The current geological epoch, the Holocene started about 11,500 years ago when the glaciers began to retreat. The concentration of ¹⁰Be in ice halved abruptly at that time, because the annual precipitation of snow increased by a factor of about two (Beer et al. 2012). This warm climate epoch is commonly considered as an interglacial, and it was once thought to have been climatically stable (Dansgaard et. al. 1993). This conventional view was primarily supported by the relative isotope temperature stability on the Greenland summit, which appears to be fairly invariable on centennial and longer time scales during the Holocene (Bütikofer 2007). However, well-dated paleo-climatic proxies such as ice cores, tree-rings and sediment records show that significant climate variations also occurred during most of the Holocene, although with basically weaker amplitudes than during glacial times (Sarnthein et al. 2003, Bütikofer 2007).

Over the past several decades, many extensive paleoclimate studies have testified to the considerable climate fluctuations in the Holocene (Mayewski et al. 2004, Dergachev et al. 2007). Subsequently, Holocene climate variability on the centennial-millennial time scale has been demonstrated by many authors. The debate about Holocene climate cycles on the millennial-scale was primarily initiated by the investigations of Bond & Lotti (1997). Since then, many studies have uncovered evidence of repeated climate oscillations of ~2500, ~1500, and ~1000 years (Debret et al. 2007). As well, climate cycles of the centennial-scale which have periods of ~520, ~350, and ~210 years have been discussed in a number of studies. Nevertheless, there is surprisingly little systematic knowledge about climate variability during this period (Mayewski et al. 2004). The debate on their forcing cause and factors related with the climate cycles is still ongoing.

To seek a more comprehensive demonstration, therefore, we have studied the Holocene climate cycles and features using the wavelet analysis of climate proxies of ¹⁴C, ¹⁰Be, and ¹⁸O, and compared these with the results of different studies.

In this work, we carried out the wavelet analysis using the high-resolution records of ¹⁴C, ¹⁰Be, and ¹⁸O, which have been already released in the literature and the World Data Center for Paleo-climatology. Specifically, we obtained the ¹⁴C records from INTCAL09 (Reimer et al. 2009), total solar irradiance reconstruction data of ¹⁰Be from GRIP ice core (Steinhilber et al. 2009), and ¹⁸O records from GRIP & NGRIP (Vinther et al. 2006) and GISP2 (Stuiver et al. 1995) ice cores. These records used in this work are listed in Table 1.

Continuous Wavelet Transform (CWT) analysis provides a time evolution of the spectral features of the fluctuations. The CWT of a discrete time sequence x_n is defined as the convolution of x_n with a scaled and translated version of wavelet function as below.

where ψ is the wavelet function, (*) indicates the complex conjugate, s is the scale factor, and N is the number of data points (Torrence & Compo 1998). By varying the wavelet scale s and translating along the localized time index n, one can construct a “scalogram” showing how wave amplitude varies in the frequency-time space. Because the wavelet function ψ is in general complex, the wavelet transform W_n(s) is also complex. The transform can then be divided into the real part, R{W_n(s))}, and the imaginary part, I{W_n(s)}, or amplitude, |W_n(s)|, and phase, tan^-1[I{W_n(s)}/R{W_n(s)}]. The resulting plot of the amplitude |W_n(s)| is a contour map of amplitude in frequency-time domain. In this process of constructing a scalogram, the scale factor s is properly converted to frequency f. For the CWT analysis, here, we employed the Morlet wavelet as the wavelet function, consisting of a plane wave modulated by a Gaussian as below:

where η is a dimensionless time parameter and ω₀ is a dimensionless frequency. For this work, we took ω₀ = 10, as this choice offers a good trade-off between the frequency and temporal resolutions for the time series analyzed herein (Debret et al. 2007).

For this work, we employ the wavelet analysis to determine the time evolution of the wave activities associated with the different climatic proxies, offering specific information regarding which wave cycles are dominant or subdominant in which time intervals. This is in contrast to the simple power spectral analysis, which provides only frequencies of the major wave power peaks.

