot reconstructions using indirect proxies yielded indicative results: tests based on cosmogenic radionuclides (Asvestari et al. 2017), including \(^{44}\)Ti in measured in the fallen meteorites (see Fig. 19) as well as geomagnetic and heliospheric proxies (Lockwood et al. 2016a, b) favor the ‘lower’ reconstructions (Hoyt and Schatten 1998; Usoskin et al. 2016b) against the ‘high’ reconstructions (Svalgaard and Schatten 2016). On the other hand, comparison with the solar open magnetic field models (Owens et al. 2016) cannot distinguish between different series.

The current situation with the sunspot number series is developing quickly and can hardly be resolved now. The old ‘classical’ WSN and GSN series need to be corrected for apparent inhomogeneities. Yet, newly emerging revisions of the sunspot series are mutually inconsistent and require efforts of the solar community on a consensus approach. On the other hand, the scientific community needs a ‘consensus’ series of solar activity, and the work in this direction is under way. Currently, probably the most reliable information of solar activity before the twentieth century can be obtained from cosmogenic isotope data (Sect. 3). This review will be updated as the situation progresses.

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.

Several geomagnetic activity indices have been proposed recently. One is the IDV (interdiurnal variability) index (Svalgaard and Cliver 2005; Lockwood et al. 2013) based on Bartels’ historical u-index of geomagnetic activity, related to the difference between successive daily values of the horizontal or vertical component of the geomagnetic field. Another index is IHV (inter-hourly variability) calculated from the absolute differences between successive hourly values of the horizontal component of the geomagnetic field during night hours to minimize the effect of the daily curve (Svalgaard et al. 2004; Mursula and Martini 2006). Some details of the derivation and use of these indirect indices for long-term solar-activity studies can be found, e.g., in the Living Review by Lockwood (2013).

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. 1999; Wang et al. 2005; Krivova et al. 2007).

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 Sect. 3.

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 eighteenth 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 2007; 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 fourth 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 1978; 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 (\(^{1}{/}_{200}\,\textendash \,^{1}{/}_{1000}\)) 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 1983) 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 1992; 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 and latitude extent 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 eighteenth 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 (\(37.5^{\circ }\) N \(127^{\circ }\) E), which is currently less than \(30^{\circ }\), was about \(40^{\circ }\) 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 1998; Rigozo et al. 2001; Zharkova et al. 2015). 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 (1955, 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 \(\varDelta ^{14}\)C data with this series succeeded in convincing the scientific community that secular variations of \(^{14}\)C 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.

2.4.1 Quasi-periodicities

2.4.2 Randomness versus 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 1990a; Oliver and Ballester 1996; Frick et al. 1997), 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 1965; Sonett 1983; Vitinsky et al. 1986) 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 2000; Charbonneau 2001; 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 2000; 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. 2008; 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). However, this result was criticized by Poluianov and Usoskin (2014) as being an artifact of an inappropriate analysis (aliasing effect of incorrect smoothing). In addition, Cauquoin et al. (2014) have shown that such periodicities were not observed in \(^{10}\)Be data 330 kyr ago.

Different numeric tests, such as an analysis of the Lyapunov exponents (Ostryakov and Usoskin 1990b; Mundt et al. 1991; Kremliovsky 1995; Sello 2000), Kolmogorov entropy (Carbonell et al. 1994; 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., a 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 1990; Ostryakov and Usoskin 1990b; 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, 2001) suggest that one considers 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 2010), which include stochastic processes (e.g., Weiss et al. 1984; Feynman and Gabriel 1990; Schmalz and Stix 1991; Moss et al. 1992; Hoyng 1993; Brooke and Moss 1994; Lawrence et al. 1995; Schmitt et al. 1996; Charbonneau and Dikpati 2000; 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 \(\alpha \)-effect, which is the result of the electromotive force averaged over turbulent vortices, and thus can contain a fluctuating contribution (e.g., Hoyng 1993; Ossendrijver et al. 1996; Brandenburg and Spiegel 2008; Moss et al. 2008). Note that a significant fluctuating component (with the amplitude more than 100% of the regular component) is essential in all these models.

2.4.3 A note on solar activity predictions

Randomness (see Sect. 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., reviews by Pesnell 2012, 2016), 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. 2006). Detailed review of the solar activity prediction methods and results have been recently provided by (Hathaway 2009; Petrovay 2010; Pesnell 2012).

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. 1999). 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 2003; Kane 2007) because of chaotic/stochastic behavior (see Sect. 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 so far 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. 2006; 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 that, even millennia ahead (Zharkova et al. 2015), 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., Sect. 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. 2004; Abreu et al. 2008; Lockwood et al. 2011), however, these are not really true predictions but rather acknowledgements of the fact that the Modern Grand maximum (Usoskin et al. 2003c; Solanki et al. 2004) has ceased. Similar caution can be made about predictions of a Grand minimum (e.g., Lockwood et al. 2011; Miyahara et al. 2010)—a grand minimum should appear soon or later, but presently we are hardly able to predict its occurrence.

3.1.1 Heliospheric modulation of cosmic rays

The flux of cosmic rays (highly energetic fully ionized nuclei) is considered roughly constant (at least at the time scales relevant for the present study) in the vicinity of the Solar system. However, before reaching the vicinity of Earth, galactic cosmic rays experience complicated transport in the heliosphere that leads to modulation of their flux. Heliospheric transport of GCR is described by Parker’s theory (Parker 1965; Toptygin 1985) and includes four basic processes: the diffusion of particles due to their scattering on magnetic inhomogeneities, the convection of particles by out-blowing solar wind, adiabatic energy losses in expanding solar wind, drifts of particles in the magnetic field, including the gradient-curvature drift in the regular heliospheric magnetic field, and the drift along the heliospheric current sheet, which is a thin magnetic interface between the two heliomagnetic hemispheres (Potgieter 2013). Because of variable solar-magnetic activity, CR flux in the vicinity of Earth is strongly modulated (see Fig. 4). The most prominent feature in CR modulation is the 11-year cycle, which is in inverse relation to solar activity. The 11-year cycle in CR is delayed (from a month up to two years) with respect to the sunspots (Usoskin et al. 1998). The time profile of cosmic-ray flux as measured by a neutron monitor (NM) is shown in Fig. 4 (panel b) together with the sunspot numbers (panel a). Besides the inverse relation between them, some other features can also be noted. A 22-year cyclicity manifests itself in cosmic-ray modulation through the alteration of sharp and flat maxima in cosmic-ray data, originated from the charge-dependent drift mechanism. One may also note short-term fluctuations, which are not directly related to sunspot numbers but are driven by interplanetary transients caused by solar eruptive events, e.g., flares or CMEs. An interesting feature is related to the recent decade. The CR flux in 2009 was the highest ever recorded by NMs (Moraal and Stoker 2010), as caused by the favorable heliospheric conditions (unusually weak heliospheric magnetic field and the flat heliospheric current sheet) (McDonald et al. 2010). On the other hand, the sunspot minimum was comparable to other minima. The level of CR modulation during the cycle 24 was moderate, much more shallow than for the previous cycles, reflecting the weak solar cycle 24. For the previous 50 years of high and roughly-stable solar activity, no trends have been observed in CR data; however, as will be discussed later, the overall level of CR has changed significantly on the centurial-millennial timescales.

3.1.2 Geomagnetic shielding

The shielding effect is the strongest at the geomagnetic equator, where the present-day value of \(P_{\mathrm {c}}\) may reach up to 17 GV in the region of India. There is almost no cutoff in the geomagnetic polar regions (\(\lambda _G\ge 60^\circ \)). However, even in the latter case the atmospheric cutoff becomes important, i.e., particles must have rigidity above 0.5 GV in order to initiate the atmospheric cascade which can reach ground (see Sect. 3.1.3).

The geomagnetic field is seemingly stable on the short-term scale, but it changes essentially on centurial-to-millennial timescales (e.g., Korte and Constable 2006). Such past changes can be evaluated based on measurements of the residual magnetization of independently-dated samples. These can be paleo- (i.e., natural stratified archives such as lake or marine sediments or volcanic lava) or archaeological (e.g., clay bricks that preserve magnetization upon baking) samples. Most paleo-magnetic data preserve not only the magnetic field intensity but also the direction of the local field, while archeo-magnetic samples provide information on the intensity only. Using a large database of such samples, it is possible to reconstruct (under reasonable assumptions) the large-scale magnetic field of the Earth. Data available provides good global coverage for the last three millennia, allowing for a reliable paleomagnetic reconstruction of the true dipole moment (DM) or virtual axial dipole moment² (VADM) and its orientation (Licht et al. 2013). Less precise, but still reliable reconstructions of the DM and its orientation are possible for the last seven millennia (Knudsen et al. 2008; Usoskin et al. 2016a). Directional paleomagnetic reconstruction are less reliable on a longer timescale, because of the spatial sparseness of the paleo/archeo-magnetic samples in the earlier part of the Holocene (Korte et al. 2011). Some paleomagnetic reconstructions are shown in Fig. 5. All paleomagnetic models depict a similar long-term trend—an enhanced intensity during the period between 1500 BC and 500 AD and a significantly lower field before that.

Changes in the dipole moment M inversely modulate the flux of CR at Earth, with strong effects in tropical regions and globally. The migration of the geomagnetic axis, which changes the geomagnetic latitude \(\lambda _G\) of a given geographical location is also important; while not affecting the global flux of CR, it can dramatically change the CR effect regionally, especially at middle and high latitude. These changes affect the flux of CR impinging on the Earth’s atmosphere both locally and globally and must be taken into account when reconstructing solar activity from terrestrial proxy data (Usoskin et al. 2008, 2010). Accounting for these effects is quite straightforward provided the geomagnetic changes in the past are known independently, e.g., from archeo and paleo-magnetic studies (Donadini et al. 2010). However, because of progressively increasing uncertainties of paleomagnetic reconstructions back in time, it presently forms the main difficulty for the proxy method on the long-term scale (Snowball and Muscheler 2007), especially in the early part of the Holocene. On the other hand, the geomagnetic field variations are relatively well known for the last few millennia (Genevey et al. 2008; Korte and Constable 2008; Knudsen et al. 2008; Licht et al. 2013).

