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Geomagnetic jerks as chaotic fluctuations of the Earth’s magnetic field
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E. Qamili1,2, A. De Santis1,3,*, A. Isac4, M. Mandea5, B. Duka6
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Istituto Nazionale di Geofisica e Vulcanologia, Roma, Italy
Scuola di Dottorato in Scienze Polari, Università degli Studi di Siena, Siena, Italy
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Università G. D’Annunzio, Chieti, Italy
Geological Institute of Romania, Bucharest, Romania
Directorate for Strategy and Programmes, Centre National d'Etudes Spatiale, Paris, France
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Faculty of Natural Sciences, University of Tirana, Tirana, Albania
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Author to whom correspondence should be addressed: [email protected]
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Abstract
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The recent geomagnetic field is chaotic and can be characterised by a mean exponential time scale
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τ after which it is no longer predictable. The field is also ergodic so time analyses can substitute the
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more difficult phase space analyses. Taking advantage of these two properties of the Earth's
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magnetic field, a scheme of processing global geomagnetic models in time is presented in order to
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estimate fluctuations of the time scale τ. Considering that the capability to predict the geomagnetic
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field is reduced over geomagnetic jerks occurrence periods, here we propose a method to detect
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these events. This approach considers that epochs characterised by relative minima of fluctuations
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of the estimates of τ, representing periods when the geomagnetic field is less predictable, are
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possible jerks occurrence times. We analyse the last 400 years of the geomagnetic field (data from
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Gufm1, IGRF and CM4 models) to detect epochs with lower values of the time scale as coinciding
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with the occurrence of jerks. Most of the well known jerks are confirmed through this method and a
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few others have been detected.
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Keywords: Geomagnetic field; Geomagnetic Jerks; Ergodicity; Chaos
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1. Introduction
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The first time derivative of the geomagnetic field, i.e. the secular variation (SV), represents the
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temporal evolution of the Earth’s core magnetic field on time scales longer than about one year. The
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most rapid features on the slope of this magnetic secular variation are the so-called geomagnetic
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jerks [Courtillot et al., 1978] or as recently suggested, geomagnetic rapid secular fluctuations
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[Olsen and Mandea, 2008; Mandea and Olsen, 2009] which have time scales from several months
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to few years [Macmillan, 2007]. These events are observed in the magnetic records as sudden V-
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shaped changes in the slope of SV, as an abrupt change in the second time derivative i.e. secular
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acceleration, of the geomagnetic field, or equivalently, as an Dirac-delta function in the third time
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derivative [Mandea et al., 2010]. Usually geomagnetic jerks are particularly visible in the eastward
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(Y) component, which is supposed to be the least affected geomagnetic field component by external
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fields.
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In the past, different types of analysis have been introduced to allow the identification of
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geomagnetic jerks. Methods of detection of jerks are various but mostly based on spectral
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techniques [e.g. Mandea et al., 2010 and references therein]. We propose here a new method based
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on the so-called Nonlinear Forecasting Approach [NFA; Sugihara and May, 1991]. This technique
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is normally able to detect a possible exponential divergence of some prediction from the real signal
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in the phase space which is reconstructed from the time-delay of the original signal. An important
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parameter is the mean exponential characteristic time τ, after which no reliable prediction can be
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made [Barraclough and De Santis, 1997; De Santis et al., 2002].
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Combining this technique with the most classical and widely used in geomagnetism (Spherical
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Harmonic Analysis: SHA) we can perform our analysis in the usual time domain taking advantage
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of the geomagnetic field ergodicity [De Santis et al., 2011]. This study deals with the detection and,
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possibly, confirmation of the presence of geomagnetic jerks by means of NFA in time in order to
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detect those epochs when the geomagnetic field appears more chaotic, i.e. those less predictable
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periods that are characterised by smaller characteristic time τ. We then consider these times a
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possible estimates of jerks occurrence.
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The organization of this paper is as follows. After this introduction, the next section describes more
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in detail the geomagnetic jerk phenomena. Section 3 is dedicated to the nonlinear forecasting
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approach in time. The following section introduces the global geomagnetic models used for this
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analysis. It shows the temporal behaviour of errors between predicted and definitive geomagnetic
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models. The trend in time of these errors is used to detect geomagnetic jerks. Finally, we discuss the
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results.