The wavelet results for the ¹⁴C, ¹⁰Be, and ¹⁸O records are summarized in Fig.1, and the major periods determined from the wavelet analysis are listed in Table 2. In Fig.1, the original data are also shown above each of the scaolograms. In addition, the plots on the right of each scalogram represent the wavelet amplitudes averaged over the entire time period. To compare these with the spectral results obtained by other authors, we listed the major peaks derived from spectral analysis of the ¹⁴C and ¹⁰Be records in Table 3.

The wavelet results of Fig. 1 and Table 2 show that the spectral features and major periodicities of ¹⁴C and ¹⁰Be records are similar to each other, while the variations of the ¹⁸O records are quite different from them. In addition, each and every cycle shows a climate signal in a different time interval with the different amplitude, and its period is also varying with the selected records and observation sites, respectively.

In the wavelet results of ¹⁴C and ¹⁰Be, the most significant cycle is certainly Hallstatt cycle of ~2180-2310 years in the Holocene time frame. However, the ~970, ~500-520, ~350-360, and ~210-220 year cycles also show distinct signals in their respective time intervals. In particular, the de Vries cycle of ~210-220 years shows strong amplitude in the intervals that coincide with the occurrence of grand solar minima. Meanwhile, the Eddy cycle of ~970 years is mostly prominent before the mid-Holocene.

On the other hand, the climate cycles derived from the ¹⁸O records show different characteristics depending on the observation sites. The Eddy cycle of ~900-1000 years is the most significant one in the records of NGRIP and GISP2 ice cores but it is not in the record of GRIP, while all of the three records indicate that it is dominant in the early half of the Holocene. The ~1910-1920 year and ~550 year cycles are significant in the records of GRIP and GISP2 ice cores but they are much weaker or hard to identify in the record of NGRIP.

Table 3 shows the periods of major spectral peaks for ¹⁴C and ¹⁰Be derived by other researchers. These previous results were all obtained from the simple spectral analysis rather than a wavelet method. To compare these previous results with ours for the most representative cycles such as Hallstatt, Eddy, and de Vries cycles, we have summarized both results in Table 4.

Table 4 indicates that our wavelet results are overall in good agreement with the previous spectral results for each cycle. However, the period of each climate cycle is variable in both analyses up to several percentages, in particular for the Hallstatt cycle. Fig. 2 shows the amplitude variations of the ¹⁸O records observed in the 3 different sites in Greenland (GRIP, NGRIP and GISP2). In Fig. 2, our visual inspection indicates that the climate variations of ¹⁸O records of 3 different sites are similar, being consistent with the wavelet results in Figs. 1(c-e). The ice core temperature data of GISP2 follows the general trend. Clearly detailed fluctuations are different among the observation sites but the overall pattern is similar, including the abrupt drop around 8200 BP and the major decline between ~10000BP and ~11000BP. As well, we could confirm such similarity in the periodic variation (represented in Fig. 1 and Table 3) such as Eddy cycle which shows the dominant signals from early to mid-Holocene in all the 3 different observation sites.

In 1982, Chuck Sonnet employed the power spectrum analysis to demonstrate that there was a ~2000 year periodicity in the rather limited ¹⁴C data available at that time. He and Paul Damon called this the “Hallstatt cycle” and Sonnet’s result was later validated using the ¹⁰Be data (Beer et al. 2012). After the discovery of Hallstatt cycle, its properties were discussed by Damon & Linick (1986), Damon (1988), Damon et al. (1990). Subsequently, Vasiliev & Dergachev (2002) confirmed the existence of amplitude modulation with a period of ~2400 years using the power spectrum, time spectrum, and bispectrum analyses of ¹⁴C data. As well, McCracken et al. (2005) deduced the 2300 year periodicity from the Fourier analysis of ¹⁰Be data. Through this study, we derived the Hallstatt cycle of 2180 and 2312 years from the ¹⁰Be and ¹⁴C data, and the similar cycles of ~1900, ~2500 and ~2750 years from the ¹⁸O data of GISP2, GRIP and NGRIP ice cores, respectively.