Open image in new window

3.1.3 Cosmic-ray–induced atmospheric cascade

When an energetic CR particle enters the atmosphere, it first moves straight in the upper layers, suffering mostly from ionization energy losses that lead to the ionization of the ambient rarefied air and gradual deceleration of the particles. However, after traversing some amount of matter (the nuclear interaction mean-free path is on the order of 100 g/cm\(^{2}\) for a proton in the air) the CR particle may collide with a nucleus in the atmosphere, producing a number of secondaries. These secondaries have their own fate in the atmosphere, in particular they may suffer further collisions and interactions forming an atmospheric cascade (e.g., Dorman 2004). Because of the thickness of the Earth’s atmosphere (1033 g/cm\(^{2}\) at sea level) the number of subsequent interactions can be large, leading to a fully-developed cascade (also called an air shower) consisting of secondary rather than primary particles. A schematic view of the atmospheric cascade is shown in Fig. 6. Three main components can be separated in the cascade:

• The “hadronic” nucleonic component is formed by the products of nuclear collisions of primary cosmic rays and their secondaries with the atmospheric nuclei, and consists mostly of superthermal protons and neutrons.

• The “soft” or electromagnetic component consists of electrons, positrons and photons.

• The “hard” or muon component consists mostly of muons; pions are short lived and decay almost immediately upon production, feeding muons and the “soft” component.

The development of the cascade depends mostly on the amount of matter traversed and is usually linked to residual atmospheric depth, which is very close to the static barometric pressure, rather than to the actual altitude, that may vary depending on the exact atmospheric density profile.

Open image in new window

Cosmogenic isotopes are a by-product of the hadronic branch of the cascade (details are given below). Accordingly, in order to evaluate cosmic-ray flux from the cosmogenic isotope data, one needs to know the physics of cascade development. Several models have been developed for this cascade, in particular its hadronic branch with emphasis on the generation of cosmogenic isotope production. The first models were simplified quasi-analytical (e.g., Lingenfelter 1963; O’Brien and Burke 1973) or semi-empirical models (e.g., Castagnoli and Lal 1980). With the fast advance of computing facilities it became possible to exploit the best numerical method suitable for such problems—Monte-Carlo (e.g., Masarik and Beer 1999, 2009; Webber and Higbie 2003; Webber et al. 2007; Usoskin and Kovaltsov 2008b; Kovaltsov and Usoskin 2010; Kovaltsov et al. 2012; Argento et al. 2013). The fact that models, based on different independent Monte-Carlo packages, namely, a general GEANT tool and a specific CORSIKA code, yield similar results provides additional verification of the approach.

3.1.4 Transport and deposition

A scheme for the transport and redistribution of the two most useful cosmogenic isotopes, \(^{14}\)C and \(^{10}\)Be, is shown in Fig. 7. After a more-or-less similar production, the two isotopes follow quite different fates, as discussed in detail in Sects. 3.2.3 and 3.3.3. Therefore, expected terrestrial effects are quite different for the isotopes and comparing them with each other can help in disentangling solar and climatic effects (see Sect. 3.7.3). A reader can find great detail also in a book by Beer et al. (2012).

Open image in new window

3.2.1 Measurements

Radiocarbon is usually measured in tree rings, which allows an absolute dating of the samples by means of dendrochronology. Using a complicated technique, the \(^{14}\)C activity³ A is measured in an independently dated sample, which is then corrected for age as

$$\begin{aligned} A^* = A\cdot \exp {\left( \frac{0.693\, t}{T_{1/2}}\right) }, \end{aligned}$$

(5)

where t and \(T_{1/2}\) are the age of the sample and the half-life of the isotope, respectively. Then the relative deviation from the standard activity \(A_o\) of oxalic acid (the National Bureau of Standards) is calculated:

$$\begin{aligned} \delta ^{14}\mathrm {C} = \left( \frac{A^*-A_o}{A_o}\right) \cdot 1000 . \end{aligned}$$

(6)

After correction for the carbon isotope fractionating (account for the \(^{13}\)C isotope) of the sample, the radiocarbon value of \(\varDelta ^{14}\)C is calculated (see details in Stuiver and Pollach 1977).

$$\begin{aligned} \varDelta ^{14}\mathrm {C} = \delta ^{14}\mathrm {C} - (2\cdot \delta ^{13}\mathrm {C} +50)\cdot (1+\delta ^{14}\mathrm {C}/1000) , \end{aligned}$$

(7)

where \(\delta ^{13}\)C is the per mille deviation of the \(^{13}\)C content in the sample from that in the standard belemnite sample calculated similarly to Eq. (6). The value of \(\varDelta ^{14}\)C (measured in per mille ‰) is further used as the index of radiocarbon relative activity. The series of \(\varDelta ^{14}\)C for the Holocene is presented in Fig. 8a as published by the IntCal13 collaboration of 21 dating laboratories as a result of systematic precise measurements of dated samples from around the world (Reimer et al. 2013) http://www.radiocarbon.org/IntCal13.htm. Panel b depicts the production rate \(Q_\mathrm{14C}\) reconstructed by Roth and Joos (2013) using the most up-to-date carbon cycle model.

Open image in new window

A potentially interesting approach has been made by Lal et al. (2005), who measured the amount of \(^{14}\)C directly produced by CR in polar ice. Although this method is free of the carbon-cycle influence, the first results, while being in general agreement with other methods, are not precise.

3.2.2 Production

The connection between the cosmogenic-isotope–production rate, Q, at a given location (quantified via the geomagnetic latitude \(\lambda _{\mathrm {G}}\)) and the cosmic-ray flux is given by

$$\begin{aligned} Q = \int ^\infty _{P_{\mathrm {c}}(\lambda _{\mathrm {G}})} S(P,\phi )\ Y(P)\ dP , \end{aligned}$$

(9)

where \(P_{\mathrm {c}}\) is the local cosmic-ray–rigidity cutoff (see Sect. 3.1.2), \(S(P,\phi )\) is the differential energy spectrum of CR (see Sect. 3.1.1) and Y(P) is the differential yield function of cosmogenic isotope production, calculated using a Monte-Carlo simulation of the cosmic-ray–induced atmospheric cascade (Kovaltsov et al. 2012; Poluianov et al. 2016). Because of the global nature of the carbon cycle and its long attenuation time, the radiocarbon is globally mixed before the final deposition, and Eq. (9) should be integrated over the globe. The yield function Y(P) of the \(^{14}\)C production is shown in Fig. 9a together with those for \(^{10}\)Be (see Sect. 3.3.2) and for a ground-based neutron monitor (NM), which is the main instrument for studying cosmic-ray variability during the modern epoch. One can see that the yield function increases with the energy of CR. On the other hand, the energy spectrum of CR decreases with energy. Accordingly, the differential production rate (i.e., the product of the yield function and the spectrum, \(F=Y\cdot S\)—the integrand of Eq. 9), shown in Fig. 9b, is more informative. The differential production rate reflects the sensitivity to cosmic rays, and the total production rate is simply an integral of F over energy above the geomagnetic threshold.

Open image in new window

Thanks to the development of atmospheric cascade models (Sect. 3.1.3), there are numerical models that allow one to compute the radiocarbon production rate as a function of the modulation potential \(\phi \) and the geomagnetic dipole moment M. The overall production of \(^{14}\)C is shown in Fig. 10.

The production rate of radiocarbon, \(Q_{^{14}\mathrm {C}}\), can vary as affected by different factors (see, e.g., Damon and Sonett 1991):

• Variations of the cosmic-ray flux on a geological timescale due to the changing galactic background (e.g., a nearby supernova explosion or crossing the dense galactic arm).

• Secular-to-millennial variations are caused by the slowly-changing geomagnetic field. This is an important component of the variability, which needs to be independently evaluated from paleo and archeo-magnetic studies.

• Modulation of cosmic rays in the heliosphere by solar magnetic activity. This variation is the primary aim of the present method.

• Short-term variability of CR on a daily scale (suppression due to interplanetary transients or enhancement due to solar energetic-particle events) can be hardly resolved in radiocarbon data.

Open image in new window

Therefore, the production rate of \(^{14}\)C in the atmosphere can be modelled for a given time (namely, the modulation potential and geomagnetic dipole moment) and location. The global production rate Q is then obtained as a result of global averaging.

Until recently there was a discrepancy between the modeled global-average \(^{14}\)C production rates of \(2.0\,\textendash \,2.3\ \mathrm{atoms}\ \mathrm{cm}^{-2}\ \mathrm{s}^{-1}\) (see, e.g., O’Brien 1979; Masarik and Beer 1999; Goslar 2001; Usoskin et al. 2006b, and references therein) and the steady-state production calculated from the \(^{14}\)C inventory in the carbon-cycle model (see Sect. 3.2.3) being \(1.6{\,\textendash \,}1.8\ \mathrm{atoms}\ \mathrm{cm}^{-2}\ \mathrm{s}^{-1}\) for the pre-industrial period (e.g.,Goslar 2001, and references therein). The situation has been resolved recently with a new numerical model (Kovaltsov et al. 2012; Poluianov et al. 2016). For example, the mean global-averaged \(^{14}\)C production rate for the period 1750–1900 is estimated from measurements as \(1.75 \pm 0.01\ \mathrm{atoms}\ \mathrm{cm}^{-2}\ \mathrm{s}^{-1}\) (Roth and Joos 2013). The production model (Usoskin et al. 2016a; Poluianov et al. 2016) yields for the same period theoretical production rate \(1.71{\,\textendash \,}1.76\ \mathrm{atoms}\ \mathrm{cm}^{-2}\ \mathrm{s}^{-1}\), depending on the solar activity and geomagnetic field reconstructions used, being thus in an excellent agreement with the data.

