A&A 509, A5 (2010) Astronomy DOI: 10.1051/0004-6361/200913095 & c ESO 2010 Astrophysics The chaotic solar cycle I. Analysis of cosmogenic 14C-data A. Hanslmeier1 and R. Brajša2 1 Inst. für Physik, Geophysik Astrophysik und Meteorologie, Univ.-Platz 5, 8010 Graz, Austria e-mail: [email protected] 2 Hvar Observatory, Faculty of Geodesy, University of Zagreb, Kaciˇ ceva´ 26, 10000 Zagreb, Croatia e-mail: [email protected] Received 9 August 2009 / Accepted 18 September 2009 ABSTRACT Context. The study of solar activity over long time intervals using proxies. Aims. The periodicity of the solar activity cycle is studied. The solar activity cycle is governed by a complex dynamo mechanism. Methods of nonlinear dynamics enable us to learn more about the regular and chaotic behavior of solar activity. Methods. By applying methods of nonlinear dynamics, the solar activity cycle is studied by using solar activity proxies that have been reaching into the past for over 10 000 years. The complexity of the system is expressed by several parameters of nonlinear dynamics, such as embedding dimension or false nearest neighbors, and the method of delay coordinates is applied to the time series. Results. The solar activity cycle is found to be on the edge of chaotic behavior. This can explain the observed intermittent period of longer lasting solar activity minima. Filtering the data by eliminating variations below a certain period (the periods 380 yr and 57 yr were used) yields a far more regular behavior of solar activity. Conclusions. The solar time series of solar activity proxies used here clearly shows that solar activity behaves differently from random data. The unfiltered data exhibit a complex dynamics requiring an embedding dimension >15. The variations become more regular when filtering the data. The results also indicate that solar activity proxies are influenced by other than solar variations and reflect solar activity only on longer time scales. Key words. solar-terrestrial relations – Sun: activity – Sun: general 1. Introduction Nonlinear effects in the solar activity have been studied in theoretical models and reviewed by several authors (Stix The Sun is the only star where its activity cycle has been stud- 1981; Ruzmaikin 1985; Hoyng 1990, 1992; Rosner & Weiss ied in detail and can be traced back in time over periods span- 1992; Wilson 1994; Stix 2002; Ossendrijver 2003; Rüdiger & ning several 10 000 years by using proxies for solar activity. Hollerbach 2004). In particular, the question of whether the so- Such long time series are needed to answer the question whether lar cycle is an example of deterministic chaos has been examined the solar activity and the underlying dynamo mechanism is of in different papers based on theoretical considerations and data periodic, chaotic or stochastic nature. In a chaotic system a analysis (Schmalz & Stix 1991; Jennings 1992; Ruzmaikin et al. strong sensitivity on initial conditions exists. For slightly differ- 1992; Carbonell et al. 1994; Tobias 1996; Charbonneau 2005; ent initial conditions (or perturbations of the system) the solu- Letellier et al. 2006; Volobuev 2006). Brandenburg & Spiegel tions diverge exponentially giving the impression of a stochastic (2008) introduce on/off intermittency into a mean field dynamo behavior. A stochastic process is one whose behavior is non- model by imposing stochastic fluctuations in either the α-effect deterministic in that a system’s subsequent state is determined (a measure for the effect of convection and coriolis forces) or both by the process’s predictable actions and by a random ele- through the inclusion of a fluctuating electromotive force. Small- ment. Up to now it has not been clear whether the solar dynamo scale fluctuations with time scales in the order of 0.3–3 years, behaves like a chaotic, or a stochastic system. Chaotic phenom- which are sufficiently strong, can produce long-term variations ena in astrophysics and cosmology, mainly for dynamics in the in the system on time scales of hundreds of years. They find no solar system and galactic dynamics, as well as applications to suppression of magnetic activity in both solar hemispheres and cosmology, such as properties of cosmic microwave background point out that the relation between the level of solar magnetic ac- radiation, were reviewed by Gurzadyan (2002), while the phe- tivity and sunspot numbers may be never understood completely. nomena showing evidence of the nonlinear dynamics in solar physics are summarized by Hanslmeier (1997). These theoretical models help to explain the observed peri- Direct solar activity observations are only available for a few ods of strongly reduced solar activity, as for the most often cited centuries and proxies such as observations of aurorae, cosmo- the Maunder minimum (1645–1715) which was, e.g., described genic