Here we consider the 14C-based decade reconstruction of sunspot numbers (shown in Figure 21). It is identical to that shown in Figure 17, but includes also a Gleissberg 1-2-2-2-1 filter in order to suppress noise and short-term fluctuations. This series forms the basis for the forthcoming analysis, while differences related to the use of other reconstructions are discussed.
In order to discuss spectral features of long-term solar-activity dynamics, we show in Figure 22 a wavelet spectral decomposition of the sunspot number reconstruction throughout the Holocene shown in Figure 21. The left-hand panels show the conventional wavelet decomposition in the time-frequency domain, while the right-hand panels depict the global spectrum, namely, an integral over the time domain, which is comparable to a Fourier spectrum. The peak in the global spectrum at about an 80-year period corresponds to the Gleissberg periodicity, known from a simple Fourier analysis of the 14C series (Peristykh and Damon, 2003). The peak at an approximately 150 year period does not correspond to a persistent periodicity, but is formed by a few time intervals (mostly 6000 – 4000 BC) and can be related to another “branch” of the secular cycle, according to Ogurtsov et al. (2002). The de Vries/Suess cycle, with a period of about 210 years, is prominent in the global spectrum, but it is intermittent and tends to become strong with around 2400 clustering time (Usoskin and Kovaltsov, 2004). Another variation with a period of around 350 years can be observed after 6000 BC (cf. Steinhilber et al., 2012). Variations with a characteristic time of 600 – 700 years are intermittent and can be hardly regarded as a typical feature of solar activity. There is also a weak millennial periodicity (Steinhilber et al., 2012). Of special interest is the 2000 – 2400 year Hallstatt cycle (see, e.g., Vitinsky et al., 1986; Damon and Sonett, 1991; Vasiliev and Dergachev, 2002), which is relatively stable and mostly manifests itself as a modulation of long-term solar activity, leading to the clustering of grand minima (Usoskin et al., 2007).
On the other hand, an analysis of the occurrence of grand minima (see Section 4.2) shows no clear periodicity except for a marginal 2400 year clustering, implying that the occurrence of grand minima and maxima is not a result of long-term cyclic variability but is defined by stochastic/chaotic processes.
A very particular type of solar activity is the grand minimum, when solar activity is greatly reduced. The most famous is the Maunder minimum in the late 17th century, which is discussed below in some detail (for a detailed review see the book by Soon and Yaskell, 2003). Grand minima are believed to correspond to a special state of the dynamo (Sokoloff, 2004; Miyahara et al., 2006b), and its very existence poses a challenge for the solar-dynamo theory. It is noteworthy that dynamo models do not agree on how often such episodes occur in the sun’s history and whether their appearance is regular or random. For example, the commonly used mean-field dynamo yields a fairly-regular 11-year cycle (Charbonneau, 2010), while dynamo models including a stochastic driver predict the intermittency of solar magnetic activity (Choudhuri, 1992; Schüssler et al., 1994; Schmitt et al., 1996; Ossendrijver, 2000; Weiss and Tobias, 2000; Mininni et al., 2001; Charbonneau, 2001). Most of the models predict purely random occurrence of the grand minima, without any intrinsic long-term memory (Moss et al., 2008). Although cosmogenic isotope data suggest the possible existence of such memory (Usoskin et al., 2007), statistics is not sufficient to distinguish between the two cases (Usoskin et al., 2009d).
The Maunder minimum 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., 2001a). There is some uncertainty in the definition of its duration; 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). The Maunder minimum was amazingly well covered (more than 95% of days) by direct sunspot observations (Hoyt and Schatten, 1996), especially in its late phase (Ribes and Nesme-Ribes, 1993). On the other hand, sunspots appeared rarely (during 2% of the days) and seemingly sporadically, without an indication of the 11-year cycle (Usoskin and Mursula, 2003). This makes it almost impossible to apply standard methods of time-series analysis to sunspot data during the Maunder minimum (e.g., Frick et al., 1997)). Therefore, special methods such as the distribution of spotless days vs. days with sunspots (e.g., Harvey and White, 1999; Kovaltsov et al., 2004) or an analysis of sparsely-occurring events (Usoskin et al., 2000) should be applied in this case. Using these methods, Usoskin et al. (2001a) 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) 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 14C 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 10Be 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., 2001a). A recent theoretical study (Owens et al., 2012; Wang and Sheeley Jr, 2012) confirms that such a phase change between cosmic rays and solar activity can appear for very weak cycles.
The time behavior of sunspot activity during the Maunder minimum yielded the following general scenario (Vitinsky et al., 1986; Ribes and Nesme-Ribes, 1993; Sokoloff and Nesme-Ribes, 1994; Usoskin et al., 2000, 2001a; Miyahara et al., 2006b). Transition from the normal high activity to the deep minimum did not have any apparent precursor. On the other hand, 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). 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. It is interesting to note that such a qualitative evolution of a grand minimum is consistent with predictions of the stochastically-forced return map (Charbonneau, 2001).