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2. Geomagnetic jerks
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Several geomagnetic jerks have been noted as occurring over the 20th and 21th centuries. The first
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geomagnetic jerk has been detected at the end of 70s’ by Courtillot et al. [1978]. Since then,
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applying different analysis techniques to geomagnetic series from worldwide observatories, many
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other events have been detected, in particular around 1901, 1913, 1925, 1932, 1949, 1958, 1969,
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1978, 1986, 1991, 1999 and 2003 [Mandea et al., 2010 and references therein]. Among them, some
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geomagnetic jerks are characterized by a very large scale extension, sometimes considered global
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(1969, 1978, 1991, 1999), other events (1901, 1913, 1925) have possibly a similar extension but the
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irregularity in data distribution does not allow to confirm such an aspect, while other four (1932,
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1949, 1958, 1986) have not been detected everywhere at the Earth’s surface [e.g. Mandea et al.,
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2010 and references therein]. The occurrence of 2003 jerk, which is the first event detected by
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means of high-quality and global coverage data from three magnetic satellites, provided a clear
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picture on the geomagnetic jerk characteristics, showing the regional nature of the 2003 event that
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was most obvious in the vertical component [Olsen and Mandea, 2007; Mandea et al., 2010]. The
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authors hypothesized that in general, geomagnetic jerks haven’t global extension. But recently
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Pinheiro et al. [2011], from an analysis of error bars in the time occurrence of geomagnetic jerks,
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confirmed the worldwide characteristic of 1969, 1978 and 1991 geomagnetic jerks while the 1999
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event was detected only locally. It is interesting to notice that the geomagnetic jerks detected over
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the twentieth century are characterized by a mean repeat time interval of around 9 years, although
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there is the recent tendency to detect more densely occurrences of jerks (see 2003 jerk). The latter
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feature has been also considered a possible precursor for an imminent geomagnetic field reversal or
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excursion [De Santis, 2007].
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Going back in time, unfortunately, complete time series data prior to the 20th century are not
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available so is extremely difficult to detect such events. Nevertheless some individual time series of
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declination and inclination have been computed for London, Rome, Edinburgh, Paris, Bucharest
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and Munich [Malin and Bullard, 1981; Cafarella et al., 1992; Barraclough, 1995; Alexandrescu et
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al., 1996; Soare et al., 1998; Korte et al., 2009]. Using these data other four geomagnetic jerks have
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been identified in 1700, 1730, 1750 and 1870 [Alexandrescu et al., 1997]. Korte et al. [2009],
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studying more than 600 historical declination values form the southern Germany and surrounding
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areas, confirmed the presence of some of the geomagnetic jerks (1700, 1730, 1750) detected by
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Alexandrescu et al. [1997] but with some time offset (around 10 years). In their study the authors
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suggest geomagnetic jerks in the following epochs: 1410, 1448, 1508, 1598, 1603, 1661, 1693,
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1708, 1741, 1763, 1861, 1889 and 1932.
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On the definition and on the characteristics of geomagnetic jerks (i.e., the date at which it
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occurs, the time duration of the impulse, and/or the space distribution of events) there has been
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much debate within the geomagnetic community. As said before, a fundamental point is the
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interpretation of the so-called worldwide jerks. Different authors [e.g. Alexandrescu et al., 1996]
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suggest that geomagnetic jerks are phenomena visible over the whole Earth’s surface but
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characterized by a time interval of a few years between the two hemispheres. Based on an analysis
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of three jerks Le Huy et al. [1998] found an anti-correlated character between the Earth’s surface
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signatures of two successive jerks [see also Chulliat et al., 2010] that was contradicted by the
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hypothesis that geomagnetic jerks are not worldwide in occurrence, or at least, most of them are not
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global [Olsen and Mandea, 2007].
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Now, it is firmly established the internal origin of geomagnetic jerks [Jackson and Finlay, 2007
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but see also Alldredge, 1984], but their physical origin is still under discussion. Different authors
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have argued an origin caused by changes in the flow patterns of the Earth’s liquid outer core [e.g.
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Waddington et al., 1995; Olsen and Mandea, 2008]. For instance, Bloxham et al. [2002] have
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showed that jerks can be explained in terms of a combination of a steady flow with a simple time-
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dependent axisymmetric and equatorial symmetric zonal flow with typical periods of several
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decades, which is consistent with torsional oscillations in the fluid outer core [Buffett et al., 2009].
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More recently, different authors [e.g. Olsen and Mandea, 2008; Wardinski et al., 2008], analysing
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satellite and land observatory data, have found geomagnetic rapid secular fluctuations for very short
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time scales less than a couple of years [e.g. Mandea and Olsen, 2009], which have been also called
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as geomagnetic jerks proposing that the torsional oscillations may also explain these observed
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sudden changes in core surface flows. However, the torsional oscillations may not be the exclusive
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cause of the core surface flows because the time scale of the torsional oscillations is of the order of
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several decades and the reoccurrence period of geomagnetic jerks is shorter than 10 yr [Holme and
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de Viron, 2005; Wardinski et al., 2008]. On the other hand, Gibert et al. [1998] proposed a possible
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correlation between geomagnetic jerks and changes in phase of the Chandler wobble within at most
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3.5 years, a correlation supported by recent analyses made by Gibert and Le Mouël [2008].