isotopes, growth of certain plants like corals, etc. have by Eddy (1976). The solar activity was not zero during that pe- to be used to obtain a longer time series of the solar activity. riod, and Ribes & Nesme-Ribes (1993) presented results from An overview of these proxies can be found, e.g., in Hanslmeier observations of sunspots during that period made at the Paris (2007). observatory. Article published by EDP Sciences Page 1 of 7 A&A 509, A5 (2010) Weiss (1988) studied secular solar and geomagnetic varia- cycle and introduced large-amplitude stochastic fluctuations in tions in the past 10 000 years. Feynman & Gabriel (1990) studied either or both the meridional flow and poloidal source terms in the 88 year cycle and the Maunder minimum and their conclu- the model. Solar cycle-like oscillatory behavior persists even for sion was that there is evidence for a chaotically behaving Sun, fluctuation amplitudes as high as 300%. because the solar dynamo is chaotic and is operating in a re- In this paper we study a proxy for solar activity, the isotope gion close to the transition between period doubling and chaos. 14C. This will enable us to use a long time series for the inves- Since Maunder-type minima reoccur irregularly during the mil- tigations. The abundance of this isotope depends on the cosmic lennia, it appears that the Sun remains close to this transition to ray flux, which is lower when the Sun is in its most active phase. and from chaos. Tapping et al. (2007) developed a model indi- There is an anticorrelation in the production of cosmogenic iso- cating a lower mean irradiance during the Maunder minimum of topes and solar activity. Geomagnetic field variations also cause 1 ± 0.4 W m−2 lower than the mean irradiance over the last solar irregularities in the 14C data. To estimate possible short term ge- activity cycle. omagnetic influence, the data were convolved by random func- The mean monthly sunspot index, for a period of 235 years tions of different amplitudes. was studied by Pavlos et al. (1992). The existence of a low- dimensional strange attractor was shown and a correlation di- 2. Data and data analysis mension (about 4.5) was given. Spiegel (1993) and Spiegel & Zahn (1996) propose that the solar butterfly diagram represents 2.1. Data the motion of activity waves that are very stable nonlinear soli- The sunspot number was reconstructed based on dendrochrono- tary waves. The waves are driven by an instability under the 14 convection zone in a layer called the tachocline. Charbonneau logically dated C radiocarbon data covering the past (2001) studied one-dimensional maps to which dynamo equa- 11 400 years (see Solanki et al. 2004). Our data consist of a tions can be reduced. A strong odd-even signal is found to be a series of reconstructed 10-year averaged sunspot numbers with good precursor to the transition from bursting to quiescent be- their 68% uncertainty. Years are given BP (before present), i.e., havior. the calendar AD year, Yad, is related to the BP year, Ybp, as Yad = 1950-Ybp. The data are available in tabular form. The tab- Consolini et al. (2006) make a principal-components (PC) ulated years correspond to centers of the corresponding 10-year analysis of the solar cycle. Their main results can be summarized intervals. Negative values are artifacts and are consistent with as: the main 11-year cycle can be described by only 2 modes, zero within the error limits. These 14C radiocarbon data are used a full description of the chaotic nature of solar cycle, however, for applying different methods of nonlinear dynamics. requires up to 28 modes. The Lyapunov exponents of the first 28 As stated in the introduction, the variations seen in the data PCs are evaluated showing the existence of two different chaotic have not only a solar origin but also depend on geomagnetic field regimes. variations. The data were corrected for palaeogeomagnetic vari- Chaos and intermittency in the solar cycle were reviewed by ations. However, to test the robustness of our results, field vari- Spiegel (2009). In this paper a description of the behavior of ations were simulated by adding random data of different am- the solar cycle (both in time and spatial domain) by means of plitudes to the original values and then studying how the results waves is suggested. If the solar activity cycle resembles a non- changed. linear dynamo, this can be approximated by a complex system of Lorenz equations (Jones et al. 1985). From a sixth order sys- tem of ordinary differential equations plane nonlinear dynamo 2.2. Methods of nonlinear dynamics waves can be obtained. These can model the activity of stars, and a sequence of bifurcations is observed that ends in chaos as 2.2.1. Time series analysis the dynamo number D is increased. Letellier et al. (2006) studied First we calculated the mutual information.