Physics Letters A 377 (2013) 443–447

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Physics Letters A

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A bi-stable SOC model for Earth’s magnetic field reversals

A.R.R. Papa a,b,∗, M.A. do Espírito Santo a,c, C.S. Barbosa a, D. Oliva a

a Observatório Nacional, General José Cristino 77, 20921-400, Rio de Janeiro, RJ, Brazilb Universidade do Estado do Rio de Janeiro, São Francisco Xavier 524, 20559-900 RJ, Rio de Janeiro, Brazilc Instituto Federal do Rio de Janeiro, Antônio Barreiros 212, 27213-100, Volta Redonda, RJ, Brazil

a r t i c l e i n f o a b s t r a c t

Article history:Received 25 May 2012Received in revised form 13 November 2012Accepted 20 December 2012Available online 21 December 2012Communicated by A.R. Bishop

We introduce a simple model for Earth’s magnetic field reversals. The model consists in random nodessimulating vortices in the liquid core which through a simple updating algorithm converge to a self-organized critical state, with inter-reversal time probability distributions functions in the form of power-laws for long persistence times (as supposed to be in actual reversals). A detailed description of reversalsshould not be expected. However, we hope to reach a profounder knowledge on reversals through someof the basic characteristic that are well reproduced. The work opens several future research trends.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

One of the most rooted believes among common people is thatthe compass needle points to the North. The Earth has a magneticfield imposing this behavior. However, it has not been like thatduring the whole Earth’s history. In different geological epochs theneedle would have pointed in the opposite direction. The magneticfield of the Earth has changed its direction many times duringthe last 160 million years (Myr). Each of these changes is calleda reversal. Their existence has been documented through a largenumber of experiments in the oceanic floor and other locationswhere the magnetization at the time of their formation has beenrecorded in rocks. Reversals are, together with earthquakes, someof the more astonishing events on Earth.

The interest in the study of geomagnetic reversals comes fromthe contribution they can add to the comprehension of impor-tant processes in Earth’s evolution. These processes include, amongothers, the internal dynamics of the planet and the ecosphere. Itis very interesting the coincidence in time of some reversals se-quences and tectonics events. This is the case, for example, of thechange in the frequency of reversals localized around 40 Myr agoand the change in the direction of propagation of the Japanese hotspot occurred at approximately the same epoch. This correlationas well as others related to tectonics has been pointed previouslyby some authors [1]. Some studies have shown a possible connec-tion of long-term changes in reversion rate to plume activity in themantle–core interface [2], with changes in internal–external nuclei

* Corresponding author at: Observatório Nacional, General José Cristino 77,20921-400, Rio de Janeiro, RJ, Brazil. Tel.: +55 (21) 35049119; fax: +55 (21)25807081.

E-mail address: papa@on.br (A.R.R. Papa).

0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.physleta.2012.12.018

interfaces [3] and with the arrival of cold material to the core–mantle boundary [4].

On the other hand, under what we call normal conditions (forexample, in our days, away from an impending or recent rever-sal) the Earth’s magnetic field acts as a shield against the streamof charged particles coming from space (mainly the sun), avoid-ing that most of them reach the Earth’s surface, a fact that wouldhave dire consequences for living organisms. However, in timesof geomagnetic reversals, the Earth’s magnetic field (at least itsdipole component) decreases to values close to zero, which allowsa greater number of charged particles reaching the surface. Thesestreams of particles must have interfered on the biological cyclesof the Earth [5].

The sequence of geomagnetic reversals from around 180 Myrago to our days is presented in Fig. 1. It represents in a binaryfashion the geomagnetic reversals scale of Cande and Kent [6,7]published in the middle 90s of the last century. The frequency dis-tribution f (�t) of periods �t between reversals seems to follow,at least for large persistence times [8], an equation of the form:

f (�t) = k.�ta (1)

known as power-law. In Eq. (1), k is a proportionality constantand a is the exponent of the power-law which also correspondsto the slope of the f (�t) versus �t graph in log–log plots. A fit tothis type of mathematical dependence for the sequence of rever-sals gives a value around −1.5 for a [9].

However the exact dependence of the curve of probability den-sity of persistence times is still the subject of discussions. Someauthors claim that the distribution is exponential [10] whereassome others claim that it is closer to a Levy distribution [11] oreven to a Tsallis distribution [12]. The small data set on reversalsmust keep this discussion alive for a long time.