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 14C (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 4-th 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 18th 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.
The presence of grand minima in solar activity on the long-term scale has been mentioned numerously (e.g., Eddy, 1977a; Solanki et al., 2004), using the radioisotope 14C data in tree rings. For example, Eddy (1977b) identified major excursions in the detrended 14C record as grand minima and maxima of solar activity and presented a list of six grand minima and five grand maxima for the last 5000 years (see Table 1). Stuiver and Braziunas (1989) and Stuiver et al. (1991) also studied grand minima as systematic excesses of the high-pass filtered 14C data and suggested that the minima are generally of two distinct types: short minima of duration 50 – 80 years (called Maunder-type) and longer minima collectively called Spörer-like minima. Using the same method of identifying grand minima as significant peaks in high-pass filtered 14C series, Voss et al. (1996) provided a list of 29 such events for the past 8000 years. A similar analysis of bumps in the 14C production rate was presented recently by Goslar (2003). However, such studies retained a qualitative element, since they are based on high-pass–filtered 14C data and thus implicitly assume that 14C variability can be divided into short-term solar variations and long-term changes attributed solely to the slowly-changing geomagnetic field. This method ignores any possible long-term changes in solar activity on timescales longer than 500 years (Voss et al., 1996). The modern approach, based on physics-based modelling (Section 3), allows for the quantitative reconstruction of the solar activity level in the past, and thus, for a more realistic definition of the periods of grand minima or maxima.
|6||–360||60||a, b, c, d)|
|7||–765||90||a, b, c, d)|
|9||–2860||60||a, c, d)|
|10||–3335||70||a, b, c, d)|
|11||–3500||40||a, b, c, d)|
|12||–3625||50||a, b, d)|
|13||–3940||60||a, c, d)|
|15||–4325||50||a, c, d)|
|16||–5260||140||a, b, d)|
|20||–5985||30||a, c, d)|
|21||–6215||30||c, d, e)|
|22||–6400||80||a, c, d, e)|
|23||–7035||50||a, c, d)|
|25||–7515||150||a, c, d)|
a) According to Stuiver and Quay (1980); Stuiver and Braziunas (1989).
b) According to Eddy (1977a,b).
c) According to Goslar (2003).
d) According to Usoskin et al. (2007).
e) Exact duration is uncertain.
A list of 27 grand minima, identified in the quantitative solar-activity reconstruction of the last 11,000 years, shown in Figure 21, is presented in Table 1 (after Usoskin et al., 2007). The cumulative duration of the grand minima is about 1900 years, indicating that the sun in its present evolutionary stage spends (17%) of its time in a quiet state, corresponding to grand minima. Note that the definition of grand minima is quite robust.
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). 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 -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 1) 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. Section 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 (Section 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 1 is shown in Figure 23. 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.
In the last decades we were living in a period of a very active sun with a level of activity that is unprecedentedly 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 Figure 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 for the period 1750 – 1900 was 35 ± 9 (39 ± 6, if the Dalton minimum in 1797 – 1828 is not counted), it stands high at the level of 75 ± 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 first shown by Usoskin et al. (2003c) and Solanki et al. (2004), 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, or perhaps even to a low-activity stage, comparable to the Dalton minimum in the turn of 18 – 19th centuries (e.g., Lockwood et al., 2011). 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.
The question of how often grand maxima occur and how strong they are, cannot be studied using the 400-year-long series of direct observations. An increase in solar activity around 1200 AD, also related to the Medieval temperature optimum, is sometimes qualitatively regarded as a grand maximum (Wilson, 1988b; de Meyer, 1998), but its magnitude is lower than the modern maximum (Usoskin et al., 2003c). Accordingly, it was not included in a list of grand maxima by Eddy (1977a,b).
† Center and duration of the modern maximum are preliminary since it is still ongoing.
A quantitative analysis is only possible using proxy data, especially cosmogenic isotope records. Using a physics-based analysis of solar-activity series reconstructed from 10Be 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 14C 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. This result is confirmed by the most recent composite reconstruction by Steinhilber et al. (2012). The result by Solanki et al. (2004) has been disputed by Muscheler et al. (2005) who claimed that equally high (or even higher) solar-activity bursts occurred several times during the last millennium, circa 1200 AD, 1600 AD and at the end of the 19th century. We note that the latter claimed peak (ca. 1860) is not confirmed by direct solar or geomagnetic data. However, as argued by Solanki et al. (2005), the level of solar activity reconstructed by Muscheler et al. (2005) was overestimated because of an erroneous normalization to the data of ground-based ionization chambers (see also McCracken and Beer, 2007). This indicates that 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.