3.2.3 Transport and deposition

Upon production cosmogenic radiocarbon gets quickly oxidized to carbon dioxide \(\hbox {CO}_{2}\) and takes part in the regular carbon cycle of interrelated systems: atmosphere-biosphere-ocean (Fig. 7). Because of the long residence time, radiocarbon becomes globally mixed in the atmosphere and involved in an exchange with the ocean. It is common to distinguish between an upper layer of the ocean, which can directly exchange \(\hbox {CO}_{2}\) with the air and deeper layers. The measured \(\varDelta ^{14}\)C comes from the biosphere (trees), which receives radiocarbon from the atmosphere. Therefore, the processes involved in the carbon cycle are quite complicated. The carbon cycle is usually described using a box model (Oeschger et al. 1974; Siegenthaler et al. 1980), where it is represented by fluxes between different carbon reservoirs and mixing within the ocean reservoir(s), as shown in Fig. 11. Production and radioactive decay are also included in box models. Free parameters in a typical box model are the \(^{14}\)C production rate Q, the air-sea exchange rate (expressed as turnover rate \(\kappa \)), and the vertical–eddy-diffusion coefficient K, which quantifies ocean ventilation. Starting from the original representation (Oeschger et al. 1974), a variety of box models have been developed, which take into account subdivisions of the ocean reservoir and direct exchange between the deep ocean and the atmosphere at high latitudes. More complex models, including a diffusive approach, are able to simulate more realistic scenarios, but they require knowledge of a large number of model parameters. These parameters can be evaluated for the present time using the bomb test–studying the transport and distribution of the radiocarbon produced during the atmospheric nuclear tests. However, for long-term studies, only the production rate is considered variable, while the gas-exchange rate and ocean mixing are kept constant. Under such assumptions, there is no sense in subdividing reservoirs or processes, and a simple carbon box model is sufficient.

Open image in new window

Using the carbon cycle model and assuming that all its parameters are constant in time, one can evaluate the production rate Q from the measured \(\varDelta ^{14}\)C data. This assumption is well validated for the the Holocene (Damon et al. 1978; Stuiver et al. 1991) as there is no evidence of considerable oceanic change or other natural variability of the carbon cycle (Gerber et al. 2002), and accordingly all variations of \(\varDelta ^{14}\)C predominantly reflect the production rate. This is supported by the strong similarity of the fluctuations of the \(^{10}\)Be data in polar ice cores (Sect. 3.3) compared to \(^{14}\)C, despite their completely different geochemical fate (Bard et al. 1997; Steinhilber et al. 2012). However, the changes in the carbon cycle during the last glaciation and deglaciation were dramatic, especially regarding ocean ventilation; this and the lack of independent information about the carbon cycle parameters, make it hardly possible to qualitatively estimate solar activity from \(^{14}\)C before the Holocene.

A new-generation carbon cycle model has been developed moving from static box-exchange models to a dynamical model. Roth and Joos (2013) presented a fully featured model of intermediate complexity, named Bern3D-LPJ, which includes in addition to the dynamical atmosphere, a 3D dynamic ocean, ocean sediments, and a vegetation models. So far this is the most sophisticated and complete carbon cycle model. A multi-millennial reconstruction of the \(^{14}\)C production rate, obtained as a result of application of this model to the IntCal09 radiocarbon data (Reimer et al. 2009) is shown in Fig. 8b.

First attempts to extract information on production-rate variations from measured \(\varDelta ^{14}\)C were based on simple frequency separations of the signals. All slow changes were ascribed to climatic and geomagnetic variations, while short-term fluctuations were believed to be of solar origin. This was done by removing the long-term trend from the \(\varDelta ^{14}\)C series and claiming the residual as being a series of solar variability (e.g., Peristykh and Damon 2003). This oversimplified approach was natural at earlier times, before the development of carbon cycle models, but later it was replaced by the inversion of the carbon cycle (i.e., the reconstruction of the production rate from the measured \(^{14}\)C concentration). Although mathematically this problem can be solved correctly as a system of linear differential equations, the presence of fluctuating noise with large magnitude makes it not straightforward, since the time derivative cannot be reliably identified leading thus to possible amplification of the high-frequency noise in \(\varDelta ^{14}\)C data. One traditional approach (e.g., Stuiver and Quay 1980) is based on an iterative procedure, first assuming a constant production rate, and then fitting the calculated \(\varDelta ^{14}\)C variations to the actual measurements using a feedback scheme. A concurrent approach based on the presentation of the carbon cycle as a Fourier filter (Usoskin and Kromer 2005) produces similar results. Roughly speaking, the carbon cycle acts as an attenuating and delaying filter for the \(^{14}\)C signal (see Fig. 12). The higher the frequency is, the greater the signal is attenuated. In particular, the large 11-year solar cycle expected in the \(^{14}\)C is attenuated by a factor of hundred in the measured \(\varDelta ^{14}\)C data, making it hardly detectable. Because of the slow oceanic response, the \(^{14}\)C data is also delayed with respect to the production signal. The production rate \(Q_{^{14}\mathrm {C}}\) for the Holocene is shown in Fig. 8 and depicts both short-term fluctuations as well as slower variations, mostly due to geomagnetic field changes (see Sect. 3.2.5).

Open image in new window

3.2.4 The Suess effect and nuclear bomb tests

Unfortunately, cosmogenic \(^{14}\)C data cannot be easily used for the last century, primarily because of the extensive burning of fossil fuels. Since fossil fuels do not contain \(^{14}\)C, the produced \(\hbox {CO}_{2}\) dilutes the atmospheric \(^{14}\hbox {CO}_{2}\) concentration with respect to the pre-industrial epoch. Therefore, the measured \(\varDelta ^{14}\)C cannot be straightforwardly translated into the production rate Q after the late nineteenth century, and a special correction for fossil fuel burning is needed. This effect, known as the Suess effect (e.g., Suess 1955), can be up to −25‰ in \(\varDelta ^{14}\)C in 1950 (Tans et al. 1979), which is an order of magnitude larger than the amplitude of the 11-year cycle of a few ‰. Moreover, while the cosmogenic production of \(^{14}\)C is roughly homogeneous over the globe and time, the use of fossil fuels is highly nonuniform (e.g., de Jong and Mook 1982) both spatially (developed countries, in the northern hemisphere) and temporarily (World Wars, Great Depression, industrialization, etc.). This makes it very difficult to perform an absolute normalization of the radiocarbon production to the direct measurements. Sophisticated numerical models (e.g., Sabine et al. 2004; Mikaloff Fletcher et al. 2006) aim to account for the Suess effect and make good progress. However, the results obtained indicate that the determination of the Suess effect does not yet reach the accuracy required for the precise modelling and reconstruction of the \(^{14}\)C production for the industrial epoch. Note that the atmospheric concentration of another carbon isotope \(^{13}\)C is partly affected by land use, which has also been modified during the last century.

Another anthropogenic activity greatly disturbing the natural variability of \(^{14}\)C is related to the atmospheric nuclear bomb tests actively performed in the 1960s. For example, the radiocarbon concentration nearly doubled in the early 1960s in the northern hemisphere after nuclear tests performed by the USSR and the USA in 1961 (Damon et al. 1978). On one hand, such sources of momentary spot injections of radioactive tracers (including \(^{14}\)C) provide a good opportunity to verify and calibrate the exchange parameters for different carbon-cycle reservoirs and circulation models (e.g., Bard et al. 1987; Sweeney et al. 2007). Thus, the present-day carbon cycle is more-or-less known. On the other hand, the extensive additional production of isotopes during nuclear tests makes it hardly possible to use the \(^{14}\)C as a proxy for solar activity after the 1950s (Joos 1994).

These anthropogenic effects do not allow one to make a straightforward link between pre-industrial data and direct experiments performed during more recent decades. Therefore, the question of the absolute normalization of \(^{14}\)C model is still open (see, e.g., the discussion in Solanki et al. 2004, 2005; Muscheler et al. 2005).

3.2.5 The effect of the geomagnetic field

As discussed in Sect. 3.1.2, knowledge of geomagnetic shielding is an important aspect of the cosmogenic isotope method. Since radiocarbon is globally mixed in the atmosphere before deposition, its production is affected by changes in the geomagnetic dipole moment M, while magnetic-axis migration plays hardly any role in \(^{14}\)C data.

The crucial role of paleomagnetic reconstructions has long been known (e.g., Elsasser et al. 1956; Kigoshi and Hasegawa 1966). Many earlier corrections for possible geomagnetic-field changes were performed by detrending the measured \(\varDelta ^{14}\)C abundance or production rate Q (Stuiver and Quay 1980; Voss et al. 1996; Peristykh and Damon 2003), under the assumption that geomagnetic and solar signals can be disentangled from the production in the frequency domain. Accordingly, the temporal series of either measured \(\varDelta ^{14}\)C or its production rate Q is decomposed into the slow changing trend and faster oscillations. The trend is supposed to be entirely due to geomagnetic changes, while the oscillations are ascribed to solar variability. Such a method, however, obliterates all information on possible long-term variations of solar activity. On the other hand, this also misinterprets the short-term (centennial timescale) variations of the geomagnetic field which are essential (e.g., Licht et al. 2013). Accordingly, the frequency-domain decomposition may lead to erroneous results. A direct correction for the geomagnetic field effect should be used instead.

Simplified empirical correction factors were also often used (e.g., Stuiver and Quay 1980; Stuiver et al. 1991). The modern approach is based on a physics-based model (e.g., Solanki et al. 2004; Vonmoos et al. 2006) and allows the quantitative reconstruction of solar activity, explicitly using independent reconstructions of the geomagnetic field. In this case the major source of errors in solar activity reconstructions is related to uncertainties in the paleomagnetic data (Snowball and Muscheler 2007). These errors are insignificant for the last several millennia (Licht et al. 2013; Usoskin et al. 2016a), but become increasingly important for earlier times.

3.3.1 Measurements

The cosmogenic isotope \(^{10}\)Be is useful for long-term studies of solar activity because of its long half-life of around \(1.5 \times 10^{6}\) years. Its concentration is usually measured in stratified ice cores allowing for independent dating. The \(^{10}\)Be/\(^{9}\)Be ratio needs to be precisely measured at an accuracy better than \(10^{-13}\). This can be done using AMS (Accelerator Mass Spectrometry) technique, which makes the measurements complicated and expensive. Correction for the decay is straightforward and does not include isotope fractionating. From the measured samples, first the \(^{10}\)Be concentration is defined, usually in units of \(10^{4}\) atoms/g. Sometimes, a correction for the snow precipitation amount is considered leading to the observable \(^{10}\)Be flux, which is the number of atoms, precipitating to the surface per cm\(^{2}\) per second.