444 A.R.R. Papa et al. / Physics Letters A 377 (2013) 443–447

Fig. 1. A binary representation (see text) of geomagnetic reversals from around160 Myr ago to our days [6,7]. We have arbitrarily assumed −1 as the current polar-ization. There was a particularly long period with no reversals from 120 to 80 Myr.The period from 165 Myr to 120 Myr was very similar to the period from 40 to0 Myr (in both, the number of reversals and the average duration of reversals).

The fact that periods between geomagnetic reversals (calledchrons) distributions follow dependences of the form in Eq. (1)could be the fingerprint of a self-organized critical mechanism astheir source [8,13]. However, self-organized criticality is not theunique physical mechanism able of producing power-law distribu-tion functions (see for example the work by Meirelles et al. [14]and references therein).

A naïve definition of reversals could be the change in directionof the Earth’s magnetic field. A more rigorous definition relatesreversals to the changes in sign of the g0

1 coefficient of the repre-sentation of the geomagnetic field in spherical harmonics. Thereare other components (quadrupolar, for example) that can con-tribute relevantly to the field. However, even this more complexdefinition doesn’t fulfill specific requirements for the Earth. De-pending on sites where samples are collected to analyze reversalsthey present magnetizations in the opposite direction of samplescoming from other sites. To permit that the magnetization presentthe same direction around the whole Earth is normally requireda given time (around 5000 years in the average). This is the rea-son to only consider reversals those with a certain stability whichtranslate as those which last for a long enough time. The rest arecalled excursions.

Fig. 1 suggests that the sequence of Earth’s magnetic field re-versals represents a non-equilibrium process. The mean time inter-val between successive reversals increases with geological time (atleast during the period from 80 Myr ago to our days). The reversalseries, even considering a long period of time (around 160 Myr), issmall (around 300 reversals) which brings serious problems whenwe try to characterize reversals statistically. Even worse is the casefor the record of magnetic field intensities which exists just foraround 10 Myr [15] and seems to follow a frequency distributionbetween a bi-normal and a bi-log-normal dependence. All theseresults must be treated carefully because they will certainly sufferchanges with the acquisition of newest data.

Geomagnetic reversals have called the attention of many sci-entists during the last seventy years and this attention has beentranslated in many types of approaches to the problem which in-clude from purely theoretical to experimental works [6–8,16–24].Long-term trends and non-stationary characteristics of the rever-sals record as well as the scarcity of data have been advanced as aserious difficulty to determine in a consistent way the type of law

governing the series distribution, if there exist any [9,16]. Their in-trinsic nature is still under discussion and will certainly motivatemany others scientific efforts.

Jerks [25], rapid changes in the time variation of the magneticfield measured at the Earth’s surface, deserve a mention apart.They have an intrinsic internal origin but at the same time notalways are simultaneously detected at different locations. In someextreme cases they are detected in some places and never in other.These facts point to the importance of spatial scales in jerks stud-ies. Our work addresses its attention to a mean field model andconsequently, jerks are out of our scope and should be the aim offuture efforts.

There is a vast literature on models which have been (and stillare being) used to model magnetic field reversals. Among manyothers we can easily enumerate geodynamo models (like well-known Rikitake model [26]), shell models (like GOY models [27])and MHD models, from the computational point of view. Therehave been also many attempts of experimental emulation of theEarth’s core, among which we simply mention the liquid siliconexperience known as VKS experiment [28] and the Riga experi-ment [29].

In this work we introduce the simplest model that resumesome of the main characteristics of the Earth’s liquid core and re-versals described above and that have been the subject of previouscited works. At the same time the model owns a considerable orig-inality. Without appealing to a detailed description of edie vorticesand their mutual interaction in the external core, we representthem by random numbers. The weakest vortices are systematicallychanged and substituted by new ones. So are those in the neigh-borhood of the weakest.

The rest of the Letter is organized as follows: first, we presentthe model in Section 2, later on we present the results and a dis-cussion of our simulations (Section 3) and finally, the conclusionsof our work and some trends of currently running researches arepresented in Section 4.

2. The model

The Earth’s liquid core as well as the electric current structureson its volume are simulated through a square network of side Land L × L nodes. This square lattice is interpreted as the projectionof vortices at the Earth’s equator plane. Focusing our interest inthe equatorial plane, and consequently in a 2D model representinga 3D problem, is justified by the well-known result on convectivecells that accompany the Earth’s rotation axis through the liquidcore of the planet [30,31]. In principle most of the information car-ried by convective cells can be recovered at the equatorial plane.

