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) 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.
The time behavior of sunspot activity during the Maunder minimum yields 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 was sudden (within a few years) without any apparent precursor. 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. 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 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. Update
The presence of grand minima in solar activity on the long-term scale has been mentioned numerously (e.g., Eddy, 1977a; Usoskin et al., 2003c; 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 17, 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 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). All 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 significantly 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 4.2). 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 4.5.1). 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. Update
A histogram of the duration of grand minima from Table 1 is shown in Figure 20. 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.
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