Geomagnetic jerks as chaotic fluctuations of the Earth's magnetic field

Authors


Corresponding author: A. De Santis, Istituto Nazionale di Geofisica e Vulcanologia, Roma, Italy. (angelo.desantis@ingv.it)

Abstract

[1] The geomagnetic field is chaotic and can be characterized by a mean exponential time scale < τ > after which it is no longer predictable. It is also ergodic, so time analyses can substitute the more difficult phase space analyses. Taking advantage of these two properties of the Earth's magnetic field, a scheme of processing global geomagnetic models in time is presented, to estimate fluctuations of the time scale τ. Here considering that the capability to predict the geomagnetic field is reduced over periods of geomagnetic jerks, we propose a method to detect these events over a long time span. This approach considers that epochs characterized by relative minima of fluctuations in time scale τ, i.e., those periods when a geomagnetic field is less predictable, are possible jerk occurrence dates. We analyze the last 400 years of the geomagnetic field (covered by the Gufm1 model) to detect minima of fluctuations, i.e., epochs characterized by low values of the time scale. Most of the well known jerks are confirmed through this method and a few others have been suggested. Finally, we also identify some short periods when the field is less chaotic (more predictable) than usual, naming these periods as steady state geomagnetic regime, to underline their opposite behavior with respect to jerks.

1 Introduction

[2] The first time derivative of the geomagnetic field, i.e., the secular variation (SV), represents the temporal evolution of the Earth's core magnetic field. The most rapid features in the changes of slopes of this magnetic secular variation are the so-called geomagnetic jerks [Courtillot et al., 1978] or as recently suggested, geomagnetic rapid secular fluctuations [Olsen and Mandea, 2008; Mandea and Olsen, 2009], which have time scales from several months to a few years [Macmillan, 2007]. These events are observed in magnetic data as sudden V-shaped changes in the slope of SV, in other words, as an abrupt change in the second time derivative, i.e., the secular acceleration series, or equivalently, as a Dirac-delta function in the third time derivative [Mandea et al., 2010]. Usually, geomagnetic jerks are particularly visible in the eastward (Y) component, which is supposed to be the less affected by external fields.

[3] In the past, different types of analysis have been introduced to allow the identification of geomagnetic jerks. Methods of detection of geomagnetic jerks are various, mostly based on spectral techniques [e.g., Mandea et al., 2010, and references therein], or techniques applied mostly to direct magnetic measurements or global geomagnetic models [e.g., Chambodut and Mandea, 2005]. We propose here a new method that is not a spectral one, because it is based on the so-called nonlinear forecasting approach [NFA; Sugihara and May, 1990]. This technique is able to detect a possible exponential divergence of some prediction from the real signal in the phase space, which is reconstructed from the time delay of the original signal. An important parameter is the mean exponential characteristic time τ, after which no reliable prediction can be made [Barraclough and De Santis, 1997; De Santis et al., 2002].

[4] Combining this technique with the most classical and widely used in geomagnetism, i.e., the spherical harmonic analysis, we can perform our analysis in the usual time domain by taking advantage of the geomagnetic field ergodicity [De Santis et al., 2011]. This study deals with the detection and, possibly, confirmation of the presence of geomagnetic jerks by means of NFA in time, to find those epochs when the geomagnetic field appears more chaotic, i.e., those less predictable periods that are characterized by smaller characteristic time τ. We then consider these epochs as possible estimates of jerk occurrence dates.

[5] The organization of this paper is as follows. After this introduction, section 2 describes more in detail the geomagnetic jerk phenomena. Section 3 is dedicated to the nonlinear chaotic analysis of the geomagnetic field, followed by section 4, where the global geomagnetic model used for this analysis is introduced. In section 5 we describe the NFA in the time domain whereas section 6 shows the temporal behavior of errors between predicted and actual geomagnetic models: the trend in time of these errors is used to detect geomagnetic jerks. Finally, we discuss the results in section 7.