There exist different \(^{10}\)Be series suitable for studies of long-term solar activity, coming from ice cores in Greenland and Antarctica. They have been obtained from different cores with different resolutions, and include data from Milcent, Greenland (Beer et al. 1983); Camp Century, Greenland (Beer et al. 1988); Dye 3, Greenland (Beer et al. 1990); Dome Concordia and South Pole, Antarctica (Raisbeck et al. 1990); GRIP, Greenland (Yiou et al. 1997); GISP2, Greenland (Finkel and Nishiizumi 1997); Dome Fuji, Antarctica (Horiuchi et al. 2007, 2008; Miyake et al. 2015); Dronning Maud Land, Antarctica (Ruth et al. 2007); NGRIP (North Greenland Ice Core Project), Greenland (Berggren et al. 2009); NEEM (North Greenland Eemian Ice Drilling), Greenland (Sigl et al. 2015); West Antarctic Ice Sheet Divide Ice Core (WAIS/WDC), Antarctica (Sigl et al. 2015), etc.

We note that data on \(^{10}\)Be in other archives, e.g., lake sediments, is usually more complicated to interpret because of the potential influence of the climate (Horiuchi et al. 1999; Belmaker et al. 2008).

Details of the \(^{10}\)Be series and their comparison with each other can be found in Beer (2000), Muscheler et al. (2007), and Beer et al. (2012).

3.3.2 Production

The isotope \(^{10}\)Be is produced as a result of spallation of atmospheric nitrogen and oxygen (carbon is less abundant in the atmosphere and makes a negligible contribution) by the nucleonic component of the cosmic-ray–induced atmospheric cascade (Sect. 3.1.3).

A small contribution may also exist from photo-nuclear reactions (Bezuglov et al. 2012). The cross section (a few mb) of the spallation reactions is almost independent of the energy of impacting particles and has a threshold of about 15 MeV. Thus, the production of \(^{10}\)Be is defined mostly by the multiplicity of the nucleonic component, which increases with the energy of primary cosmic rays (see Fig. 9). Maximum production occurs at an altitude of 10–15 km due to a balance between the total energy of the cascade (which increases with altitude) and the number of secondaries (decreasing with altitude). Most of the global \(^{10}\)Be is produced in the stratosphere (55–70%) and the rest in the troposphere (Lal and Peters 1967; Masarik and Beer 1999, 2009; Usoskin and Kovaltsov 2008b; Kovaltsov and Usoskin 2010).

Computation of \(^{10}\)Be isotope production is straightforward, provided a model of the atmospheric cascade is available. The first consistent model was developed by D. Lal et al. (Bhandari et al. 1966; Lal and Peters 1967; Lal and Suess 1968), using an empirical approach based on fitting simplified model calculations to measurements of the isotope concentrations and “star” (inelastic nuclear collisions) formations in the atmosphere. Next was an analytical model by O’Brien (1979), who solved the problem of the GCR-induced cascade in the atmosphere using an analytical stationary approximation in the form of the Boltzmann equation. Those models were based on calculating the rate of inelastic collisions or “stars” and then applying the mean spallation yield per “star”. A new step in the modelling of isotope production was made by Masarik and Beer (1999), who performed a full Monte-Carlo simulation of a GCR-initiated cascade in the atmosphere and used cross sections of spallation reactions directly instead of the average “star” efficiency. Modern models (Webber and Higbie 2003; Webber et al. 2007; Usoskin and Kovaltsov 2008b; Kovaltsov and Usoskin 2010; Poluianov et al. 2016) are based on a full Monte-Carlo simulation of the atmospheric cascade, using improved cross sections. The global production rate of \(^{10}\)Be is about \(0.02{\,\textendash \,}0.04\ \mathrm{atoms}\ \mathrm{cm}^{-2}\ \mathrm{s}^{-1}\) (Masarik and Beer 1999; Webber et al. 2007; Kovaltsov and Usoskin 2010; Poluianov et al. 2016), which is lower than that for \(^{14}\)C (about \(2\ \mathrm{atoms}\ \mathrm{cm}^{-2}\ \mathrm{s}^{-1}\); see Sect. 3.2.2) by two orders of magnitude. The yield function of \(^{10}\)Be production is shown in Fig. 9a and the differential production rate in Fig. 9b. One can see that the peak of \(^{10}\)Be sensitivity, especially in polar regions, is shifted towards lower energies (below 1 GeV) compared with a neutron monitor. This implies that the \(^{10}\)Be isotope is relatively more sensitive to less energetic CR and is, therefore, more affected by solar energetic particles (Usoskin et al. 2006b). Comparison of model computations with direct beryllium production experiments (Usoskin and Kovaltsov 2008b; Kovaltsov and Usoskin 2010), and also the results of modelling of the short-living \(^{7}\)Be isotope (Usoskin et al. 2009a) suggest that some numerical models (Masarik and Beer 1999; Webber and Higbie 2003; Webber et al. 2007) tend to underestimate the production.

Although the production of \(^{10}\)Be can be more-or-less precisely modelled, a simple normalization “surface”, similar to that shown in Fig. 10 for \(^{14}\)C, is not easy to produce because of partial mixing in the atmosphere (see Sect. 3.3.3). Simplified models, assuming either only global (e.g., Beer 2000) or polar production (Bard et al. 1997; Usoskin et al. 2004), have been used until recently. However, it has been recognized that a more realistic model of the limited atmospheric mixing should be used. Without detailed knowledge of \(^{10}\)Be transport in the atmosphere, it is impossible to relate the quantitatively-measured concentration to the production (as done for \(^{14}\)C using the carbon cycle), and one has to assume that the measured abundance is proportional (with an unknown coefficient) to the production rate in a specific geographical region (see Sect. 3.3.3).

3.3.3 Atmospheric transport

After production, the \(^{10}\)Be isotope has a seemingly simple (Fig. 7) but difficult-to-account-for fate in the atmosphere. Its atmospheric residence time depends on scavenging, stratosphere-troposphere exchange and inter-tropospheric mixing (e.g., McHargue and Damon 1991). Soon after production, the isotope is thought to become attached to atmospheric aerosols and follows their fate (Beer et al. 2012). In addition, it may be removed from the lower troposphere by wet deposition (rain and snow). The mean residence time of the aerosol-bound radionuclide in the atmosphere is quite different for the troposphere, being a few weeks, and stratosphere, where it is one to two years (Raisbeck et al. 1981). Accordingly, \(^{10}\)Be produced in the troposphere is deposited mostly locally, i.e., in the polar regions, while stratospheric \(^{10}\)Be can be partly or totally mixed. In addition, because of the seasonal (usually Spring) intrusion of stratospheric air into the troposphere at mid-latitudes, there is an additional contribution of stratospheric \(^{10}\)Be. Therefore, the measured \(^{10}\)Be concentration (or flux) in polar ice is modulated not only by production but also by climate/precipitation effects (e.g., Steig et al. 1996; Bard et al. 1997). This led Lal (1987) to the extreme conclusion that variations of polar \(^{10}\)Be reflect a meteorological, rather than solar, signal. However, comparison between Greenland and Antarctic \(^{10}\)Be series and between \(^{10}\)Be and \(^{14}\)C data (e.g., Bard et al. 1997; Horiuchi et al. 2008; Beer et al. 2012; Steinhilber et al. 2012) suggests that the beryllium data mostly depicts production variations (i.e., solar signal) on top of which some meteorological effects can be superposed (see also Sect. 3.7.3).

Open image in new window

Since both assumptions of the global and purely-local polar production of \(^{10}\)Be archived in polar ice are over-simplified, several attempts have been made to overcome this problem. For instance, McCracken (2004) proposed several simple mathematical models of partial atmospheric mixing (without division in the troposphere and stratosphere) and compared them with observed data. From this semi-empirical approach McCracken concluded that M2 (full mixing above \(60^{\circ }\) latitude and a limited mixing between \(40^{\circ }\) and \(60^{\circ }\) latitude) is a reasonable model for Antarctica. Vonmoos et al. (2006) assumed that the production of \(^{10}\)Be recorded in Greenland is related to the entire hemisphere in the stratosphere (i.e, global stratospheric mixing) but is limited to latitudes above \(40^{\circ }\) latitude in the troposphere (partial tropospheric mixing). This approach uses either semi-empirical or indirect arguments in choosing the unknown degree of mixing.

Recent efforts in employing modern atmospheric 3D circulation models for simulations of \(^{10}\)Be transport and deposition, including realistic air-mass transport and dry-vs-wet deposition (Field et al. 2006; Heikkilä et al. 2008, 2009), look more promising. An example of \(^{10}\)Be deposition computed on the world grid using the NASA GISS model (Field et al. 2006) is shown in Fig. 13. Precision of the models allows one to distinguish local effects, e.g., for Greenland (Heikkilä et al. 2008). A simulation performed by combining a detailed \(^{10}\)Be-production model with an air-dynamics model can result in an absolute model relating production and deposition of the radionuclide. The validity and usefulness of this approach has been demonstrated by Usoskin et al. (2009a), who directly modeled production (using the CRAC model—Usoskin and Kovaltsov 2008b) and transport (using the GISS ModelE—Koch et al. 2006) of a short-living beryllium isotope \(^{7}\)Be and showed that such a combined model is able to correctly reproduce both the absolute level and temporal variations of the \(^{7}\)Be concentration measured in near ground air around the globe. Keeping in mind the similarity between production and transport of the two beryllium isotopes, \(^{7}\)Be and \(^{10}\)Be, this serves as support for the advanced modelling of \(^{10}\)Be transport. A similar general agreement between measured and modelled seasonal variability has been recently found for \(^{10}\)Be in an Antarctic ice core (Pedro et al. 2011).

3.3.4 Effect of the geomagnetic field

In order to properly account for geomagnetic changes (Sect. 3.1.2), one needs to know the effective region in which the radionuclide is produced before being stored in the archive analyzed. For instance, if the concentration of \(^{10}\)Be measured in polar ice reflects mainly the isotope’s production in the polar atmosphere (as, e.g., assumed by Usoskin et al. 2003c), no strong geomagnetic signal is expected to be observed, since the geographical poles are mostly related to high geomagnetic latitudes. On the other hand, assuming global mixing of atmospheric \(^{10}\)Be before deposition in polar ice (e.g., Masarik and Beer 1999), one expects that only changes in the geomagnetic dipole moment affect will the signal. However, because of partial mixing, which can be different in the stratosphere and troposphere, taking into account migration and displacement of the geomagnetic dipole axis may be essential for a reliable reconstruction of solar variability from \(^{10}\)Be data (McCracken 2004). Therefore, only a full combination of the transport and production models, the latter explicitly including geomagnetic effects estimated from paleomagnetic reconstructions, can adequately account for geomagnetic changes and separate the solar signal. These forms a new generation of physics-based models for the cosmogenic-isotope proxy method. We note that paleomagnetic data should ideally not only provide the dipole moment (VADM or VDM) but should also provide estimates of the geomagnetic axis attitude and displacement of the dipole center (Korte et al. 2011).

