A statistical analysis of polarity reversals of the geomagnetic field
Introduction
Local paleomagnetic measurements of the geomagnetic field Merrill et al., 1996, Hollerbach, 2003, Cande and Kent, 1995 are currently used to extract informations about the geomagnetic dipole, thus providing information regarding the geodynamo process. Unlike the solar magnetic field, where the polarity reversals are strictly periodic, with a main period of 22 years, geomagnetic measurements reveal a sequence of sudden and occasional global polarity reversals in the last 160 Myear. The typical duration of reversals is a few thousand years, that is much shorter than the typical time interval between successive reversals, which may range from up to years Merrill et al., 1996, Cande and Kent, 1995, Valet and Meynadier, 1993. Despite the considerable work performed both on data analysis and on theoretical modeling, the main fundamental questions concerning the relation between the Earth’s magnetic dynamo process and polarity reversals remain still unanswered Merrill et al., 1996, Roberts and Glatzmaier, 2000, Moffatt, 1978, Stefani and Gerbeth, 2005. The nature of the triggers (external or internal to Earth) and the physical mechanisms giving rise to the reversals, the reason for the long time variations in the average reversal rate (cf. e.g. Hollerbach, 2003, Yamazaki and Oda, 2002), are all open problems.
The sequence of geomagnetic reversals displays a behaviour which seems to be the result of a chaotic (or stochastic) process. An example of reversal dataset, namely the well known CK95 database (Cande and Kent, 1995), is shown in Fig. 1. In principle, the geodynamo should be described by 3D global magnetohydrodynamics (MHD) that self-consistently solve for the fluid flow, thermodynamics and magnetic fields with all nonlinear feedbacks (for a review see Roberts and Glatzmaier, 2000 and references therein, and the results of the recent 15.2 TFlops simulation of geodynamo on the Earth Simulator Kageyama et al., 2004, Kageyama and Yoshida, 2005). However, even the most advanced numerical codes developed so far do not have a high enough resolution to be confident that the critical dynamics is being captured by the simulation, and moreover they are able to simulate only short series of spontaneous reversals. Non-regular reversals can be observed also in several types of simplified models, such as purely deterministic toy models mimicking the dynamics of the dynamo effect with just few modes Rikitake, 1958, Crossley et al., 1986, Turcotte, 1992, models of noise-induced switchings between two metastable states Schmitt et al., 2001, Hoyng et al., 2002, Hoyng and Duistermaat, 2004, or mean-field dynamo models with a noise-perturbed profile Giesecke et al., 2005, Stefani and Gerbeth, 2005.
A statistical approach is extremely useful for the characterization of a complex process such as the time evolution of the geomagnetic field. Notwithstanding the paucity of datasets, it was commonly assumed that the reversal sequence is produced by an underlying Poisson process. This assumption relies on the fact that the distribution of persistence times, defined as the time intervals between two consecutive reversals seems to follow an exponential Merrill et al., 1996, Hoyng et al., 2002, McFadden and Merrill, 1997, Constable, 2000, namely , where represents the reversal occurrence rate. However, also other distributions are invoked in the literature, see e.g. Jonkers (2003), where the author claims that the frequency distribution of time intervals between Cenozoic geomagnetic reversals approximates a power law for large . Due to the small number of events in the datasets, the probability distribution function , shown in Fig. 1 for the CK95 dataset, does not clearly show, at a first look, a clear behaviour.
Another interesting feature which has been recently brought to the attention of the scientific community by Constable (2000) is the time dependence of the occurrence rate of events . The author showed that a Poisson model with a monotonic varying rate, either increasing or decreasing, fails in describing the reversal process. Nevertheless, reversals could perhaps be modeled as a renewal Poisson process with a rate that must change sign at some interval before 158 Myear (Constable, 2000). In any case, modeling the variations of over the entire time interval is a very delicate issue (cf. also McFadden and Merrill, 1984, Gallet and Hulot, 1997, McFadden and Merrill, 1997). Moreover, when the occurrence rate depends on time, it is difficult to determine the Poisson character of events and to give a clear physical interpretation to the persistence time distribution (Feller, 1968).
Here, starting from the above empirical evidences, and using a simple statistical test on some databases, we investigate whether a conjecture based on the occurrence of a Poisson process for reversals is correct or not. We show in Section 2 that this is indeed not the case, and that geomagnetic reversals are clustered in time, a result which can be attributed to the presence of memory in the process generating polarity reversals. In Section 3 the results obtained from data analysis are compared with those arising from some dynamical models of the geodynamo. Conclusions are given in Section 4.
It is also worth mentioning that the problem studied in this paper is of broader interest, as abrupt flow reversals have been observed also in the large-scale circulation during turbulent Rayleigh-Benard convection Benzi, 2005, Fontanele Araujo et al., 2005, Tsuji et al., 2005, Brown et al., 2005, or in the wind direction in atmosphere (van Doorn et al., 2000). In all these cases, it is assumed that reversals are Poisson events.
Section snippets
Local Poisson hypothesis
Following the empirical evidence that the rate of reversals is not constant, we can test, as a zeroth order hypothesis, whether the reversal sequence is consistent with a Local Poisson Process. More precisely, we will test the hypothesis (hypothesis ) that the geomagnetic reversals are originated by a time-varying Poisson process. Since the reversals rate is not known, the test must be independent on the rate . This can be done through a method introduced some years ago in cosmology (Bi
Modeling geodynamo
In the previous section we showed that clustering is present in the data (cf. Section 2). This indicates that the process underlying the polarity reversals is characterized by memory effects, due to the presence of long-range correlations. The fact that the dynamics of the fluid earth core is affected by its history, with generation of correlations in the reversal sequence, is not surprising from a physical point of view. Indeed, a similar behaviour has been recently observed for the solar
Conclusion
In this paper the statistical properties of persistence times between geomagnetic reversals have been investigated. We performed a statistical test which showed that geomagnetic reversals are produced by an underlying process that is far from being locally Poissonian, as conjectured by Constable (2000). Thus, the sequence of geomagnetic reversals is characterized by time correlations. As spontaneous reversals of the geodynamo field have been observed in high resolution numerical simulations
Acknowledgement
We acknowledge useful discussions with A. Noullez.
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