2 Geomagnetic Jerks

[6] Several geomagnetic jerks have been noted as occurring over the 20th and 21st centuries. The first geomagnetic jerk has been detected for the end of the 1970s [Courtillot et al., 1978; Malin and Hodder, 1982]. Since then, applying different analysis techniques to geomagnetic series from worldwide observatories, many other events have been detected, in particular around 1901, 1913, 1925, 1932, 1949, 1958, 1969, 1978, 1986, 1991, 1999, and 2003 [Mandea et al., 2010, and references therein]. Among them, some geomagnetic jerks are characterized by a very large spatial scale of extension, sometimes considered global (1969, 1978, 1991, 1999), other events (1901, 1913, 1925) have possibly a similar extension but the irregularity in data distribution does not allow to confirm such an aspect, while other four (1932, 1949, 1958, 1986) are considered regional events because they have not been detected everywhere at the Earth's surface [e.g., Mandea et al., 2010, and references therein]. The occurrence of the 2003 jerk, which is the first event detected by means of high-quality and global coverage data from three magnetic satellites, provided a clear picture on the geomagnetic jerk characteristics, showing the regional nature of the 2003 event that is most obvious in the vertical component [Olsen and Mandea, 2007; Mandea et al., 2010]. The authors hypothesized that in general, geomagnetic jerks have no global extension. But recently Pinheiro et al. [2011], from an analysis of error bars in the time occurrence of geomagnetic jerks, suggested the worldwide characteristic of 1969, 1978, and 1991 geomagnetic jerks, while the 1999 event was detected only regionally. In a more recent paper [Duka et al., 2012], it was found that the 1969, 1978, and 1999 events can be considered global events, while the 1991 event has a regional extension. It is interesting to notice that the geomagnetic jerks detected over the 20th century are characterized by a mean repeat time interval of around 9 years, although, thanks to a better ground and satellite data coverage, there is a recent tendency to detect more densely occurrences of jerks.

[7] Going back in time, unfortunately, complete magnetic three-component time series data prior to the second half of the 19th century are not available so it is extremely difficult to detect such events. Nevertheless, some individual time series of declination and inclination have been collected for London, Rome, Edinburgh, Paris, Bucharest and Munich [Malin and Bullard, 1981; Cafarella et al., 1992; Barraclough, 1995; Alexandrescu et al., 1996a; Soare et al., 1998; Korte et al., 2009]. Using these data other four geomagnetic jerks have been identified in 1700, 1730, 1750, and 1870 [Alexandrescu et al., 1997]. Korte et al. [2009], studying more than 600 historical declination values from the southern Germany and surrounding areas, confirm the presence of some of the geomagnetic jerks (1700, 1730, 1750), events which are also suggested by Alexandrescu et al. [1997], but with some time offset (of some years) for all the earlier events present in the Munich curve, because of the uncertainty in dating these events precisely. In their study, Korte et al. [2009] suggested geomagnetic jerks around the following epochs: 1410, 1448, 1508, 1598, 1603, 1661, 1693, 1708, 1741, 1763, 1861, 1889, and 1932 but using some caution for some of them, especially for the earliest events.

[8] On the definition and on characteristics of geomagnetic jerks (i.e., the date when they occur, the time duration of the impulse, and/or the space distribution of events) there has been much debate within the geomagnetic community. As said before, a fundamental point is the interpretation of the so-called worldwide jerks [Pinheiro and Jackson, 2008]. Different authors [e.g., Alexandrescu et al., 1996b] suggest that geomagnetic jerks are phenomena visible over the whole Earth's surface, but characterized by an eventual time lag of a few years between the two hemispheres. Based on an analysis of three jerks Le Huy et al. [1998] found an anticorrelated character between the Earth's surface signatures of two successive jerks [see also Chulliat et al., 2010], thereafter questionable by the hypothesis that geomagnetic jerks are not worldwide in distribution, or at least, some of them are not global [Olsen and Mandea, 2007; Pinheiro et al., 2011]. The very last statement has been possible because magnetic satellite data have become available, and because a global coverage has been achieved, the uneven distribution of magnetic observatories is no longer a key factor in interpretation.