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Mandea et al. [2000] have proposed that geomagnetic jerks could be indicators that anticipate
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the changes in the Earth’s rotation rate. The comprehension of geomagnetic jerks and of their
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distribution in space and time can help us to better understand the origin of the magnetic field and
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its variations, the evaluation of the electrical conductivity of the lower mantle and its possible
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lateral heterogeneity [Mandea-Alexandrescu et al., 1999; Pinheiro and Jackson, 2008; Nagao et al.,
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2003], and the verification of some hypotheses regarding the internal structure of the Earth.
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However, some of the above features are still under discussion.
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3. Nonlinear chaotic analysis of the geomagnetic field
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Many methods, used to find possible nonlinearity and chaos characterizing a system, are based on
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reconstruction of the phase space, i.e. a generalised reference coordinate space, where a dynamical
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system can be represented by the trajectory of the point, with the independent variables as variations
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along the axes.
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As a chaotic system, the recent geomagnetic field is sensitive to initial conditions [Barraclough and
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De Santis, 1997; De Santis et al., 2002; De Santis et al., 2004; De Santis and Qamili, 2010] and the
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average divergence of initially close trajectories propagates exponentially with time, i.e. ε(t) ≈ ε0 eKt
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(with K>0). This implicates the limitation of time on which these systems can be predicted and that
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the prediction error increases exponentially with time. K is the Kolmogorov Entropy which
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quantifies the rate of information loss in a chaotic process [e.g. Wales, 1991], such as, for instance,
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in the case of a transmitting signal in a chaotic medium [Schuster and Just, 2005] and it is inversely
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proportional to the mean length of the time over which a chaotic system is predictable. K entropy is
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zero for completely deterministic/regular system and infinite for random systems, but finite and
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larger than zero for chaotic processes. This implies that after a mean characteristic time τ=1/K no
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reliable predictions can be made. According to Taken’s theorem [Takens, 1991] it would be
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possible, by a time delay embedding, to reconstruct a phase space topologically equivalent with the
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original one from a single observable quantity.
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However, De Santis et al. [2011] demonstrated the ergodicity of the recent geomagnetic field, i.e.
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the equivalence of time averages to phase space averages. Under these conditions, the same authors
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have applied the NFA in the time domain without any reconstruction of the phase space. They have
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simply analysed the divergence of the errors between predicted and definitive global geomagnetic
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models. The accuracy of the actual models let us to confirm that these models can not provide
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reliable predictions after around τ=6 years.
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4. Global models used in the analyses
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Global models of the geomagnetic field help us to study the recent and past geomagnetic field and
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predict its short term evolution in time. These models provide sets of Gauss coefficients, g n and
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hnm , at successive epochs or some temporal functions of them. The models may also provide a set of
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predictive coefficients to estimate close future values of the field [De Santis et al., 2011]. The
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geomagnetic global models analysed in this work are briefly described below.
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The IGRF model (International Geomagnetic Reference Field; Finlay et al. [2010]) is a series of
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mathematical models of the Earth's main field and its secular variation in terms of a spherical
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harmonic (SH) expansion of the geomagnetic potential up to degree and order N=10 till 1995 and
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N=13 from 2000 and SV up to degree N=8. These models, whose Gauss coefficients are available
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every five years, cover the time span from 1900 to 2010 and are mostly based on observatory and
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satellite data.
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Gufm1 is a SH model up to degree and order N=14 based on historical ground and marine data
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for the interval from 1590 to 1990 [Jackson, et al., 2000]. The time-dependent field model is
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parameterized spatially in terms of spherical harmonics and temporally in B-splines.
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CM4 (Comprehensive Model; Sabaka et al. [2004]) is based on observatory quietest day data
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for the period 1960-2002 and on POGO, MAGSAT, CHAMP and Ørsted satellite properly selected
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data. The internal (core and lithospheric) magnetic fields are represented by a degree and order up
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to N=65 spherical harmonic expansion. Also for this model as the previous one, the temporal
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changes of the coefficients (but limited up to degree N=13) are described by cubic B-splines.
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In the next section we will consider time intervals of 5 years from 1600 to 1980 (each time
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extrapolating the field for the successive 10 years) for Gufm1 and from 1990 to 2000 (each time
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extrapolating the field for the successive 5 years) for IGRF and CM4.
