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Cosmogenic isotopes have frequently been employed as proxies of ancient cosmic ray fluxes. On the basis of periodicities of the ^{10}Be time series (using data from both the South and North Poles) and the ^{14}C time series (with data from Intercal-98), we offer evidence of the existence of cosmic ray fluctuations with a periodicity of around 30 years. Results were obtained by using the wavelet transformation spectral technique, signal reconstruction by autoregressive spectral analysis (ARMA), and the Lomb-Scargle periodogram method. This 30-year periodicity seems to be significant in nature because several solar and climatic indexes exhibit the same modulation, which may indicate that the 30-year frequency of cosmic rays is probably a modulator agent for terrestrial phenomena, reflecting the control source, namely, solar activity.

The importance of cosmic ray variations was pointed out long ago in a vast compendium of relevant research [

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The application of wavelet analysis to the paleoclimatic proxy data [^{10}Be at the North Pole). Furthermore, some properties of hurricanes, such as their total cyclonal energy and the tropical storms appearance along the Atlantic coast of México, together with other properties linked to hurricanes show such a 30-year cycle [

The upper panel shows time series of category-4 hurricane versus ^{10}Be. The coherence between both series appears in the middle panel, and the global wavelet spectrum (GWS) appears in the right-hand part of the figure, where the red dashed line indicates the border, also with the reliability of 95%. The scale color bar at the right indicates the level of coherence.

Curiously, not only in nature but in several areas of contemporary human activity, such as business and commerce, 30-year cycle indexes are also found. (e.g.,

In the present work, we present evidence of the existence of a thirty-year periodicity for the cosmic rays: preliminary results were presented at the 2008COSPAR meeting in Montreal.

The spectral analysis of cosmic ray data from neutron monitors has been widely studied through several different methods, for instance [

One of the main problems in determining significant long-term periodicities in the flux of cosmic rays is that the time series of data are relatively very short, as they have been available only for the last five to six decades, when data on cosmic rays (CR) from different stations throughout the world began to be organized and homologated. Because of this restriction, a proxy for cosmic rays has often been employed, one of which in our case allows us in a deterministic way to provide evidence of a 30-year cycle for cosmic rays.

The cosmogenic isotopes Beryllium-10 (^{10}Be) and Carbon-14 (^{14}C) are conventionally considered to be proxies for cosmic rays, in such a way that an adequate spectral analysis may reveal important periodicities. These cosmogenic isotopes are mainly produced by galactic cosmic ray flux modulated by changes in interplanetary and geomagnetic magnetic fields. The analysis of cosmogenic isotopes stored in natural archives, such as ^{10}Be in polar ice cores and ^{14}C in tree rings, provides a means of extending our knowledge of solar variability over much longer periods (e.g., [^{14}C and ^{10}Be concentrations reflect the production rate, which is modulated by not only solar activity but also by atmospheric transport and deposition processes [

Data on ^{10}Be and ^{14}C can be obtained for periods of thousands of years: we use the INTCAL 98 (^{14}C time series and the ^{10}Be time series from [

The spectral techniques for analyzing periodicities of cosmophysical phenomena are very varied. The simplest technique for investigating periodicities is the Fourier Transform (FT). Although useful for stationary time series, this method is not appropriate for time series that do not fulfill the steady state condition, as is the case with cosmogenic isotopes.

In order to find the time evolution of the main frequencies of the time series, we apply the wavelet method using the Morlet mother wavelet ([

In Figures

The upper panel shows the time series of the Be^{10} from the South Pole. The wavelet spectrum appears in the middle panel, and the global wavelet spectra appear in the right-most part of the figure, where the red dashed line indicates the border of 95% reliability.

It can be seen in Figure

Figure ^{14}C; it can be seen that precisely at the 30-yrs. frequency there appears a small hump relative to the importance of the other periodicities. It would then be natural to ask how to know if such periodicity really does exist with a good reliability.

The upper panel shows the time series for the ^{14}C from the South Pole. The wavelet spectrum appears in the middle panel, and the global wavelet Spectrum appears in the right-hand part of the figure, where the red dashed line indicates a border of reliability of 95%.

This turns out to be a very complex problem, to give relevance to the periodicity of 30 years; on the other hand, it is necessary for one side to filter that signal and, on the other hand, to eliminate masking frequencies higher or lower than 30 years. It is precisely in such a situation that the ^{10}Be and ^{14}C at the North Pole appears very clearly, both with high reliability, far above 95%.

The upper panel shows the (^{10}Be) time series in the North Pole. The wavelet spectrum is a continuous one throughout all the entire time scale. The global spectrum at the right-hand panel shows that the 30-year periodicity has an extremely high confidence level.

The upper panel shows the (^{14}C) time series. The wavelet spectrum is a continuous one throughout the entire time scale. The global spectrum at the right panel shows that the 30-years periodicity has a confidence level higher than 95%.