[9] Currently, the internal origin of geomagnetic jerks is firmly established (Malin and Hodder [1982]; Jackson and Finlay [2007], but see also Alldredge [1984]), their physical origin being still under discussion. Different authors have argued an origin caused by changes in the flow patterns of the Earth's liquid outer core [e.g., Waddington et al., 1995; Bloxham et al., 2002; Olsen and Mandea, 2008]. For instance, Bloxham et al. [2002] have shown that jerks can be explained in terms of a combination of a steady flow with a simple time-dependent axisymmetric and equatorial symmetric zonal flow with typical periods of several decades, which is consistent with torsional oscillations in the fluid outer core [Buffett et al., 2009]. More recently, other authors [e.g., Olsen and Mandea, 2008; Wardinski et al., 2008], analyzing satellite and ground observatory data, have found geomagnetic rapid secular fluctuations with very short time scales, less than a couple of years [e.g., Mandea and Olsen, 2009], which have been also called as geomagnetic jerks, proposing that the torsional oscillations may also explain these observed sudden changes in core surface flows. Other authors [Gillet et al., 2010] focused their study on these topics to understand and explain the processes in the outer core through the variations in the length-of-day (ΔLOD). The association between a corresponding behavior in the LOD variations and geomagnetic jerks is well known [Jault et al., 1988; Holme and de Viron, 2005], i.e., an abrupt change in SV should be matched by an equivalent abrupt change in the core flow, which in turn is associated with torsional oscillations and the same abrupt change should be detected in the LOD. In particular, Gillet et al. [2010] analyzed independent changes in ΔLOD data and the predictions from the average of an ensemble of core flow models over the time spam 1925–1990 and found periods of 6–7 years, for the torsional oscillations in the fluid core. However, the torsional oscillations may not be the exclusive cause of the core surface flows because the time scale of the torsional oscillations could also be of the order of several decades [Braginskiy, 1970], while the reoccurrence period of geomagnetic jerks is shorter than 10 years [Holme and de Viron, 2005; Wardinski et al., 2008]. On the other hand, Gibert et al. [1998] proposed a possible correlation between geomagnetic jerks and changes in phase of the Chandler wobble within at most 3.5 years, a correlation supported by an analysis made by Bellanger et al. [2002]. Mandea et al. [2000] have proposed that geomagnetic jerks could be indicators that anticipate the changes in the Earth's rotation rate.

[10] The comprehension of geomagnetic jerks and of their distribution in space and time can help us to better understand the origin of the magnetic field and its variations, the evaluation of the electrical conductivity of the lower mantle and its possible lateral heterogeneity [Alexandrescu Mandea et al., 1999; Pinheiro and Jackson, 2008; Nagao et al., 2003], and the verification of some hypotheses regarding the internal structure of the Earth. However, some of the above features are still under discussion.

3 Nonlinear Chaotic Analysis of the Geomagnetic Field

[11] Many methods, used to find possible nonlinearity and chaos characterizing a system are based on a reconstruction of the phase space, i.e., a generalized reference coordinate space, where a dynamical system can be represented by the trajectory of a single point, with the independent variables as variations along the axes [Schuster and Just, 2005].

[12] As a chaotic system, the recent geomagnetic field is sensitive to initial conditions [Barraclough and De Santis, 1997; De Santis et al., 2002; De Santis et al., 2004; De Santis and Qamili, 2010] and the average divergence ε(t) of initially close trajectories in the phase space propagates exponentially with time t (the rigorous formula imposes a limit of t → 0; [Barraclough and De Santis, 1997])

display math(1)

where K is the Kolmogorov or K-Entropy (with K > 0), which is a quantity that measures the degree of chaos in the dynamics of a system. Since the prediction error increases exponentially with time, this implies a strict limitation of time on which these systems can be predicted. The Kolmogorov Entropy quantifies the rate of information loss in a chaotic process [e.g., Wales, 1991], such as, for instance, in the case of a transmitting signal in a chaotic medium [Schuster and Just, 2005], and it is inversely proportional to the mean length of the time over which a chaotic system is predictable. K-Entropy is zero for completely deterministic/regular systems and infinite for random systems, but finite and larger than zero for chaotic processes. This implies that after a mean characteristic time τ = 1/K no reliable predictions can be made. According to Takens' theorem [Takens, 1981] it would be possible, by a time delay embedding, to reconstruct a phase space topologically equivalent with the original one from a single observable quantity.

[13] However, De Santis et al. [2011] demonstrated the ergodicity of the recent geomagnetic field, i.e., the equivalence of time averages to phase space averages [e.g., Eckmann and Ruelle, 1985]. Under these conditions, the same authors have applied the NFA in the time domain without any reconstruction of the phase space: they have simply analyzed the divergence of the errors between predicted and definitive global geomagnetic models. The accuracy of the actual models lead us to confirm that they cannot provide reliable predictions after some τ = 6 (±3) years.