Meanwhile, in the publications of Damon & Sonett (1991) and Damon & Jirikowie (1992), the ~2400 year and ~210 year cycles are considered to be the fundamental ones with most of the other secular cycles discovered in radiocarbon data being harmonic components of the longest cycle (Vasiliev & Dergachev 2002). Our results from the ¹⁴C and ¹⁰Be data also show that the Hallstatt and de Vries cycles are dominant during the Holocene, and the de Vries cycle of ~210-220 years corresponds to the occurrence of grand solar minima presented by Usoskin et al. (2007). This feature was also indicated by Abreu et al. (2012) who noticed that the amplitude of the ~210 year cycle is greatest during the periods when grand solar minima are more frequent. Related with this feature, there is an interesting demonstration as follows: When the estimated solar modulation function by Castagnoli & Lal (1980), Ф_CL80 is filtered with a 1000-year running average, groups of deep minima occur at 400 BP, 3100 BP, 5300 BP and 7300 BP. Thus, these deep minima clusters occur with a periodicity of (7300-400)/3 = 2300 years. That is, the ~2300 year periodicity first detected by Sonnet was a consequence of 2300 year recurrence of clusters of grand minima (Beer et al. 2012). This is not precisely consistent with, but seems to be overall in a qualitative agreement with our wavelet results in Figs. 1a and 1b (i.e., ~ 500, ~2500, ~5300 and ~7300 BP). In conclusion, therefore, we may infer that the Hallstatt and de Vries cycles are in close relation with the appearance of group of grand minima.

In addition, Eddy cycle also shows some difference between wavelet and spectral analysis, and also among the ¹⁴C, ¹⁰Be, and ¹⁸O records. Interestingly, however, all the records of ¹⁴C, ¹⁰Be, and ¹⁸O reveal that Eddy cycle is predominant during the early half of the Holocene.

In this paper, we focused on determining Holocene climate cycles, especially on the centennial to millennial time scales. Through our analyses, we confirmed most major periodicities of centennial and millennial scales which have been discussed in the previous studies. In conclusion, this work reveals the following features:

1) The dominant cycles and features of 14C and 10Be data show strong similarity and the same overall patterns throughout the Holocene. But the climate variations of 18O records show somewhat different characteristics compared to 14C and 10Be. 2) The wavelet results of 14C and 10Be show that the most conspicuous cycles are the ~2180-2310 and ~970 year periodicities on the millennial scale, and the ~350-360 and ~210-220 year cycles on the centennial scale. In particular, the Hallstatt cycle of ~2180-2310 years is significant throughout the Holocene, whereas the Eddy cycle of ~970 years is conspicuous in the early half of the Holocene 3) In contrast, 18O records show that the predominant climate cycles are ~1900-2000 and ~900-1000 year periodicities on the millennial scale, and the ~550-600 year cycle on the centennial scale. The period of ~1900, ~2500 and ~2750 years derived from the 18O records of GISP2, GRIP and NGRIP ice cores, respectively, might correspond to the Hallstatt cycles. The Eddy cycle derived from the 3 different sites shows the period of ~900-1000 years in the early half of the Holocene 4) The de Vries cycle of ~210-220 years shows distinct signals in the intervals centered on ~500, ~2500, ~5300 and ~7300 BP, being separated by ~ 2270 years on average. Each of them coincides with the occurrence of grand solar minima. Thus, it is likely that the Hallstatt and de Vries cycles derived from the 14C and 10Be records are closely related with the appearance or disappearance of grand solar minima, and also, there is a link that implies a connection of the millennial and centennial cycles between them. Therefore, it seems that both of the Hallstatt and de Vries cycles are directly connected with the solar activity. 5) On the other hand, abrupt event around 8200 BP appeared in the 18O records, but did not show in the 14C and 10Be data. It is considered that it is probably not related with solar forcing. 6) Lastly, this work indicates that the climate variations such as the Hallstatt, Eddy, and de Vries cycles are not only stationary oscillations, but also vary with the proxy data, observational sites, and even analysis methods to some extent.