3.5.1 Regression models

Mathematical regression is the most apparent and often used (even recently) method of solar-activity reconstruction from proxy data (see, e.g., Stuiver and Quay 1980; Ogurtsov 2004). The reconstruction of solar activity is performed in two consecutive steps. First, a phenomenological regression (either linear or nonlinear) is built between a proxy data set and a direct solar-activity index for the available “training” period (e.g., since 1750 for WSN or since 1610 for GSN). Then this regression is extrapolated backwards to evaluate SN from the proxy data. The main shortcoming of the regression method is that it depends on the time resolution and choice of the “training” period. The former is illustrated by Fig. 14, which shows the scatter plot of the \(^{10}\)Be concentration versus GSN for the annual and 11-year smoothed data. One can see that the slope of the \(^{10}\)Be-vs-GSN relation (about –500 g/atom) within individual cycles is significantly different from the slope of the long-term relation (about –100 g/atom), i.e., individual cycles do not lie on the line of the 11-year averaged cycles. Moreover, the slope of the regression for individual 11-year cycles varies essentially depending on the solar activity level. Therefore, a formal regression built using the annual data for 1610–1985 yields a much stronger GSN-vs-\(^{10}\)Be dependence than for the cycle-averaged data (see Fig. 14b), leading to a potentially-erroneous evaluation of the sunspot number from the \(^{10}\)Be proxy data.

Open image in new window

It is equally dangerous to evaluate other solar/heliospheric/terrestrial indices from sunspot numbers by extrapolating an empirical relation obtained for the last few decades back in time. This is because the last decades (after the 1950s), which are well covered by direct observations of solar, terrestrial and heliospheric parameters, corresponded to a very high level of solar activity. After a steep rise in activity level between the late 19th and mid 20th centuries, the activity remained at a roughly constant high level, being totally dominated by the 11-year cycle without a long-term trends. Accordingly, all empirical relations built based on data for this period are focused on the 11-year variability and can overlook possible long-term trends (Mursula et al. 2003). This may affect all regression-based reconstructions, whose results cannot be independently (directly or indirectly) tested. In particular, this may be related to solar irradiance reconstructions, which are often based on regression-like models, built and verified using data for the last three solar cycles, when there was no strong trend in solar activity.

Open image in new window

As an example let us consider an attempt (Belov et al. 2006) to reconstruct cosmic-ray intensity since 1610 from sunspot numbers using a (nonlinear) regression. The regression between the count rate of a neutron monitor and sunspot numbers (Fig. 4), established for the last 50 years is highly significant. Based on that, Belov et al. (2006) extrapolated the regression back in time to produce a reconstruction of cosmic-ray intensity (quantified in NM count rate) to 1560 (see Fig. 15). One can see that there is no notable long-term trend in the reconstruction, and the fact that all CR maxima essentially lie at the same level, from the Maunder minimum to modern times, is noteworthy. It would be difficult to dispute such a result if there was no direct test for CR levels in the past. Independent reconstructions based on cosmogenic isotopes or theoretical considerations (e.g., Usoskin et al. 2002a; Scherer et al. 2004; Scherer and Fichtner 2004; McCracken and Beer 2007) provide clear evidence that cosmic-ray intensity was essentially higher during the Maunder minimum than nowadays. This example shows how easy it is to overlook an essential feature in a reconstruction based on a regression extrapolated far beyond the period it is based on. Fortunately, for this particular case we do have independent information that can prevent us from making big errors. In many other cases, however, such information does not exist (e.g., for total or spectral solar irradiance), and those who make such unverifiable reconstructions should be careful about the validity of their models beyond the range of the established relations.

3.5.2 Reconstruction of heliospheric parameters

The modulation potential \(\phi \) (see Sect. 3.1.1) is directly related to cosmogenic isotope production in the atmosphere. It is a parameter describing the spectrum of galactic cosmic rays (see the definition and full description of this index in Usoskin et al. 2005) in the force-field approximation and is sometimes used as a stand-alone index of solar (or, actually, heliospheric) activity. We note that, provided the isotope production rate Q is estimated and geomagnetic changes can be properly accounted for, it is straightforward to obtain a time series of the modulation potential, using, e.g., the relation shown in Fig. 10.

Several reconstructions of modulation potential for the last few millenia are shown in Fig. 16. While being quite consistent in the relative changes, they differ in the absolute level and fine details, mostly because of the ambiguity of the exact value of the modulation potential (see discussion in Sect. 3.1.1).

Reconstructions of solar activity often end at this point, representing solar activity by the modulation potential, as some authors (e.g., Beer et al. 2003; Vonmoos et al. 2006; Muscheler et al. 2007) believe that further steps (see Sect. 3.5.3) may introduce additional uncertainties. However, since \(\phi \) is a heliospheric, rather than solar, index, the same uncertainties remain when using it as an index of solar activity. Moreover, the modulation potential is a model-dependent quantity (see discussion in Sect. 3.1.1) and therefore does not provide an unambiguous measure of heliospheric activity. In addition, the modulation potential is not a physical index but rather a formal fitting parameter to describe the GCR spectrum near Earth and, thus, is not a universal solar-activity index.

Open image in new window

Modulation of GCR in the heliosphere (see Sect. 3.1.1) is mostly defined by the turbulent heliospheric magnetic field (HMF), which ultimately originates from the sun and is thus related to solar activity. It has been shown, using a theoretical model of the heliospheric transport of cosmic rays (e.g., Usoskin et al. 2002a), that on the long-term scale (beyond the 11-year solar cycle) the modulation potential \(\phi \) is closely related to the open solar magnetic flux \(F_{o}\), which is a physical quantity describing the solar magnetic variability (e.g., Solanki et al. 2000; Krivova et al. 2007).

Sometimes, instead of the open magnetic flux, the mean HMF intensity at Earth orbit, B, is used as a heliospheric index (Caballero-Lopez and Moraal 2004; McCracken 2007; Steinhilber et al. 2010). Note that B is linearly related to \(F_{o}\) assuming constant solar-wind speed, which is valid on long-term scales. An example of HMF reconstruction for the last 600 years is shown in Fig. 17. In addition, the count rate of a “pseudo” neutron monitor (i.e., a count rate of a neutron monitor if it was operated in the past) is considered as a solar/heliospheric index (e.g., Beer 2000; McCracken and Beer 2007).

Open image in new window

3.5.3 A link to sunspot numbers

The open solar magnetic flux \(F_{o}\) described above is related to the solar surface magnetic phenomena such as sunspots or faculae. Modern physics-based models allow one to calculate the open solar magnetic flux from data of solar observation, in particular sunspots (Solanki et al. 2000, 2002; Krivova et al. 2007; Owens et al. 2012) or geomagnetic activity indices (Lockwood et al. 2014b). Besides the solar active regions, the model includes ephemeral regions. Although these models are based on physical principals, they contain some unknowns like the decay time of the open flux, which cannot be measured or theoretically calculated and has to be found by means of fitting the model to data. This free parameter has been determined by requiring the model output to reproduce the best available data sets for the last 30 years with the help of a genetic algorithm. Inversion of the model, i.e., the computation of sunspot numbers for given \(F_{o}\) values is formally a straightforward solution of a system of linear differential equations, however, the presence of noise in the real data makes it only possible in a numerical-statistical way (see, e.g., Usoskin et al. 2004, 2007). By inverting this model one can compute the sunspot-number series corresponding to the reconstructed open flux, thus forging the final link in a chain quantitatively connecting solar activity to the measured cosmogenic isotope abundance. A sunspot-number series reconstructed for the Holocene using \(^{14}\)C isotope data is shown in Fig. 18.

Open image in new window

As very important for the climate research, the variations of the total solar irradiance (TSI) are sometimes reconstructed from the solar proxy data (Steinhilber et al. 2009; Vieira et al. 2011). However, the absolute range of the TSI variability on the centennial-millennial time scales still remains unknown (Schmidt et al. 2012).

3.7.1 Comparison with direct data

The most direct verification of solar-activity reconstruction is a comparison with the actual GSN sunspot data for the last few centuries. However, regression-based models (see Sect. 3.5.1) cannot be tested in this way, since it would require a long set of independent direct data outside the “training” interval. It is usual to include all available data into the “training” period to increase the statistics of the regression, which rules out the possibility of testing the model. On the other hand, such a comparison to the actual GSN since 1610 can be regarded as a direct test for a physics-based model since it does not include phenomenological links over the same time interval. The period of the last four centuries is pretty good for testing purposes since it includes the whole range of solar activity levels from the nearly spotless Maunder minimum to the modern period of a very active sun. However, because of the uncertainties in the sunspot number series (see Sect. 2.2.1), this method shows only an approximate agreement, and direct sunspot numbers cannot serve as the ultimate basis to verify the cosmogenic-based reconstructions. On the contrary, the latter can be used to verify the sunspot number data.

Models focused on the reconstruction of heliospheric parameters (HMF or the modulation potential \(\phi \)) cannot be verified in this manner since no heliospheric data exists before the middle of the twentieth century. Comparison to direct cosmic-ray data after the 1950s (or, with caveats, after the 1930s—McCracken and Beer 2007) is less conclusive, since the latter are of shorter length and correspond to a period of high solar activity, leading to larger uncertainties during grand minima. Moreover, \(^{14}\)C data cannot be tested in this way because of the anthropogenic (Suess) effect and nuclear tests (Sect. 3.2.4).

It is important that some (semi)empirical relations forming the basis for the proxy method are established for the recent decades of high solar activity. The end of the Modern grand maximum of activity and the current moderate level of activity, characterized by the highest ever observed cosmic ray flux as recorded by ground-based neutron monitors, the very low level of the HMF and geomagnetic activity, should help to verify the connections between solar activity, cosmic ray fluxes, geomagnetic activity, the heliospheric magnetic field, and open field. Since some of these connections are somewhat controversial, these extreme conditions should help to quantify them better.