4 Gufm1 Model

[14] Global models of the geomagnetic field allows us to represent and, in turn, to study the recent and past geomagnetic field and predict its short-term evolution. These models provide sets of Gauss coefficients, inline image and inline image, at successive epochs, or some temporal functions of them. The models may also provide a set of predictive coefficients (coefficients of secular variation) to estimate close future values of the field [Finlay et al., 2010]. The geomagnetic global model used in this work is the Gufm1 model [Jackson et al., 2000] and it is briefly described below. We chose this model because of its coherence and its significant length of validity in time.

[15] Gufm1 is a spatial spherical harmonic expansion model up to degree and order N = 14 based on historical ground and marine data for the interval from 1590 to 1990 [Jackson et al., 2000]. The time-dependent field model is parameterized spatially in terms of spherical harmonics and temporally in B-splines. The model coefficients are expanded in fourth-order B-splines basis functions, whose 163 nodes are regularly separated by 2.5 years.

[16] In the following, we analyze Gufm1 data year-by-year for the full available period of validity. Here, the idea is to show the temporal evolution of errors calculated by differences between predictive and definitive model values. Since the Gufm1 model does not give a predictive field, we estimate the predictive SV extrapolating the past SV based on the previous l years into the subsequent j years, with l, j = 5, 10; this explains why we obtain a set of results l + j years shorter than the period covered by Gufm1. Then, on the basis of this SV, we produce a predicted field and compare it with the real Gufm1 field values. To apply equation (1) in the limit of a short time interval, we need that j is not too long; this is the reason that we do not use a time longer than 10 years. In the next section we provide more details on this method.

5 Nonlinear Forecasting Approach in the Time Domain

[17] What is common for all geomagnetic field models is that any extrapolation of their calculated geomagnetic field outside the typical time of validity would cause large errors. We can estimate these errors ε, from predictive (inline image and inline image) and actual model Gauss coefficients (inline image and inline image) at the Earth's surface, considering the cumulative Mauersberger/Lowes spectrum, as [Maus et al., 2008; Lesur and Wardinski, 2009]

display math(2)

where inline image is the actual model coefficient while inline image is the predictive coefficient; N is the maximum degree/order of the spherical harmonic expansion of the geomagnetic field potential. Although the way we estimate the predictive model coefficients affects the value of ε, it does not affect the exponential growth in time of this error.

[18] Imposing the same initial value for both predicted (extrapolated) and definitive (actual) models, each exponential growth should have an offset equal to –ε0. In this case the error we actually calculate reduces to

display math(3)

where ε0 is a constant that measures the initial difference between prediction and actual value, while τ is the characteristic time of growth that is related to K-Entropy when the system under study is ergodic and chaotic [De Santis et al., 2011]. For our convenience, hereafter we call ε'(t) simply as ε(t), although it is estimated with equation (3). Figure 1 shows a few examples of the ε(t) evolution for l, j = 10 years and some time-windows (1600–1610, 1685–1695, 1695–1705, 1725–1735, and 1940–1950). From this figure it is evident that all the considered segments have a clear exponential growth with a comparable time scale exponent. The divergence of errors for all the considered segments is a clear evidence of a chaotic geomagnetic field and the mean time < τ > of 7 years agrees, within the estimated uncertainties, with the results from previous analyses [De Santis et al., 2011]. It is important to note that equation (3) must be taken in its statistical meaning, i.e., analyzing an ensemble of subsequent diverging predicted-definitive differences we can estimate < τ > ≈T = 1/K, where < τ > is the mean value of the characteristic time associated with the chaotic character of the field.

Figure 1.

Examples of the evolution of ε considering different periods of time, together with the corresponding characteristics time of prediction. The indicated error is the corresponding standard deviation of each exponential fit. The indicated epochs are the intervals of time considered when the extrapolation is made, while the predictive SV is calculated from the Gauss coefficients of the previous 10 years.