To each node we have initially assigned a random value be-tween −1 and 1 to simulate both, the accumulated magnetic en-ergy (the absolute value) at each of the simulated positions andthe magnetic moment orientation (the plus or minus sign). Wehave looked then for the lowest absolute value through the wholesystem and changed it and its four nearest neighbors by new ran-dom values between −1 and 1 (this defines our time step). In thisway (picking the lowest one) we simulate a continuous energy fluxto the core bulk and at the same time the possible absorption ofsmaller vortices by larger ones. The strengthening of smaller vor-tices is also allowed because the lowest absolute value is eventu-ally substituted by larger absolute values with a finite probability.In this way we also simulate the creation of new vortices. On theother hand, the assignment of new random values, to the nodewith the lowest absolute value and its four nearest neighbors, alsoworks as a release of energy out of the system.

The process of search and replace the lower absolute value isrepeated many times (between 106 and 108 times or more de-pending on the size L) in order to obtain stationary distributions

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over which we implement our calculations. Note that in any case,we are working in some kind of mean field approximation. Wedo not detail the amount of energy lost or gained by a particularnode. The updating algorithm works just in the average (as in thecase of the magnetization that we define in what follows).

We define the magnetization M (which corresponds to the totaldipolar moment) for the system as:

M = 1

N

N∑

i=1

si (2)

where the sum runs over all the nodes and N = L × L is the totalnumber of nodes. It can take values between −1 and 1, corre-sponding to all nodes in the −1 value and to all nodes in the 1value, respectively.

As a consequence of the updating rule our model could be clas-sified as a generalization of the Bak–Sneppen model [5] whichis one of the simplest models presenting self-organized criticalityand, at the same time, has served as general basis to model sev-eral physical phenomena [5,14,22–24]. The Bak–Sneppen model inits more abstract conception has been, itself, subject of many otherscientific papers [32–34].

It is worth to mention here that in our model we are interestedin sign changes of magnetization. This contrasts with previoususes of Bak–Sneppen-like models where the important characteris-tics were avalanches and their probability distribution functions.Avalanches in those cases are often defined as periods of con-tinuous activity (changes in node values) above or below a giventhreshold. The avalanche picture could, however, play a role also inour model: the definition of reversals just as inversions in the geo-magnetic field (in our case, change of sign in magnetization) is nota very rigorous one, because of the complexity of the geomagneticfield and their structure around the Earth globe.

We do not attempt with this simplified model to obtain a de-tailed description of reversals or the mechanisms at their basis. Welook just for some general details like the effective occurrence ofreversals in the model and the frequency distribution for intervalsbetween subsequent reversals to, once this is established, knowthe class of universal behavior displayed by the Earth’s core andits accompanying reversals.

3. Results

We have begun our simulations taking as initial states manyarbitrary configurations (random values for each node, all nodeswith the same value, and so on). Independently of this initial con-figuration the series of chosen sites is random (the node with thelowest absolute value can be anywhere in the simulated network)in the sense that there exist no correlations of any kind. However,after many time steps, and given that we are changing the low-est value and its neighbors by new random values, favoring theexistence of higher absolute values, it will become more probablythat the next lowest value to be changed appear in the vicinityof the previous one. In this way, if we study the frequency distri-bution of the distance between consequent activity (understandingby activity the change of the lowest absolute value and its near-est neighbors) small distances will be more represented than longdistances. The particular form of this dependence is in principleunknown and determined by the dynamics of the model.

After a transient the system reaches a steady state which ischaracterized by a well-liked distribution of magnetic energies innodes as well as by an inverse V-shape distribution for the low-est nodes (see Fig. 2), i.e., the nodes chosen to be changed ateach time step. The distribution of lower energies vanishes above agiven +Ec and below −Ec , while the distribution of node values is

Fig. 2. Distribution of the nodes values at the stationary state: It has a well-likedform with vertical walls at Ec = ±0.325 approximately (solid squares). Distributionof lower absolute values for nodes at the stationary state: It has an inverse V-likeshape with limit values at Ec = ±0.325 approximately (empty squares). Both distri-butions were normalized to their highest values.

practically null between these values and presents a plateau abovethe Ec and below −Ec .

Once the stationary state is reached events also show spatialcorrelation. As explained at the introductory section, in actual re-versals the space correlation could be associated to jerks (lineartemporary trends that appear in the global geomagnetic field butthat often are measured at different times or detected in somelocations but no in others) but we have not done a systematicstudy on this correlation for the present work. However, it willbe the subject of forthcoming works because the power-law it fol-lows (not shown) is one of the fingerprints of the critical stateon which we believe the system we are simulating is. We havedone many simulations with different initial conditions and in allcases the results were independent of them showing that the crit-ical state is a global attractor and consequently that the systemsself-organizes. To check for finite size effects we have implementedsimulations with different system sizes (L = 100,200 and 300). Toobtain stationary values we have discarded around 107 time stepsand averaged over more than 107 or 108 time steps. We used openboundary conditions.

