Lengths of Schwabe cycles in the seventh and eighth centuries indicated by precise measurement of carbon-14 content in tree rings

1 Introduction

[2] The activity of our Sun has been studied by various methods. The oldest continuous record of the activity of the Sun is provided by sunspot measurements, which have been made since A.D. 1610. The sunspot data from the past 400 years show that the sunspot number changes periodically, with a period of 11 years on average (Figure 1). This periodicity is called the Schwabe cycle [Schwabe, 1843; Hoyt and Schatten, 1998]. What defines the length of the Schwabe cycle is one of the most important problems in understanding the solar dynamo mechanism. However, analyzing only the sunspot record of the past 400 years is insufficient for understanding this problem. We need to know about solar activity over a much longer term. Furthermore, the sunspot activity during a time interval in the late 17th to early 18th century known as the Maunder Minimum fell to near-zero levels. The carbon-14 content in tree rings is a powerful tool for the study of solar activity before A.D. 1610 and during the Maunder Minimum because ¹⁴C is produced in the atmosphere by galactic cosmic rays, and the solar magnetic activity influences the cosmic ray flux on Earth [Beer et al., 2012; Usoskin, 2013].

[3] One of the principal sets of data on past carbon-14 contents is the IntCal data set [Reimer et al., 2009], which consists of decadal or bidecadal data on ¹⁴C content over the past 50,000 years. The IntCal data show an excursion of the ¹⁴C content during the Maunder Minimum and also other excursions in the past. These excursions are considered to indicate periods of solar inactivity; large excursions comparable to the Maunder Minimum are called “grand solar minima.” There have been some attempts to investigate the occurrence of grand solar minima in the past [Eddy, 1977; Stuiver and Quay, 1980; Voss et al., 1996; Usoskin et al., 2007]. These studies have identified 6 to 20 grand solar minima in the last 5000 years depending on the literature. The difference in the number of identified grand solar minima is due to the analytical methods used and the criteria for identifying the minima. Stuiver and Braziunas [1989] suggested that grand solar minima can be divided into two types—the Maunder type and the Spörer type, according to their duration. Figure 2 shows the main grand solar minima during the last 3000 years.

[4] Yearly ¹⁴C content data have also been examined, mainly for grand solar minima (the Maunder Minimum, the Spörer Minimum, and the minimum in the fourth century B.C. [Stuiver et al., 1998; Miyahara et al., 2004, 2006, 2010; Nagaya et al., 2012]). From these yearly ¹⁴C content measurements, it has been suggested that the length of the Schwabe cycle increased to ~14 years during the Maunder Minimum, to ~13 years during the beginning of the Spörer Minimum, and to ~16 years during the Fourth Century B.C. Minimum. An increase in the length of the Schwabe cycle during solar minima is consistent with an inverse correlation between the sunspot peak amplitude and the length of the sunspot cycle [Hathaway, 2010].

[5] Although the length of the solar cycle during the above grand minima, which have the largest amplitude, has been studied, there have been few studies of the length of the solar cycle either during other minima or during periods of normal solar activity. What we have to do at this stage, when the mechanism of the variation of the length of the Schwabe cycle has not yet been figured out, is to study extended time intervals during which we can see various types of solar activity.

[6] In this paper, we describe the measurements of carbon-14 contents from A.D. 600 to A.D. 760 with biennial resolution. This time interval includes a small grand solar minimum (A.D. 650–720, the A.D. 7 Minimum, shown in Figure 2) identified by Eddy [1977] and Usoskin et al. [2007]. We also present results of periodicity analyses of these data.

2 Methods

[7] We used a Japanese cedar tree (Cryptomeria japonica, about 2000 years old; see Figure 3b) from Yaku Island in southern Japan (30.2°N, 130.3°E, at 890 m above sea level). Each tree ring was dated absolutely by dendrochronology, using pattern matching of the time sequence of ring widths to a standard. Figure 3a shows the location of Yaku Island, and Figure 3b shows a disk cut from the tree for this study.

[8] The annual rings were carefully separated from a cut core (Figure 3c), and cellulose, which does not move between rings, was extracted using the following method: (1) washing with ethanol, followed by distilled water, in an ultrasonic bath; (2) soaking in HCl, NaOH, and HCl solutions (acid-alkali-acid treatment); (3) bleaching with hot NaClO₂/HCl to remove lignin; and (4) washing with boiling distilled water. The pretreated material was then combusted to CO₂, which was purified with cold traps, and then graphitized by hydrogen reduction. We measured the ¹⁴C content of the graphite using the accelerator mass spectrometer (AMS) at Nagoya University [Nakamura et al., 2000].