[19] From equation (2) we calculate the error over 10 year segments at 1 year step taken from differences between predictive and actual model coefficients. Since the position in time of the cubic B-splines knots is 2.5 years, we choose this shorter time step of 1 year to reduce as much as possible the influence of all the nodes in our results. As already noted above, since the Gufm1 model does not give a predictive field, we estimate the SV coefficients from the field coefficients over the l years prior to each considered epochs. Then, on the basis of this averaged SV, we produce the prediction field values for subsequent j years and compare them with the Gufm1 field values for the same period of time. Thus, we analyze the period from 1600 to 1980, with extrapolations to the successive j years at steps of 1 year. The fact that our comparison is made over j > 0 years into the future while according to equation (1) should be made in the limit of t → 0, puts some uncertainty of a few years when estimating any possible temporal characteristics, i.e., the jerk occurrences.

[20] In formulas, the above approach is translated in the following way. Each predicted model coefficient inline image at time t(yr) is estimated from the behavior of the field (in practice, of the model coefficients) over the previous Δt = l years and extrapolated into the close future from the epoch t0 as

display math(4)

where inline image. Therefore, using for instance Δt =l = 10 years, equation (4) becomes

display math(5)

[21] Here, for simplicity, we prefer to use a linear interpolation for estimating SV, but any other scheme could be in principle applied. To have a statistically robust time evolution of the mean τ value, we have performed the calculation of this quantity with some different l and j years.

6 Results and Interpretation

[22] For all the analyzed segments with the NFA in the time domain applied to the Gufm1 model, we have found a clear exponential growth from which we have obtained the characteristic time τ and its statistical error (standard deviation) every year. In Figure 2a, the blue curve is calculated by using the SV of the previous l = 10 years to produce the prediction field values for subsequent j = 10 years and compare them with the Gufm1 field values for the same period of time. The black curve is calculated by using the SV of the previous l = 5 years to produce the prediction field values for subsequent j = 5 years, while the red one is computed by using the SV of the previous l = 10 years to produce the prediction field values for subsequent j = 5 years. Since the three analyses have been made with extrapolations over different subsequent j (=5, 10) years, the blue curve anticipates some future memory longer than the other two curves (being j = 10 years instead of 5 years), so we would expect a time shift between the blue curve (j = 10 years) and the other two curves (j = 5 years). Thus, once all three curves are estimated, we deduce this time shift with a double cross-correlation, finding a value of around 2 years. Thus, the blue curve shown in Figure 2a has been actually shifted +2 years to be comparable with the other two. Of course, this could affect the next temporal estimation of the jerk occurrences, which can be given with an uncertainty of a few (say, ±2) years. For convenience, we do not show the corresponding error bars at each data point in this figure, i.e., the standard deviations of the exponential fit; however, they usually range between 5 to 10% of τ (see Figure 1). Over all the period here investigated, the curves of Figure 2a have mean values (± standard deviation) of 7, 6, and 4 years (±2 years), respectively. These values of < τ > would represent the mean time windows of predictability, i.e., that mean time after which no reliable prediction for the field can be made. The three values agree within their uncertainties with the value found by De Santis et al. [2011]. As Figure 2a shows, τ value actually fluctuates epoch by epoch around its mean value; indeed, there are periods where the τ value is relatively lower (or higher) than the mean value. These values could be attributed to: (a) worse past (or better recent) measurements with respect to the more recent (past) ones, and, consequently, (b) worse (better) models, which show a slightly worse (better) prediction. However, some clear departures from the mean trend of the τ value must have a more physical meaning that we investigate below. Also, a clear overlap of these three curves is evident, especially for the lowest values of τ, i.e., from this figure we can see that all the relative minima in τ value are mostly confirmed. We can conclude that no matter which is the used time window to calculate the SV or over which time window we fit the predictive-definitive models (given that j is short enough in order to preserve the condition t → 0 in equation (1)), the trend of τ is always almost the same (see details given in this figure by different colors curves). Following this approach, we attempt to confirm (and possibly to detect) the presence of well-known (and also unknown) geomagnetic jerks inspecting with more detail just the blue curve (Figure 2b) because its results are slightly smoother than the other curves. This figure also includes a small upper right inset, which shows the same curve with the error bars for each data point. Let us note that among all 381 points, values of 3 years (1774, 1775, and 1784) are not drawn because they represent the only very few ones where τ grows almost linearly, instead of being exponential. In the next section we consider these years, together with other few epochs, as special epochs with different characteristics with respect to jerks. In Figure 2b we have plotted the best fit line (with also the ± σ and +2σ lines) that takes into account the fact that, in general, past field is less predictable than the more recent one. This slightly increasing trend of the τ value in the studied period could be simply ascribed to the progressing improvement in quality of the geomagnetic measurements over time. In the same figure, clear and distinct epochs are evident when τ assumes low values (evidenced by arrows), i.e., those time scales for which the capability to predict the geomagnetic field future evolution is rather reduced.