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 11,400 years using 14C data, as shown in Table 2 (after Usoskin et al., 2007). A total of 19 grand maxima have been identified with a total duration of around 1030 years, suggesting that the sun spends around 10% 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 like grand minima, are defined by stochastic/chaotic processes. 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 75%) were not longer than 50 years, and only four grand minima (including the modern one) have been longer than 70 years (cf. Barnard et al., 2011). This suggest that the probability of the modern active-sun episode continuing is low4 (cf. Solanki et al., 2004; Abreu et al., 2008).
Reconstructions of long-term solar activity have different implications in related areas of science. The results, discussed in this overview, can be used in such diverse research disciplines as theoretical astrophysics, solar-terrestrial studies, paleo-climatology, and even archeology and geology. We will not discuss all possible implications of long-term solar activity in great detail but only briefly mention them here.
Parker, 1955), as follows (see detail in Charbonneau, 2010). Differential rotation produces a toroidal magnetic field from a poloidal one, while the “-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), 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 (Usoskin et al., 2007, Section 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 Section 4.2). This poses a new constraint on the dynamo theory, responsible for long-term solar-activity variations (Sokoloff, 2004; Moss et al., 2008).
In general, the following additional constraints can be posed on dynamo models aiming to describe the long-term (during the past 11,000 years) evolution of solar magnetic activity.
- The sun spends about of its time at moderate magnetic-activity levels, about of its time in a grand minimum and about in a grand maximum. Recent solar activity corresponds to a grand maximum, which has ceased after solar cycle 23.
- Occurrence of grand minima and maxima is not a result of long-term cyclic variations but is defined by stochastic/chaotic processes.
- Observed statistics of the occurrence of grand minima and maxima display deviation from a “memory-less” Poisson-like process, but tend to either cluster events together or produce long event-free periods. This can be interpreted in different ways, such as self-organized criticality (e.g., de Carvalho and Prado, 2000), a time-dependent Poisson process (e.g., Wheatland, 2003), or some memory in the driving process (e.g., Mega et al., 2003).
- Grand minima tend to be of two different types: short minima of Maunder type and long minima of Spörer type. This suggests that a grand minimum is a special state of the dynamo.
- Duration of grand maxima resemble a random Possion-like process, in contrast to grand minima.
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). 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 (Shapiro et al., 2011; Schmidt et al., 2012), 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, 2008b). Even a small change in cloud cover modifies the transparency/absorption/reflectance of the atmosphere and affects the amount of absorbed solar radiation, even without changes in the solar irradiance. However, the direct role of this effect is estimated to be small (Usoskin et al., 2008; Gray et al., 2010).
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., de Jager, 2005; Versteegh, 2005; 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. Even phenomenological and empirical studies suffer from large uncertainties, related to the quantitative interpretation of proxy data, temporal and spatial resolution (Versteegh, 2005). 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.
The proxy method of solar-activity reconstruction, based on cosmogenic isotopes, was developed from the radiocarbon dating method, when it was recognized that the production rate of 14C is not constant and may vary in time due to solar variability and geomagnetic field changes. Neglect of these effects can lead to inaccurate radiocarbon (or more generally, cosmogenic nuclide) dating, which is a key for, e.g., archeology and Quaternary geology. Thus, knowledge of past solar activity and geomagnetic changes allows for the improvement of the quality of calibration curves, such as the IntCal (Stuiver et al., 1998; Reimer et al., 2004, 2009) for radiocarbon, eventually leading to more precise dating.
Long-term variations in the geomagnetic field are often evaluated using cosmogenic isotope data. Knowledge of source variability due to solar modulation is important for better results.
In this section, solar activity on a longer scale is discussed, based on recent reconstructions.
According to these reconstructions, the sun has spent about 70% of its time during the Holocene, which is ongoing, in a normal state characterized by medium solar activity. About 15 – 20% of the time the sun has experienced a grand minimum, while 10 – 15% of the time has been taken up by periods of very high activity.
One of the main features of long-term solar activity is its irregular behavior, which cannot be described by a combination of quasi-periodic processes as it includes an essentially random component.
Grand minima, whose typical representative is the Maunder minimum of the late 17th century, are typical solar phenomena. A total of 27 grand minima have been identified in reconstructions of the Holocene period. Their occurrence suggests that they appear not periodically, but rather as the result of a chaotic process within clusters separated by 2000 – 2500 years. Grand minima tend to be of two distinct types: short (Maunder-like) and longer (Spörer-like). The appearance of grand minima can be reproduced by modern stochastic-driven dynamo models to some extent, but some problems still remain to be resolved.
The modern level of solar activity (after the 1940s) was very high, corresponding to a grand maximum, which are typical but rare and irregularly-spaced events in solar behavior. However, this grand maximum has ceased after solar cycle 23. The duration of grand maxima resembles a random Possion-like process, in contrast to grand minima.
These observational features of the long-term behavior of solar activity have important implications, especially for the development of theoretical solar-dynamo models and for solar-terrestrial studies.