[9] Since AMS gives only relative values of the ¹⁴C content, standard samples (NIST SRM4990C oxalic acid, the new National Bureau of Standards standard) were measured in the same run for absolute calibration. Blank samples (commercial oxalic acid from Wako Pure Chemical Industries) were also measured to determine the background. In this measurement, six National Institute of Standards and Technology (NIST) standard samples and two blank samples were converted to graphite, and their ¹⁴C contents were measured in the same run. The standard deviation of the ¹⁴C content of the six NIST standard samples was consistent with the statistical error. The concentration of ¹⁴C expressed as Δ¹⁴C, which is a value corrected for age and isotopic fractionation, was calculated according to the method of Stuiver and Polach [1977]. The typical precision of the ¹⁴C content measurements was 2.9‰.

3 Results

[10] We carried out two series of biennial measurements of the ¹⁴C content from A.D. 600 to 760 (exceptional cases were A.D. 600, 638, 736, and 748, for which there was only one series, and A.D. 676, 678, 686, and 698, for which there were three series). The data for overlapping years matched within measurement errors, confirming that the series of measurements were reproducible. Figure 4a shows the variation of Δ¹⁴C for the period from A.D. 600 to 760 after the series of data were combined in the form of a weighted average. The typical errors after combined are 2.0‰.

[11] To extract the periodic components in the frequency band of the Schwabe cycle, we applied an S transform to the data shown in Figure 4a, which is a type of wavelet analysis [Stockwell et al., 1996]. The results of S transform are shown in Figure 4b. From a result of wavelet analysis, signals of frequency band of the Schwabe cycle (about 8–13 years) are detected. However, it must be noted that the variation in the ¹⁴C content in tree rings reflects the time-shifted and the attenuated solar activity due to the global carbon cycle. If these effects change the periodic components in the Δ¹⁴C, the periodic analysis is meaningless. Therefore, we have calculated a ¹⁴C production by passing a three-box carbon cycle model [Nakamura et al., 1987] to our Δ¹⁴C data and applied the S transform on the obtained ¹⁴C production data. This result is shown in Figure 4d. Since the timing of detected periodicities in Figure 4d appeared as almost identical to that in Figure 4b, we have concluded that the carbon cycle does not largely have an influence on the frequency band of the Schwabe cycle shown in Figure 4b.

[12] The attenuation rate for the period of the Schwabe cycle (i.e., 11 years) is calculated to be about 1/100 using the carbon cycle model [Siegenthaler et al., 1980]. If the average carbon-14 production change is ±15%, the amplitude of the Schwabe cycle corresponds to 1.5‰ in the ¹⁴C content [Nagaya et al., 2012]. Since the typical measurement errors of our result are 2.0‰, the significance of the wavelet analysis against the measurement errors is important to discuss the Schwabe cycle length. The confidence level of this wavelet analysis with respect to measurement errors is shown in Figure 4c, which is calculated using the method of Nagaya [2012] and Nagaya et al. [2012]. The outline of the simulation procedures is as follows. We generated 10,000 normally distributed random error series (E_i(t)) whose standard deviation is equal to the errors of the Δ¹⁴C.

where e(t) is the error series of the originalΔ¹⁴C time series and R(t) is the normally distributed random number series. Then, we applied the S transform to each E_i(t) and got the probability density distribution of a power spectrum for each wavelet cell. Each distribution is fitted by an exponential distribution. The significance of the original time series (h(t)) is calculated as a ratio between its power spectrum and the standard deviation of fitted exponential distribution [Nagaya et al., 2012]. The color in Figure 4c represents the significance in sigma.

[13] We also applied these procedures to the production rate. An error series of the production rate is estimated by a Monte Carlo simulation. 10,000 Δ¹⁴C series (H_i(t)), which are summations of the original time series (h(t)) and the error series (E_i(t)), are converted to the production rates using the three-box carbon cycle model.

[14] The errors of the production rate are calculated as standard deviations for each age (t). Figure 4e shows the confidence level of the S transform result in Figure 4d.