Figure 2.

(a) Comparison between three different curves of the time of predictability < τ >, calculated every year, from Gufm1 model. The blue curve is obtained by using the SV of the previous l = 10 years to produce the prediction field values for subsequent j = 10 years, and compare them with the Gufm1 field values for the same period of time. The black curve is calculated by using the SV of the previous l = 5 years to produce the prediction field values for subsequent j = 5 years, while the red one is calculated by using the SV of the previous l = 10 years to produce the prediction field values for subsequent j = 5 years. For comparing year-by-year all the three curves, the blue one has been shifted by +2 years (see text). The errors are not shown for pictorial convenience; they can be estimated for each epoch from the standard deviation of each exponential fit. They range usually from 5% to 10% of the corresponding τ. (b) Estimation of the time of predictability < τ > every year (open circles) over the period 1600–1980 from Gufm1 model (the predictive SV is estimated over previous l = 10 years and extrapolated over the successive j = 10 years). The small upper right inset shows the same curve with the error bars for each data point. The epochs of already noted geomagnetic jerks are indicated by red arrows, those for which new possible events are suggested by blue arrows, and SSGRs are drawn in black. The mean trend is indicated by the best fit line across the data points: the slight increase in time can be ascribed by a better quality of data and model with time.

[23] Here, we make the straightforward hypothesis that there is a jerk when τ assumes a (relative) minimum value with respect to the surrounding temporal values. This definition must be completed with the following two conditions: (i) the time interval that is surrounding the potential jerk cannot be too short (say, it should be more than 4–5 years, which is roughly half of the mean time of occurrence between two successive jerks); (ii) the corresponding τ value should be lower than the best fit line.

[24] Considering the above assumptions, we confirm the presence of all known geomagnetic jerks between 1600 and 1980. In particular, we have detected these events in the epochs: 1603, 1663, 1703, 1733, 1751, 1763, 1770, 1810, 1868–1870, 1888, 1900, 1916, 1925, 1933, 1948, 1963, 1972, and 1976 (red arrows; actually the known 1959 event, that we have associated to 1963, could be even associated to the relative minimum of 1958, but which is not below the mean trend of the τ value, so not satisfying completely our definition of jerk). In addition, there are also other periods (i.e., at 1635, 1672, 1678, 1719, 1779, 1789, 1826, 1838, 1844, 1853, 1882, 1908, and 1942; blue arrows) where τ assumes a relative minimum value but that, to our knowledge, do not correspond to known geomagnetic jerks. It is important to underline the fact that although some blue arrows are within the lower half range of an average interval of time prediction (within the statistical error of one standard deviation, i.e., 1σ, with respect to the best fit line of Figure 2b), and they could be even “normal” (lower) oscillations around the mean, nevertheless, we keep them, being close to large extremes and satisfying our definition of jerk. It is important to say that some of these events must be considered with carefulness, because the τ value in these epochs is within one standard deviation with respect to the mean trend.

[25] Here we also identify some short periods when the field is less chaotic (more predictable) than usual: for instance in the epochs around 1670, 1777, 1786, 1862, 1878, 1957, and 1969 the τ value is significantly larger than the typical background (τ ≥ mean value +2σ) indicated by the dashed line of Figure 2b. We name these epochs as periods of steady state geomagnetic regime (SSGR), to emphasize their opposite behavior in time with respect to geomagnetic jerks. Although we cannot completely exclude that this feature of the field might be somehow related to the poor quality of the older data (this could be possible for some of the older SSGRs, (i.e., 1670, 1777, and 1786), this statement is no longer possible to invoke for more recent times. Thus, SSGR, in the same fashion of its apparently opposite temporal characteristics, i.e., the jerk, is a real intermittent feature of the geomagnetic field.