The way to discern whether the periodicity of 30 years found in the ^{10}Be time series really reflects a comic ray fluctuation or is merely a local phenomenon of the cosmogenic isotope at earth level is to compare the behavior of ^{10}Be at both the North and South Poles. Since concentrations are quite different from one pole to another, it should be expected that their behavior would also be quite different if the existence of such periodicity is a local phenomenon. However, an examination of Figure ^{10}Be at both the North and South Poles is very high, >0.9 (red color in the color bar of Figure

The upper panel shows the time series of ^{10}Be at the South Pole versus the ^{10}Be time series at the North Pole (black line). The middle panel shows a high coherence (>0.9) between both series, according to the color bar for coherence level in Figure

The upper panel shows the sunspot time series, the middle panel shows the wavelet spectrum, and the right panel shows the global spectrum, where the 30-year periodicity appears as a very small jump relative to the 11-year peak.

Cosmic ray variations are mostly caused by time variations in the interplanetary magnetic field (IMF), so periodicities in cosmic ray data must be reflected in IMF data. Unfortunately, confident data from the IMF only date from the beginning of the spacecraft era, not even two 30-year cycles. However, since presumably the main source of such modulation is found in solar activity (SA); to confirm such an hypothesis we carried out a wavelet analysis of SA by using a time series of sunspot data (

Since the 30-year periodicity appears as a small peak, (though one with higher than 95% confidence), instead of applying the Daubechies filter [

To verify the results obtained, Libin and Yudakin have calculated the mutual power spectra and coherence spectra for solar activity and temperature (Figure ^{10}Be (Figure

Power spectra density of solar activity and northern hemisphere temperature (1902–2010, blue) and power spectra density of greenland ice cores (red).

Power spectra density of solar activity and storminess (1946–2010)

Power spectra density of solar activity and lake baykal leves l (1926–2010).

Power spectra density of solar activity and cosmic ray intensity (1950–2010, blue) and power spectra density of solar activity and ^{10}Be (1482–2010, red).

All the above figures show the presence of stable 30-year fluctuations for almost all the processes. Although the coefficients of coherence are not always superior to a 95% confidence interval probability the observed peaks are almost always above 90%.

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Periodogram of the monthly means of monthly stratospheric CR measurements near Moscow [

It can be seen that an approximately 34-year cycle is present in the data. A similar indication was obtained from the data of a modulation parameter of CR from [

The highest maximum is at 11 years. The second most pronounced one is at approximately 0.031, corresponding to approximately 32 years. Here the levels of significance correspond to white noise.

We have shown the existence of a 30-year periodicity for cosmic rays. We can argue here that such a frequency is quite probably a modulator for terrestrial phenomena: it seems that in some way cosmic rays modulate climatic phenomena, such as the Atlantic Multidecal Oscillation (AMO) and sea-surface temperature (SST), and these, in turn, modulate hurricane development [

We use the inverse wavelet transform to obtain the decomposition of a signal which can be obtained from a time-scale filter [

The autoregressive moving-average model (ARMA) is one of the mathematical models used for the analysis and prediction of stationary time series in statistics. The ARMA model is a generalization of two simpler time series models—an autoregressive model (AR) and the moving average model (MA).

The ARMA (

Such a model can be interpreted as a linear multiple regression model, in which the explanatory variables are the past values of the dependent variable itself, but as a regression balance—moving averages of the elements of white noise. ARMA-processes are more complex compared to similar processes in a pure form; however the ARMA processes are characterized by fewer parameters, which is one of their advantages.

If we introduce the lag operator

Introducing the shorthand notation for polynomials of the left and right sides, the previous equation can be written as

For the process to be stationary, it is necessary for the roots of the characteristic polynomial of the autoregressive part

For example, the process ARMA (1,0) = AR (1) can be represented as an MA process of infinite order with coefficients in decreasing geometric progression:

Thus, the ARMA processes can be considered to be MA processes of infinite order with certain restrictions on the structure coefficients. There is a small number of parameters to describe the processes they enable rather than a complex structure. All stationary processes can be arbitrarily approximated by an ARMA model of a certain order with considerably fewer parameters than MA models use.

In the presence of unit roots of the p autoregressive polynomial, the process is nonstationary. Roots of less than unity in practice are not considered, since they are processes which exhibit explosive behavior. Accordingly, to test the stationary nature of a time series of basic tests, tests must be run for unit roots. If the tests confirm the presence of unit roots, then we need to analyze the difference between the original time series and a stationary process of the differences of one or two orders (usually the first order is sufficient and sometimes the second) of the ARMA-based model.

Such models are called ARIMA models (integrated ARMA) or Box-Jenkins models. The ARIMA model (

The ARIMA process (

To construct the ARMA model on a proxy data series of observations, it is necessary to determine the model order (numbers

To determine the order of the model an investigation of these characteristics of the time series can be done, seen as its autocorrelation function and partial autocorrelation function.

To determine the coefficients the method of least squares and maximum likelihood method can be used.

In the classic ARMA model, it can add exogenous factors x. In general, the model involves not only the current values of these factors but also lagged values. Such models are usually denoted ARMAX (

It should be noted that such models can be interpreted differently, for example, ADL (

V. Velasco gives special thanks to the CONACyT (project 180148) for financial support to this work.

^{10}Be in polar ice to trace the 11-year cycle of solar activity

^{10}Be record from the NGRIP ice core, Greenland