4 Discussion

[15] We can see the periodicity time distribution of signals more than three-sigma significance level in Figure 4c. A 10 year periodicity is seen before A.D. 630. Then the periodicity changes to 12–13 years from A.D. 630 to 710. It is notable that the time interval in which this longer cycle length occurs corresponds to the Maunder-type grand solar minimum, which lasted from A.D. 650 to 720 defined by Usoskin et al. [2007] (hereafter referred to as the “Seventh Century Minimum”). This 12–13 year component is the main component for most of the Seventh Century Minimum.

[16] In order to evaluate the robustness of our S-transform result, we performed a numerical experiment using an ideal sine wave which has the same conditions as our measurement's such as the time interval, the data point number, and the average of our measurement errors. A series simulating F(t) is formed as

where A₁ = 1.5 (‰) (the amplitude of the Schwabe cycle), t = 2n + 1(n = 0,1,…80), P₁ = 11 (year) (the Schwabe cycle length), A₂ = 10 (‰), P₂ = 100 (year) (secular trend; the grand solar minimum in this case), and R(t) is the normally distributed random number series. To adjust F(t) to our measurement accuracy, we used S = 2 (‰). We made 1000-F(t) series and applied the S transform on each series. The obtained results were converted to the confidence level [Nagaya et al., 2012]. We counted how many S-transform results have signals of the longer period (12–16 year) which continue for more than one cycle. The criteria of the signal detection are more than three-sigma significance level. Since the S-transform signals have the width band in frequency, we determined the frequency to be the center frequency of the wide signal. Although there are 116 results that have a one-cycle-longer period per 1000 results, there are only 26 and 8 results which have two cycles and three cycles, respectively. There is no result which has a period more than four cycle longer. Then, it is considered that a detection of a one-cycle signal does not have that high a confidence level (~12%) but that continuously detected signals (more than two cycles) have a higher confidence level (<3%). Our S-transform result during the Seventh Century Minimum shows the longer signals (12–13 years) continuing about 80 years which corresponds to six cycles. Therefore, the possibility that this longer signal is occurred by the measurement errors is extremely low.

[17] Since ¹⁴C forms ¹⁴CO₂ and is accumulated in tree rings after circulating in the atmosphere, it is possible that these ¹⁴C variations occurred not only due to solar activity but also due to climatic effects. However, it is very difficult to distinguish between these two causes from only ¹⁴C data. Then, if the detected frequency signals are due to solar activity, this is the fourth case in which an increase in the length of the Schwabe cycle during a grand solar minimum has been observed, after the Maunder Minimum [Miyahara et al., 2004], the Spörer Minimum [Miyahara et al., 2010], and the Fourth Century B.C. Minimum [Nagaya et al., 2012]. Table 1 shows a comparison between these four minima.

[18] Although the cycle length of the three Maunder-type minima (the Maunder, the Fourth Century B.C., and the A.D. Seventh Century minima) increased throughout most of the minima, that of the Spörer Minimum increased only for the preceding to the beginning of the minimum [Miyahara et al., 2010] (here Miyahara et al. [2008] used a period of the Spörer Minimum from A.D. 1416–1534 defined by Stuiver and Quay [1980]). Since the durations of Spörer-type solar minima are long (>110 years), it has been suggested that the dynamo mechanisms of Maunder-type and Spörer-type minima are different or that the Spörer type is a composite type made up of several Maunder-type minima [Miyahara et al., 2010; Usoskin et al., 2007]; however, there is little understanding of this. Then, further measurements during another Spörer-type minimum will be necessary to understand the difference of the Schwabe cycle behavior between the Maunder-type and the Spörer-type minima.

5 Conclusions

[19] We measured the ¹⁴C content in a Japanese cedar tree from A.D. 600 to 760 in alternate years with 2 year resolution. This time interval contains the Seventh Century Minimum. The S-transform analysis of the ¹⁴C data shows that the periodicity of 10 years is detected before this minimum, and the detected periodicity shifts to 12–13 years in subsequent years during most of the minimum (continuing for about ~80 years).

[20] An increase in the length of the Schwabe cycle during the Seventh Century Minimum has been found during the Maunder Minimum, the Spörer Minimum, and the Fourth Century B.C. Minimum. The fourth of these cases strengthens the evidence that the length of the Schwabe cycle increases during grand solar minima.

[21] In future work, we intend to clarify an accurate contribution of the solar activity to the frequency components of annual ¹⁴C data.