[26] When using global models like Gufm1, a very important issue arises: because these models are parameterized temporally in B-splines, the position in time of the cubic B-splines knots may play an important role in the current analysis. As already mentioned above, in the Gufm1 model the B-splines were chosen of order 4 and the nodes are 2.5 years apart from each other. Nevertheless, the influence of the cubic B-splines cannot explain all variations present in the shown curve, i.e., the behavior found in our analysis is not at the same time scale [see also Duka et al., 2012].

[27] Most of the above results indicate that a geomagnetic jerk occurs in those epochs when the main field suddenly becomes more chaotic (less predictable) than usual, being characterized by τ values lower than usual. This finding can be explained by the fact that geomagnetic jerks could be manifestations of larger fluctuations (with lower values of τ) in the SV curve of some chaotic processes that take place in the outer fluid core. As mentioned above, different authors [Braginskiy, 1970; Gillet et al., 2010], by studying the torsional oscillations in the fluid outer core, found periods of decades of 6–7 years. Gillet et al. [2010] explained this result by fast torsional oscillations in the fluid core with a fundamental mode of 6–7 years, a time that could explain the occurrence of rapid interannual flow variations and the abrupt geomagnetic jerks. Our results emphasize another possible interpretation of this temporal mode as the longer persistent time that can be allowed by the chaotic character of the field. This would also explain the apparent gap in the LOD variations and SV coherence spectrum between 6–7 and 20 years [Gillet et al., 2010; Braginskiy, 1970].

7 Discussion and Conclusions

[28] Using the NFA in the time domain, we have analyzed the temporal behavior of the difference between predictive and actual geomagnetic field model values for successive intervals from 1600 to 1980 using Gufm1 geomagnetic model. For all the considered (10 + 10) year moving windows, at steps of 1 year (Figure 2b), we have found a similar exponential temporal growth with a characteristic mean trend of the prediction time limited to 5–8 years (the former for older periods, the latter at more recent periods), but slightly fluctuating, generally within ±2.5 years. Some values are sometimes significantly lower than the surrounding values. A potential jerk must have this latter characteristic together with the other two conditions: (i) the time interval that is surrounding the potential jerk cannot be too short, and (ii) the corresponding τ value should be lower than the best fit line.

[29] From the invoked hypothesis, we have confirmed the presence of all the geomagnetic jerks detected by different authors via different methods [see Mandea et al., 2010]. For example, the 1700/1708, 1730/1741, 1750/1763, 1870/1861, 1889/1901, 1925/1932 events (each couple XX/YY, XX indicates an event from Alexandrescu et al. [1997] and YY from Korte et al. [2009], is thought to represent the same event because of the time offset suggested by Korte et al. [2009]), have been confirmed by our analysis. In particular, we have found an event around 1703, which can correspond to the 1700/1708 event. Regarding the 1750/1763 jerk that according to Korte et al. [2009] and Alexandrescu et al. [1997] represents the same event, from Figure 2b two clear minima can be seen: one in 1751 and one in 1763. Regarding the 1889/1901, we have noted an event in 1888 and another one in 1900, while for the 1925/1933 two clear minima appear around the same epochs. Thus, for all cases we have not excluded that they might actually be couples of close distinct jerks, in contrast with what was suggested by the previous studies, i.e., that these events represent the same geomagnetic jerks. We have also confirmed the 1870/1861 event around 1868–1870. Let us note that the event around 1733 must be considered with some caution (for this reason it is marked in Figure 2b); although it does assume a relative minimum with respect to the surrounding values, its characteristic value is slightly higher than the mean trend (so not satisfying our adding condition ii). Some other geomagnetic jerks, e.g., 1770, 1800–1820, detected in Paris declination curve, but questioned by Korte et al. [2009], are clearly visible in our analysis around 1770 and 1810, respectively. An event around 1603 could also be noted (Korte et al. [2009] suggests a geomagnetic jerk in this epoch) if a low value of τ is considered in Figure 2b. Moreover, we have had a relative minimum around 1663, which may correspond to the event suggested by Korte et al. [2009]. Therefore, from Figure 2b, all the well-known geomagnetic jerks prior to 1980 are clearly visible.