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5. NFA and description of results together with interpretation
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Extrapolating the geomagnetic field of all the above models, outside the typical time of validity,
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would cause very large errors. We can estimate these errors ε, from predictive and definitive model
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Gauss coefficients, as [Maus et al., 2008; De Santis et al., 2011]:
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ε=
1
N
n
n =1
m=0
∑ (n + 1)∑ ⎡⎣(cnm ) predictive − (cnm )definitive ⎤⎦
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(3)
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where (cnm )2 = (g nm )2 +(hnm )2 .
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As in De Santis et al. [2011], we impose the same initial value for both predicted and definitive
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models with an offset equal to -ε for each exponential growth. In this case all error segments can be
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represented by an exponential function s(t) = ε0 (et/τ− 1), where ε0 is a constant that measures the
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initial difference between prediction and actual value whilst τ is the characteristic time of growth
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that is related to K-entropy when the system under study is ergodic and chaotic. As in De Santis et
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al. [2011] analysing an ensemble of subsequent diverging predicted-definitive differences we can
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estimate <τ >≈T=1/K, where <τ > is the mean value of the characteristic time associated with the
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chaotic character of the field.
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From eq. (3) we calculate the error of 10-year segments at steps of 5 years taken from the
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differences between predictive and definitive models. Since the Gufm1 model does not give a
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predictive field we estimate the SV coefficients from the field coefficients in the 10 years prior to
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each considered epochs. Then, on the basis of this averaged SV, we produce the prediction field
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values for subsequent 10 years and compare them with the real Gufm1 field values for the same
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period of time. Thus, we analysed the period from 1600 to 1980, with extrapolations to the
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successive 10 years. We also extended the analysis from 1990 to 2000, with extrapolations over the
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successive 5 years, considering IGRF and CM4, the former as predictive model and the latter as
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definitive.
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For all the analysed segments we found a clear exponential growth from which we obtain the
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characteristic time τ and its standard deviation at steps of 5 years. The behaviour over time of τ with
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the corresponding standard deviation (as error bar at each epoch) is shown in Figure 1. Taking into
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account all the analysed segments we estimate a mean value <τ> = 7 years with an uncertainty of
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around 2 years. The divergence of errors for all the considered segments is a clear evidence of a
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chaotic geomagnetic field and the mean time <τ> of 7 years agrees, within the estimated
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uncertainties, with the results from previous analyses [De Santis et al., 2002; 2011]. This time
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would represent the mean time window of predictability, i.e. that mean time after which no reliable
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prediction for the field can be made. As Figure 1 shows, actually τ value fluctuates epoch by epoch
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around its mean value: indeed, there are periods where the τ value is relatively lower than the mean
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value. In addition, from the same figure it is clearly evident there are some periods when τ shows
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larger than usual values (even greater than a standard deviation). These values could be attributed
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to: a) better recent measurements with respect to the ones in the past, and, consequently, b) to better
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models, which show a slightly better prediction. Since the fact does not affect significantly the
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tendency of epochs with jerks to reduce the relative τ value (this is the reason why the most
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important thing is to look at relative and not absolute minima), we do not make any adjustment or
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correction to the data to take account of this little effect.
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Following this approach we try to confirm (and possibly to detect) the presence of well-known (and
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unknown) geomagnetic jerks. Here, we make the simple hypothesis that there is a jerk when τ
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assumes a (relative) minimum value with respect to the surrounding temporal values. In Figure 1,
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clear and distinct epochs are evident when τ assumes low values (evidenced by arrows), i.e. those
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time scales for which the capability to predict the geomagnetic field future evolution is rather
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reduced. Considering the above assumption, we confirm the presence of all known geomagnetic
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jerks between 1700 and 2000 (i.e. 1700, 1730, 1750, 1800-1820, 1870, 1901, 1913, 1925, 1949,
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1958, 1969 and 1999; red arrows). In addition, there are also other four periods (i.e. at 1720, 1825,
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1850-1855 and 1880-1885; blue arrows) where τ assumes a relative minimum value but that, to our
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knowledge, do not correspond to known geomagnetic jerks. Is important to underline the fact that
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although the most recent two blue arrows (1850-1855 and 1880-1885 events) are within the range
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of average interval of time prediction (within the statistical error) 5-7 year, so they could be even
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"normal" oscillations, nevertheless, we keep them because they are close to large extremes.