3.7.2 Meteorites and lunar rocks: A direct probe of the galactic cosmic-ray flux

Another more-or-less direct test of solar/heliospheric activity in the past comes from cosmogenic isotopes measured in lunar rock or meteorites. Cosmogenic isotopes, produced in meteoritic or lunar rocks during their exposure to CR in interplanetary space, provide a direct measure of cosmic-ray flux. Uncertainties due to imprecisely known terrestrial processes, including the geomagnetic shielding and redistribution process, are naturally avoided in this case, since the nuclides are directly produced by cosmic rays in the body of the rock, where they remain until they are measured, without any transport or redistribution. The activity of a cosmogenic isotope in meteorite/lunar rock corresponds to an integral of the balance between the isotope’s production and decay, thus representing the time-integrated CR flux over a period determined by the mean life of the radioisotope. The results of different analyses of measurements of cosmogenic isotopes in meteoritic and lunar rocks show that the average GCR flux remained roughly constant—within 10% over the last million years and within a factor of 1.5 for longer periods of up to \(10^{9}\) years (e.g., Vogt et al. 1990; Grieder 2001).

By means of measuring the abundance of relatively short-lived cosmogenic isotopes in meteorites, which fell through the ages, one can evaluate the variability of the CR flux, since the production of cosmogenic isotopes ceases after the fall of the meteorite. A nearly ideal isotope for studying centurial-scale variability is \(^{44}\)Ti with a half-life of \(59.2 \pm 0.6\) years (a lifetime of about 85 years). The isotope is produced in nuclear interactions of energetic CR with nuclei of iron and nickel in the body of a meteorite (Bonino et al. 1995; Taricco et al. 2006). Because of its mean life, \(^{44}\)Ti is relatively insensitive to variations in cosmic-ray flux on decadal (11-year Schwabe cycle) or shorter timescales, but is very sensitive to the level of CR flux and its variations on a centurial scale. Using a full model of \(^{44}\)Ti production in a stony meteorite (Michel and Neumann 1998) and data on the measured activity of cosmogenic isotope \(^{44}\)Ti in meteorites, which fell during the past 235 years (Taricco et al. 2006), provides a method to test, in a straightforward manner, reconstructions of solar activity after the Maunder minimum. First, the expected \(^{44}\)Ti activity needs to be calculated from the reconstructed series using the modulation potential, and then compared with the results of actual measurements (see Fig. 19). Since the life-time of the \(^{44}\)Ti is much longer than the 11-year cycle, this method does not allow for the reconstruction of solar/heliospheric activity, but it serves as a direct way to test existing reconstructions independently. As shown by Usoskin et al. (2006c), the \(^{44}\)Ti data confirms significant secular variations of the solar magnetic flux during the last century (cf. Lockwood et al. 1999; Solanki et al. 2000; Wang et al. 2005). Moreover, the recent sunspot number reconstructions yielding high solar activity during the 17th and 18th appear inconsistent with the data of \(^{44}\)Ti in meteorites (Fig. 19).

Open image in new window

3.7.3 Comparison between isotopes

As an indirect test of the solar-activity reconstruction, one can compare different isotopes. The idea behind this test is that two isotopes, \(^{14}\)C and \(^{10}\)Be, have essentially different terrestrial fates, so that only the production signal, namely, solar modulation of cosmic rays, can be regarded as common in the two series. Processes of transport/deposition are different (moreover, the \(^{14}\)C series is obtained as an average of the world-wide–distributed samples). The effect of changing geomagnetic fields is also different (although not completely) for the two isotopes, since radiocarbon is globally mixed, while \(^{10}\)Be is only partly mixed before being stored in an archive. Even comparison between data of the same \(^{10}\)Be isotope, but measured in far-spaced ice cores (e.g., Greenland and Antarctica), may help in separating climatic and extraterrestrial factors, since meteorology in the two opposite polar areas is quite different.

The first thorough consistent comparison between \(^{10}\)Be and \(^{14}\)C records for the last millennium was performed by Bard et al. (1997). They assumed that the measured \(^{10}\)Be concentration in Antarctica is directly related to CR variations. Accordingly, \(^{14}\)C production was considered as proportional to \(^{10}\)Be data. Then, applying a 12-box carbon-cycle model, Bard et al. (1997) computed the expected \(\varDelta ^{14}\)C synthetic record. Finally, these \(^{10}\)Be-based \(\varDelta ^{14}\)C variations were compared with the actual measurements of \(\varDelta ^{14}\)C in tree rings, which depicted a close agreement in the profile of temporal variation (coefficient of linear correlation \(r = 0.81\) with exact phasing). Despite some fine discrepancies, which can indicate periods of climatic influence in either (or both) of the series, that result has clearly proven the dominance of solar modulation of cosmogenic nuclide production variations during the last millennium. This conclusion has been confirmed (e.g., Usoskin et al. 2003c; Muscheler et al. 2007) in the sense that quantitative solar-activity reconstructions, based on \(^{10}\)Be and \(^{14}\)C data series for the last millennium, yield very similar results, which differ only in small details. However, a longer comparison over the entire Holocene timescale suggests that, while centennial variations of solar activity reconstructed from the two isotopes are very close to each other, there might be a discrepancy in the very long-term trend (Vonmoos et al. 2006; Inceoglu et al. 2015; Usoskin et al. 2016a), whose nature is not clear (climate changes, geomagnetic effects or model uncertainties).

Recently, Usoskin et al. (2009b) studied the dominance of the solar signal in different cosmogenic isotope data on different time scales. They compared the expected \(^{10}\)Be variations computed from \(^{14}\)C-based reconstruction of cosmic ray intensity with the actually measured \(^{10}\)Be abundance at the sites and found that: (1) There is good agreement between the \(^{14}\)C and \(^{10}\)Be data sets, on different timescales and at different locations, confirming the existence of a common solar signal in both isotope data; (2) The \(^{10}\)Be data are driven by the solar signal on timescales from about centennial to millennial time scales; (3) The synchronization is lost on short (\({<}100\) years) timescales, either due to local climate or chronological uncertainties (Delaygue and Bard 2011) but the solar signal becomes important even at short scales during periods of Grand minima of solar activity, (4) There is an indication of a possible systematic uncertainty in the early Holocene (cf. Vonmoos et al. 2006; Inceoglu et al. 2015; Usoskin et al. 2016a), likely due to a not-perfectly-stable thermohaline circulation. Overall, both \(^{14}\)C- and \(^{10}\)Be-based records are consistent with each other over a wide range of timescales and time intervals.

Thus, comparison of the results obtained from different sources implies that the variations of cosmogenic nuclides on the long-term scale (centuries to millennia) during the Holocene are primarily defined by the solar modulation of CR.

4.2.1 The Maunder minimum

The Maunder minimum (MM) is a representative of grand minima in solar activity (e.g., Eddy 1976), when sunspots have almost completely vanished from the solar surface, while the solar wind kept blowing, although at a reduced pace (Cliver et al. 1998; Usoskin et al. 2001b). As proposed by Lockwood and Owens (2014), the solar wind was uniform and slow, 250–275 km/s, nearly half of the modern time velocity. There is some uncertainty in the definition of the duration of MM: the “formal” duration is 1645–1715 (Eddy 1976), while its deep phase with the absence of apparent sunspot cyclic activity is often considered as 1645–1700, with the low, but very clear, solar cycle of 1700–1712 being ascribed to a recovery or transition phase (Usoskin et al. 2000). MM was amazingly well covered by direct sunspot observations (Hoyt and Schatten 1996)—more than 95% of days have formal records (however many of them are generic) and 30–50% of days have explicit data (Vaquero et al. 2015). The late part of MM after the 1680s is particularly weel covered with direct data from the French school of astronomy (Ribes and Nesme-Ribes 1993). On the other hand, sunspots appeared rarely (during \(\sim \)2% of the days) and seemingly sporadically, without an indication of the 11-year cycle.

Some recent studies suggested that the sunspot activity level might have been underestimated during MM: Zolotova and Ponyavin (2015) proposed that the annual number of sunspot groups was as high as 3–8 (the sunspot number 50–100) during MM, Svalgaard and Schatten (2016) proposed much more modest peak annual number of sunspot groups as 2–3 (25 – 40 in sunspot number) but still too high. These claims were based on the fact that original data include many generic statements of the absence of sunspots for long periods of time, and should be dismissed. However, these statements made by professional astronomers in a dedicated monitoring of the sun, should be considered seriously. A thorough analysis of all the available sunspot data made by applying ‘filters’ of different degrees of strictness was made by Vaquero et al. (2015) who concluded that the level of sunspot activity was indeed very low during MM, even if considering only explicit records. The low level of activity during MM was confirmed also by an aggregate study of other indirect data for that period (Usoskin et al. 2015): while there are known auroral observations during MM, they all are limited to high latitudes (close to the auroral oval), where polar lights occur even without strong geomagnetic storms; data of cosmogenic isotopes \(^{14}\)C measured in tree trunks and \(^{44}\)Ti in fallen meteorites clearly indicate a very high flux of cosmic rays (low solar activity) during MM.

Such low level of activity makes it almost impossible to apply standard methods of time-series analysis to sunspot data during MM (e.g., Frick et al. 1997). Therefore, special methods such as the distribution of spotless days versus days with sunspots (e.g., Harvey and White 1999; Kovaltsov et al. 2004; Vaquero et al. 2014) or an analysis of sparsely-occurring events (Usoskin et al. 2000) should be applied in this case. Using these methods, Usoskin et al. (2001b) have shown that sunspot occurrence during the Maunder minimum was gathered into two large clusters (1652–1662 and 1672–1689), with the mass centers of these clusters being in 1658 and 1679–1680. Together with the sunspot maxima before (1640) and after (1705) the deep Maunder minimum, this implies a dominant 22-year periodicity in sunspot activity throughout the Maunder minimum (Mursula et al. 2001), with a subdominant 11-year cycle emerging towards the end of the Maunder minimum (Ribes and Nesme-Ribes 1993; Mendoza 1997; Usoskin et al. 2000; Vaquero et al. 2015) and becoming dominant again after 1700. Similar behavior of a dominant 22-year cycle and a weak subdominant Schwabe cycle during the Maunder minimum has been found in other indirect solar proxy data: auroral occurrence (Křivský and Pejml 1988; Schlamminger 1990; Silverman 1992) and \(^{14}\)C data (Stuiver and Braziunas 1993; Kocharov et al. 1995; Peristykh and Damon 1998; Miyahara et al. 2006b). This is in general agreement with the concept of “immersion” of 11-year cycles during the Maunder minimum (Vitinsky et al. 1986, and references therein). This concept means that full cycles cannot be resolved and sunspot activity only appears as pulses around cycle-maximum times.