[30] Moreover, this analysis allows us to indicate some other periods (e.g., 1635, 1672–1678, 1719, 1779, 1789, 1826, 1838, 1844, 1853, 1882, 1908, and 1942), which can be linked to some unmapped geomagnetic jerks. From Figure 2b a minimum can be seen around 1648: although this event satisfies our definition of jerk, we have admitted that it is the only one not showing a V-shape in the temporal behavior of τ value. Further studies (in terms of using alternative techniques of searching jerks) will be needed in order to understand whether to refine or not our definition of jerks, adding eventually this latter characteristics.

[31] Here by processing global geomagnetic models in time, we have detected fluctuations that reduce locally the characteristic time τ, indicating a less predictable geomagnetic field. Considering the same origin of a geomagnetic jerk phenomenon, e.g., torsional oscillations in the fluid outer core, the effects on the Earth's surface may be different from place to place, explaining why there is no perfect temporal coincidence among different geomagnetic observatories. In Figure 3 we show some examples of magnetic field SV behavior over the Earth's surface as synthesized from Gufm1 model at prechosen continental sites (see Table 1) at specific epochs (with clear or not evident jerk). Here we have chosen the period 1860–1871, suggesting the occurrence of an event in 1868–1870 (Figure 3a) and the period 1610–1620 when, according to our analysis, no geomagnetic jerks are found (Figure 3b). For illustration, inspecting the Y component SV over the three continents, one can distinguish the found jerk in 1868–1869 in Europe, 1866–1869 in Africa, and 1869 in Australia; on the other hand, one can also notice another prior change of SV around 1862–1864 in Europe and Africa, without an analogous change in the Australian region. Let us underline that in our analysis no clear differences between regional and global events are evident. An interesting supposition could be that the more unpredictable (shorter prediction time) is the field corresponding to a jerk of a given epoch, the more global the jerk could be (this is however a very preliminary statement and we plan to investigate it furthermore).

Figure 3.

Two examples of magnetic field SV behavior as synthesized at prechosen continental sites and in specific epochs—(a) with clear jerk (epochs 1860–1871 considering the 1866–1868 event) or (b) not evident jerk (epochs 1610–1620, no event)—over the Earth's surface.

Table 1. Coordinates of the Prechosen Sites Considered in Figure 3
 SitesLatitudeLongitude
1Europe50°10°
2Africa–15°25°
3.Australia–25°140°

[32] We are aware that there are two intrinsic limitations of the method: (a) it uses each time temporal segments of a minimum 1 year, and (b) it is based on the extrapolations over some years into the future. This means that our method represents a coarse way to detect the geomagnetic jerks, with an uncertainty of a few years. Nevertheless, our results show that an investigation of errors between predictive (extrapolated) and actual parts of geomagnetic global field models is a useful tool to detect these events.

[33] Finally, we have also identified some short periods when the field is less chaotic (more predictable) than usual, and named them as periods of steady state geomagnetic regime: for instance, over epochs around 1670, 1777, 1786, 1862, 1878, 1957, and 1969 (±2 years) the τ value being significantly larger than the typical background (τ ≥ mean value +2σ, this explains also why we did not draw three years, i.e., 1774, 1775, and 1784 in Figures 2a and 2b, where τ grows almost linearly). During this time-span the geomagnetic field (at least as represented by Gufm1) is more predictable, in the sense that SV is always increasing or decreasing almost linearly, or in other words, the secular acceleration is practically constant.

[34] Our analyses underline that the geomagnetic field is always—on average—chaotic showing a tendency to diverge from any kind of prediction in such a way that after 6–7 years there is no real possibility to make a reliable prediction. On one hand, looking at the moments with smaller τ values, we have been able to confirm most of the known jerks and even to detect some new ones; on the other hand, inspecting the moments with larger τ values allowed us to introduce a new class of events, the SSGR, when the field is much more predictable than usual. Geomagnetic jerks and SSGRs represent the two contrary aspects of the geomagnetic field temporal variations, the complete understanding of which is still a challenge in geomagnetism.

Acknowledgments

[35] Part of the work signed by E.Q. has been made while she was completing her PhD course at Siena University. Partial funding has also been provided by a PNRA project (“Reversing Earth Magnetism”) and a PRIN 2008 MIUR-funded project (“Animal Magnetic Homing”, Unit of research SIM-MAG). We would like to thank F. Javier Pavón-Carrasco for the advise in a Matlab algorithm. The authors would like to thank Monika Korte and two anonymous reviewers for their constructive comments and suggestions.

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