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Most of the above results indicate that a geomagnetic jerk occurs in those epochs when the
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main field suddenly becomes more chaotic (less predictable) than usual, because characterized by
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τ values lower than usual. This finding can be explained by the fact that geomagnetic jerks could be
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a manifestation of larger fluctuations (with lower values of τ) in the SV curve of some chaotic
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processes that take place in the outer fluid core. Recently different authors [e.g. Gillet et al., 2010]
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have concentrated their study on these topics in order to understand and also to explain the
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processes in the outer core through the study of some variations in the length-of-day (ΔLOD) signal.
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The association between a corresponding signal in ΔLOD and geomagnetic jerks is well known
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[Mandea et al., 2010], i.e. an abrupt change in SV should be matched by an equivalent abrupt
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change in the core flow which in turn is associated with torsional oscillations and the same abrupt
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should be seen in LOD. Gillet et al. [2010] detected a 6 year periodicity between independent
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changes in ΔLOD data and the predictions from the average of an ensembles inversion of core flow
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models over the time spam 1925-1990. The authors explained this result by fast torsional
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oscillations in the fluid core with a fundamental mode of 6 years, a time that could explain the
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occurrence of rapid inter-annual flow variations and the abrupt of geomagnetic jerks. Our results
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highlights another interpretation of this temporal mode as the longer persistent time that can be
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allowed by the chaoticity of the field. This would also explain the apparent gap in the ΔLOD and
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SV coherence spectrum between 6 and 20 years.
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6. Conclusions
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Using the NFA in the time domain we have analysed the temporal behaviour of the deviation
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between predictive and definitive geomagnetic field models for successive intervals from 1600 to
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2000 using Gufm1 and CM4 (IGRF) geomagnetic models. For all the considered 10-year (5-year,
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for IGRF-CM4) moving windows, at steps of 5 years, we have found a similar exponential temporal
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growth with a characteristic time of prediction limited at around 7 years, but slightly fluctuating in
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general within ±2 years. These values are sometime lower than usual and we consider them as
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epochs of occurrence of a possible geomagnetic jerk. From the hypothesis we made above, i.e. there
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is a jerk when τ assumes a minimum value with respect to the surrounding temporal values, we
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have confirmed the presence of all the geomagnetic jerks known in the literature. For example, the
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1700/1708, 1730/1741, 1750/1763, 1870/1861, 1889/1901, 1925/1932 events (the first event is
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from Alexandrescu et al. [1997] whilst the second is from Korte et al. [2009] that is thought to
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represent the same events because of the time offset suggested by Korte et al. [2009]), have been
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confirmed here. Some geomagnetic jerks (1770, 1800-1820), proposed by Paris declination curve
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but questioned by Korte et al. [2009], are clearly visible in our analysis. Perhaps also an event at
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around 1600 could be detected (Korte et al. [2009] suggests a jerk at 1603) if we consider the low
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value of τ in Fig. 1. We also have a relative minimum at around 1670 which may correspond to the
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1661 suggested event [Korte et al., 2009]. From Fig. 1 also all the twenty century well-known
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geomagnetic jerks are clearly detectable. In this analysis we have detected some other periods (e.g.
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1720, 1825, 1850-1855, 1880-1885) that can correspond to some unknown jerks, which will
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deserve further investigations.
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Here, by processing global geomagnetic models in time, we detected fluctuations which
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reduce locally the characteristic time τ, indicating a less predictable geomagnetic field. This
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instability of the field could be produced by some processes in the Earth’s core, where the
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geomagnetic field is produced and maintained. These characteristics could be explained by torsional
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oscillations in the fluid outer core that may justify also the presence of geomagnetic jerks.
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We recognise that, due to the intrinsic limitation of the method in using each time temporal
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segments of a minimum of 5 years, the method represents a coarse way to find the geomagnetic
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jerks. Nevertheless, our results show that an investigation of errors between predictive and
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definitive parts of geomagnetic global field models is an useful tool to detect those geomagnetic
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jerks that other techniques have not been able to do and/or to confirm the presence of other already
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noted events.
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Acknowledgements
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Part of the work by EQ has been made while she was attending the PhD course at Siena University.
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Some funds have also been given by a PNRA project (“Reversing Earth Magnetism”) and a PRIN
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2008 MIUR-funded project (“Animal Magnetic Homing”, Unit of research SIM-MAG).
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Figure Caption
2
3
Figure 1. Estimates of the time of predictability <τ > every 5 years (open circles) over the period
4
1600-1980 from Gufm1 model and then, over the period 1990-2000, from IGRF (considered as
5
predictive model) and CM4 (considered as definitive model) (the last three points). The errors are
6
estimated and plotted for each epoch. The epochs of already noted geomagnetic jerks are indicated
7
by red arrows, and those for which new possible events are suggested by blue arrows.
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