An analysis of \(^{10}\)Be data (Beer et al. 1998) implied that the 11-year cycle was weak but fairly regular during the Maunder minimum, but its phase was inverted (Usoskin et al. 2001b). A recent theoretical study (Owens et al. 2012; Wang and Sheeley 2012) confirms that such a phase change between cosmic rays and solar activity can indeed appear for very weak cycles.

Until recently, it was believed (Vitinsky et al. 1986; Ribes and Nesme-Ribes 1993; Sokoloff and Nesme-Ribes 1994; Usoskin et al. 2000, 2001b; Miyahara et al. 2006b) that transition from the normal high activity to the deep minimum did not have any apparent precursor before MM. However, newly recovered data suggest that the start of the Maunder minimum might had been not very sudden but via a regular cycle of reduced height (Vaquero et al. 2011). A 22-year cycle was dominant in sunspot occurrence during the deep minimum (1645–1700), with the subdominant 11-year cycle, which became visible only in the late phase of the Maunder minimum. There is an indication that the length of solar cycle may slightly extend during and already slightly before a grand minimum (Miyahara et al. 2004; Nagaya et al. 2012), which is in agreement (note that the possible cycle maximum in 1650 discussed there was based on an erroneous data point and should be dismissed) with the results by Vaquero et al. (2015).

The 11-year Schwabe cycle started dominating solar activity after 1700. Recovery of sunspot activity from the deep minimum to normal activity was gradual, passing through a period of nearly-linear amplification of the 11-year cycle.

Although the Maunder minimum is the only one with available direct sunspot observations, its predecessor, the Spörer minimum from 1450–1550, is covered by precise bi-annual measurements of \(^{14}\)C (Miyahara et al. 2006a). An analysis of this data (Miyahara et al. 2006a, b) reveals a similar pattern with the dominant 22-year cycle and suppressed 11-year cycle, thus supporting the idea that the above general scenario may be typical for a grand minimum. A similar pattern has been recently also for an un-named grand minima in the fourth century BC (Nagaya et al. 2012).

A very important feature of sunspot activity during the Maunder minimum was its strong north-south asymmetry, as sunspots were only observed in the southern solar hemisphere during the end of the Maunder minimum (Ribes and Nesme-Ribes 1993; Sokoloff and Nesme-Ribes 1994). This observational fact has led to intensive theoretical efforts to explain a significant asymmetry of the sun’s surface magnetic field in the framework of the dynamo concept (see the review by Sokoloff 2004, and references therein). Note that a recent discovery (Arlt 2008, 2009) of the Staudacher’s original drawings of sunspots in late eighteenth century shows that similarly asymmetric sunspot occurrence existed also in the beginning of the Dalton minimum in 1790s (Usoskin et al. 2009c). However, the northern hemisphere dominated at that period contrary to the situation during the Maunder minimum.

4.2.2 Grand minima on a multi-millennial timescale

A list of 25 grand minima, identified in the quantitative solar-activity reconstruction of the last 11 000 years, shown in Fig. 20, is presented in Table 2 (after Usoskin et al. 2007, 2016a). The cumulative duration of the grand minima is about 1900 years, indicating that the sun in its present evolutionary stage spends \({\sim }{^1/_6}\) (17%) of its time in a quiet state, corresponding to grand minima. Note that the definition of grand minima is quite robust.

It was shown by Usoskin et al. (2014), using \(^{14}\)C data for the last three millennia, that grand minima correspond to a special mode of the solar dynamo which is clearly separated from the main mode of moderate activity. The probability density function (PDF) of the occurrence of decadal sunspot numbers in the reconstruction based on \(^{14}\)C for the last three millennia (Usoskin et al. 2014) is shown in Fig. 21. One can see that the PDF has a clear bimodal structure, where the main mode corresponds to the general mode of moderate activity (20–60 in decadal sunspot numbers), while the secondary maximum represents a statistically different mode of low activity (decadal sunspot numbers below 20) corresponding to Grand minima.

Open image in new window

The question of whether the occurrence of grand minima in solar activity is a regular or chaotic process is important for understanding the action of the solar-dynamo machine. Even a simple deterministic numerical dynamo model can produce events comparable with grand minima (Brandenburg et al. 1989; Käpylä et al. 2016). Such models can also simulate a sequence of grand minima occurrences, which are irregular and seemingly chaotic (e.g., Jennings and Weiss 1991; Tobias et al. 1995; Covas et al. 1998). The presence of long-term dynamics in the dynamo process is often explained in terms of the \(\alpha \)-effect, which, being a result of the electromotive force averaged over turbulent vortices, can contain a fluctuating part (e.g., Hoyng 1993; Ossendrijver et al. 1996) leading to irregularly occurring grand minima (e.g., Brandenburg and Spiegel 2008). The present dynamo models can reproduce almost all the observed features of the solar cycle under ad hoc assumptions (e.g., Pipin et al. 2012), although it is still unclear what leads to the observed variability. Most of these models predict that the occurrence of grand minima is a purely random “memoryless” Poisson-like process, with the probability of a grand minimum occurring being constant at any given time. This unambiguously leads to the exponential shape of the waiting-time distribution (waiting time is the time interval between subsequent events) for grand minima.

Usoskin et al. (2007) performed a statistical analysis of grand minima occurrence time (Table 2) and concluded that their occurrence is not a result of long-term cyclic variations, but is defined by stochastic/chaotic processes. Moreover, waiting-time distribution deviates from the exponential law. This implies that the event occurrence is still random, but the probability is nonuniform in time and depends on the previous history. In the time series it is observed as a tendency of the events to cluster together with a relatively-short waiting time, while the clusters are separated by long event-free intervals (cf. Sect. 4.1). Such behavior can be interpreted in different ways, e.g., self-organized criticality or processes related to accumulation and release of energy. This poses a strong observational constraint on theoretical models aiming to explain the long-term evolution of solar activity (Sect. 4.4.1). However, as discussed by Moss et al. (2008) and Usoskin et al. (2009d), the observed feature can be an artefact of the small statistics (only 27 grand minima are identified during the Holocene), making this result only indicative and waiting for a more detailed investigation.

A histogram of the duration of grand minima from Table 2 is shown in Fig. 22. The mean duration is 70 year but the distribution is bimodal. The minima tend to be either of a short (30–90 years) duration similar to the Maunder minimum, or rather long (\({>}100\) years), similar to the Spörer minimum, in agreement with earlier conclusions (Stuiver and Braziunas 1989). This suggests that grand minima correspond to a special state of the dynamo. Once falling into a grand minimum as a result of a stochastic/chaotic, but non-Poisson process, the dynamo is “trapped” in this state and its behavior is driven by deterministic intrinsic features.

Open image in new window

4.3.1 The modern episode of active sun

In the last decades we were living in a period of a very active sun with a level of activity that is very high for the last few centuries covered by direct solar observation. The sunspot number was growing rapidly between 1900 and 1940, with more than a doubling average group sunspot number, and has remained at that high level until recently (see Fig. 1). Note that growth comes mostly from raising the cycle maximum amplitude, while sunspot activity always returns to a very low level around solar cycle minima. While the average group sunspot number (using GSN) for the period 1750–1900 was \(35 \pm 9\) (\(39 \pm 6\), if the Dalton minimum in 1797–1828 is not counted), it stands high at the level of \(75 \pm 3\) for 1950–2000. Therefore, the modern active sun episode, which started in the 1940s, can be regarded as the modern grand maximum of solar activity, as opposed to a grand minimum (Wilson 1988b). As discussed by Clette et al. (2014, see their Figure 65), the number of spotless days during cycles 12–22 was half of that for another relatively high activity period ca. 1850. This again suggests the uniqueness of the modern grand maximum on the centennial time scale. The reality of the Modern grand maximum was independently confirmed by Ziȩba and Nieckarz (2014) who have shown, by studying active versus passive (spotless) days that cycles 17–23 were more active, compared to cycles 8–15.

Although uncertainties in sunspot numbers during the 18th and 19th centuries (see discussion in Sect. 2.2.1) make it a bit unclear on the centennial time scale, data on cosmogenic isotopes (Usoskin et al. 2003c; Solanki et al. 2004; Inceoglu et al. 2015) imply that such high activity episodes occur quite seldom.

However, as we can securely say now, after the very weak solar minimum in 2008–2009 (e.g., Gibson et al. 2011), solar activity returns to its normal moderate level in cycle # 24. Thus, the high activity episode known as the Modern grand maximum is over.

Is such high solar activity typical or is it something extraordinary? While it is broadly agreed that the modern active sun episode is a special phenomenon, the question of how (a)typical such upward bumps are from “normal” activity is a topic of hot debate.

4.3.2 Grand maxima on a multi-millennial timescale

A quantitative analysis is only possible using proxy data, especially cosmogenic isotope records. Using a physics-based analysis of solar-activity series reconstructed from \(^{10}\)Be data from polar (Greenland and Antarctica) archives, Usoskin et al. (2003c, 2004) stated that the modern maximum is unique in the last millennium. Then, using a similar analysis of the \(^{14}\)C calibrated series, Solanki et al. (2004) found that the modern activity burst is not unique, but a very rare event, with the previous burst occurring about 8 millennia ago. An update (Usoskin et al. 2006a) of this result, using a more precise paleo-magnetic reconstruction by Korte and Constable (2005) since 5000 BC, suggests that an increase of solar activity comparable with the modern episode might have taken place around 2000 BC, i.e., around 4 millennia ago, in agreement with more recent studies by Steinhilber et al. (2012) and Inceoglu et al. (2015). On the other hand, the definition of grand maxima is less robust than grand minima and is sensitive to other parameters such as geomagnetic field data or overall normalization (Usoskin et al. 2016a).

Keeping possible uncertainties in mind, let us consider a list of the largest grand maxima (the 50 year smoothed sunspot number stably exceeding 50), identified for the last eleven millennia using cosmogenic isotope data, as shown in Table 3. A total of 23 grand maxima have been identified with a total duration of around 1400 years, suggesting that the sun spends around 12% of its time in an active state. A statistical analysis of grand-maxima–occurrence time suggests that they do not follow long-term cyclic variations, but a clustering near highs of the Hallstatt cycle is observed (Usoskin et al. 2016a). The distribution of the waiting time between consecutive grand maxima is not unambiguously clear, but also hints at a deviation from exponential law. The duration of grand maxima has a smooth distribution, which nearly exponentially decreases towards longer intervals. Most of the reconstructed grand maxima (about 70%) were not longer than 50 years, and only five grand minima (including the modern one) have been longer than 70 years (cf. Barnard et al. 2011). Note, that the Modern grand maximum is over now and we are living during the epoch of moderate or even weak solar activity.