The inset of Fig. 3 presents the dependence of the magnetiza-tion on time for a short run. Note that in this particular case thereis a preponderancy of positive values for the magnetization. How-ever, positive and negative values are equivalent. The importantfact is that the model presents reversals. Fig. 3 presents the dis-tribution function of magnetization values for the data presentedin the inset. The asymmetry is clear between positive and negativevalues of magnetizations. Actually, it seems to be not a stationarydistribution function because for finite systems there is always aprobability for the system to enter a long period of time with thesame magnetization sign, able to change any supposedly stationarydistribution.

Finally, Fig. 4 presents the distribution of inter-reversals times,i.e., the distribution of times during which the magnetization keepsits sign unaltered (chrons in the geomagnetic vocabulary). Thestraight line follows a power-law with a slope ∼ −1.66, very closeto the value ∼ −1.5 [9] accepted for actual reversals. Contrary towhat could be thought this result is not the trivial return to zeroof random walks (characterized by a very accurate −1.5 exponentvalue).

The curve of probability density of persistence times fails tomimic the plateau regime obtained for small persistence timesfrom real data (see, for example, Fig. 1 of Ref. [11] or Fig. 2 of

446 A.R.R. Papa et al. / Physics Letters A 377 (2013) 443–447

Fig. 3. Distribution function of magnetization values for the simulation shown in theinset which is the magnetization versus time plot for a 100 × 100 system during ashort simulation. Note the preponderancy of positive values in this example. Notealso that the magnetization M remains in values much lower in absolute value thanthe extreme values 1 and −1.

Fig. 4. Distribution of time intervals between consecutive reversals (both from pos-itive to negative and from negative to positive). The slope of the straight line is−1.66 (0.01).

Ref. [27]). This is the price to be paid by the use of mean field up-dating algorithms, i.e., without a careful control of the released orabsorbed energy. In more controlled models (for example, adap-tations of the OFC model [35] to reversals) this plateau is ob-tained [12].

The more rigorous definition of reversals permits the applica-tion of avalanche-like studies in our model by either consideringreversals only those periods of time after a change of sign of themagnetization with continuous activity above a given threshold (inthe inset of Fig. 3, for example, let us say, above 0.2 and below−0.2) or by considering reversals only those changes of magneti-zation sign after which the magnetization remains with unchangedsign that last at least for a given predefined time. We have doneboth types of simulations and the results remain essentially thesame, however, as many of the reversals in their simplest defini-tion are rejected, to obtain the same degree of accuracy than insimulations where all sign changes are taken into account, longersimulations were needed.

We have implemented simulations including second and thirdorder neighbors in the updating algorithm and at least up to thirdnearest neighbors we have not detected changes in the slope ofFig. 4 (going beyond these limits affects a relevant fraction of sitescausing some instabilities). From our previous experience with

other Per–Bak-like models [36] the unique factor that change theslope of the curve for probability density of persistence times isthe introduction of forced non-active times which are difficult ofbeing physically justified in the reversals case.

4. Conclusions

As concluding remarks let us mention that we have intro-duced a two-dimensional self-organized critical model to simu-lated Earth’s magnetic field reversals, one of the more fascinatinggeophysical phenomena. The model presents reversals, i.e. changesin the sign of magnetization. We have obtained probability distri-butions functions for the inter-reversals time periods in the formof power-laws. This dependence is a potential evidence of having acritical system at the base of reversals. This is also the type of de-pendence expected in some previous theoretical and experimentalworks on the subject. The longest possible duration of periods be-tween catastrophic events (in our case, reversals) in those types ofsystems is uniquely determined by the size of the system whichin this case is the volume of the Earth’s liquid core. At the bestof our knowledge, it is the first SOC model representing reversalsand it has also the peculiarity of reaching a single stationary statewith two equally probably opposite magnetization states. Elementsto be introduced in future works include among others a carefulanalysis on the spatial distribution of consecutive activity and itspossible relations with jerks and a quantitative estimative on theactual ratio between maximum observed magnetization and themaximum magnetization that the systems might support, whichcorresponds in the real system the model mimics, to the wholeEarth’s core turning once a day.

Acknowledgements

The authors thank partial financial support from CNPq andCAPES (Brazilian Science Founding Agencies) and FAPERJ (Rio deJaneiro State Science Foundation). The authors sincerely acknowl-edge the managing editor and anonymous referees whose criti-cisms and comments have greatly contributed to improve the finalform of this work.

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