It is still unclear wether grand maxima correspond to a special state of solar dynamo or rather to a tail of the regular mode. Although there are some indications for the former (Usoskin et al. 2016a), they are inconclusive.

4.4.1 Theoretical constrains

The basic principles of the occurrence of the 11-year Schwabe cycle are more-or-less understood in terms of the solar dynamo, which acts, in its classical form (e.g., Parker 1955), as follows (see detail in Charbonneau 2010). Differential rotation \(\varOmega \) produces a toroidal magnetic field from a poloidal one, while the “\(\alpha \)-effect”, associated with the helicity of the velocity field or Joy’s Law tilt of active regions, produces a poloidal magnetic field from a toroidal one. This classical model results in a periodic process in the form of propagation of a toroidal field pattern in the latitudinal direction (the “butterfly diagram”). As evident from observation, the solar cycle is far from being a strictly periodic phenomenon, with essential variations in the cycle length and especially in the amplitude, varying dramatically between nearly spotless grand minima and very large values during grand maxima. The mere fact of such great variability, known from sunspot data, forced solar physicists to develop dynamo models further. Simple deterministic numerical dynamo models, developed on the basis of Parker’s migratory dynamo, can simulate events, which are seemingly comparable with grand minima/maxima occurrence (e.g., Brandenburg et al. 1989). However, since variations in the solar-activity level, as deduced from cosmogenic isotopes, appear essentially nonperiodic and irregular, appropriate models have been developed to reproduce irregularly-occurring grand minima (e.g., Jennings and Weiss 1991; Tobias et al. 1995; Covas et al. 1998). Models, including an ad hoc stochastic driver (Choudhuri 1992; Schmitt et al. 1996; Ossendrijver 2000; Weiss and Tobias 2000; Mininni et al. 2001; Charbonneau 2001; Charbonneau et al. 2004; Käpylä et al. 2016), are able to reproduce the great variability and intermittency found in the solar cycle (see the review by Charbonneau 2010). A recent statistical result of grand minima occurrence (Sect. 4.3.2) shows disagreement between observational data, depicting a degree of self-organization or “memory”, and the above dynamo model, which predicts a pure Poisson occurrence rate for grand minima (see Sect. 4.2). This poses a new constraint on the dynamo theory, responsible for long-term solar-activity variations (Sokoloff 2004; Moss et al. 2008).

4.4.2 Solar-terrestrial relations

The sun ultimately defines the climate on Earth supplying it with energy via radiation received by the terrestrial system, but the role of solar variability in climate variations is far from being clear. Solar variability can affect the Earth’s environment and climate in different ways (see, e.g., reviews by Haigh 2007; Gray et al. 2010). Variability of total solar irradiance (TSI) measured during recent decades is known to be too small to explain observed climate variations (e.g., Foukal et al. 2006; Fröhlich 2006; Yeo et al. 2014). On the other hand, there are other ways solar variability may affect the climate, e.g., an unknown long-term trend in TSI (Solanki and Krivova 2004; Wang et al. 2005) or a terrestrial amplifier of spectral irradiance variations (Shindell et al. 1999; Haigh et al. 2010). Uncertainties in the TSI/SSI reconstructions remain large (Schmidt et al. 2012; Yeo et al. 2014), making it difficult to assess climate models on the long-term scale. Alternatively, an indirect mechanism also driven by solar activity, such as ionization of the atmosphere by CR (Usoskin and Kovaltsov 2006) or the global terrestrial current system (Tinsley and Zhou 2006) can modify atmospheric properties, in particular cloud cover (Ney 1959; Svensmark 1998; Usoskin and Kovaltsov 2008a). Although the role of this direct mechanism is found to be small (Mironova et al. 2015), indirect effects of energetic particles may be still notable (e.g., Gray et al. 2010; Calisto et al. 2011; Martin-Puertas et al. 2012).

Accordingly, improved knowledge of the solar driver’s variability may help in disentangling various effects in the very complicated system that is the terrestrial climate (e.g., Gray et al. 2010). It is of particular importance to know the driving forces in the pre-industrial era, when all climate changes were natural. Knowledge of the natural variability can lead to an improved understanding of anthropogenic effects upon the Earth’s climate.

Studies of the long-term solar-terrestrial relations are mostly phenomenological, lacking a clear quantitative physical mechanism. Therefore, more precise knowledge of past solar activity, especially since it is accompanied by continuous efforts of the paleo-climatic community on improving climatic data sets, is crucial for improved understanding of the natural (including solar) variability of the terrestrial environment.

5.1.1 The event of 775 AD: the worst case scenario?

The event of ca. 775 AD was analyzed using biennial \(^{14}\)C data by Miyake et al. (2012), who suggested that the event was probably caused by \(\gamma \)-rays from an unknown nearby supernova. This event is confirmed by annual \(^{14}\)C data from a German oak tree (Usoskin et al. 2013), Russian and American tree samples (Jull et al. 2014), New Zealand trees (Güttler et al. 2013), etc., and corals from the Chinese Sea (Liu et al. 2014) According to model simulations, the production of \(^{14}\)C appears in agreement with that of \(^{10}\)Be (Usoskin et al. 2013; Melott and Thomas 2012; Pavlov et al. 2013). Although some exotic scenarios were proposed for the event: an unidentified nearby supernova (Miyake et al. 2012); a gamma-ray burst (Hambaryan and Neuhäuser 2013; Pavlov et al. 2013); or even a cometary impact on Earth (Liu et al. 2014), it is generally accepted now that it was a signature of a (probably, consequence of) extreme SEP event (Usoskin and Kovaltsov 2012; Eichler and Mordecai 2012; Usoskin et al. 2013; Melott and Thomas 2012; Thomas et al. 2013; Cliver et al. 2014). A detailed analysis performed by Mekhaldi et al. (2015) not only confirmed its solar origin but also made it possible to assess, based on data from different cosmogenic isotopes, the reconstructed integral spectrum as shown in Fig. 25 along with the fluence spectra of two extreme SEP events of the space era: the hardest event of 23-Feb-1956 (GLE # 5) with the greatest fluence of high energy SEPs; and the event of 04-Aug-1972 with the greatest measured fluence of lower energy SEPs. One can see that the energy spectrum of the 775 AD event was very hard, close to that of the 23-Feb-1956 event, but scaled up by a factor \({\approx } 40\). This implies that the fluence of high energy particles was much greater (factor of \({\approx } 40\)) in 775 AD than ever observed during the space era. On the other hand, the low-energy fluence (energy range of several tens of MeV) for the 775 AD event was only a factor of two greater than that of the event of Aug-1972, suggesting that the fluence of low-energy SEP might have a natural limit (Asvestari et al. 2016), for example related to the “streaming limit” (Reames and Ng 2010). Thus, the high fluence of high energy particles does not necessarily mean a very high fluence of the lower-energy particles.

Open image in new window

The signal of the 775 AD event was so strong and clearly visible in the ice-core \(^{10}\)Be data that it is now used as a tie point (the point with independently known date, e.g., volcanic eruption or, as in this case, the SEP event) for more precise dating of ice cores (Sigl et al. 2015).

Another similar event was found to occur in 994 AD (Miyake et al. 2013), that was roughly half of that of 775 AD and also had a very hard spectrum (Mekhaldi et al. 2015).

Do we expect that even stronger SEP events took place in the past? The 775 AD event was observed as the sharpest peak (\({\approx } 0.4\)%/years) in the decadal \(^{14}\)C IntCal dataset, while the smaller peak (\({\approx } 0.35\)%/years) of the 994 AD was only barely seen in the IntCal data. Recently, Miyake et al. (2016) measured, with (bi)annual resolution, \(^{14}\)C around four smaller peaks (\({\approx } 0.3\)%/years) since 4800 BC and found no other events, suggesting for uniqueness of the 775 AD one for the last six millennia. We note that other remaining peaks in the IntCal data are smaller than the one of 775 AD. Accordingly, an event stronger than that of 775 AD could be found only in a case of an unlikely random coincidence of the event itself with an incidental drop of \(^{14}\)C caused by other reasons, masking the spike. In particular, the event twice as strong as the 775 AD one is hardly possible to occur since it would have produced a large spike in the IntCal data which could not be missed (see Fig. 26). Thus, the 775 AD event can securely serve as the worst case scenario of the SEP event during the entire Holocene.

Open image in new window

5.1.2 Occurrence rate

The integral probability distribution of the occurrence of strong SEP events, as revealed from the cosmogenic isotope data, is shown in Fig. 27.

Open image in new window

One can see that the break in the distribution marginally hinted in the directly observed SEP events at around \(F_{30} = (5{\,\textendash \,}7) \times 10^{9}\ \mathrm{cm}^{-2}\) (nonproportionally fewer strong events observed) is confidently confirmed by the cosmogenic isotope data. In particular, no event with \(F_{30} \,{>} 2 \times 10^{10}\ \mathrm{cm}^{-2}\) was found over the last 600 years using annually resolved \(^{10}\)Be data. It is noteworthy that the idea of a possible extreme Carrington SPE of 1859 AD (McCracken et al. 2001) is discarded (see also Wolff et al. 2012). On the longer time scale of 11 millennia, no event with \(F_{30} \,{>}5 \times 10^{10}\ \mathrm{cm}^{-2}\) has been found. This gives a new strict observational constraint on the occurrence probability of extreme SPEs.

According to Usoskin and Kovaltsov (2012) practical limits can be set as \(F_{30} \approx 1\), 2–3 and \(5 \times ~10^{10}\ \mathrm{cm}^{-2}\) (10, 20–30 and 50 times greater than the SEP event of February 23, 1956), for the occurrence probability of \(10^{-2}\), \(10^{-3}\), and \(10^{-4}\ \mathrm{years}^{-1}\), respectively. The mean SEP flux is found as \({\approx } 40\) (\(\mathrm{cm}^{2}\ \mathrm{s})^{-1}\) in agreement with estimates from the lunar rocks. On average, extreme SPEs contribute about 10% to the total SEP fluence.