Physics Letters A 377 (2013) 443447Contents lists available at SciVerse ScienceDirect
Physics Letters A
A bi-stable SOC model for Earths magnetic field reversals
A.R.R. Papa a,b,, M.A. do Esprito Santo a,c, C.S. Barbosa a, D. Oliva aa Observatrio Nacional, General Jos Cristino 77, 20921-400, Rio de Janeiro, RJ, Brazilb Universidade do Estado do Rio de Janeiro, So Francisco Xavier 524, 20559-900 RJ, Rio de Janeiro, Brazilc Instituto Federal do Rio de Janeiro, Antnio 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 Earths 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 Earths 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 Earths 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 . Some studies have shown a possible connec-tion of long-term changes in reversion rate to plume activity in themantlecore interface , with changes in internalexternal nuclei
* Corresponding author at: Observatrio Nacional, General Jos Cristino 77,20921-400, Rio de Janeiro, RJ, Brazil. Tel.: +55 (21) 35049119; fax: +55 (21)25807081.
E-mail address: firstname.lastname@example.org (A.R.R. Papa).0375-9601/$ see front matter 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.physleta.2012.12.018interfaces  and with the arrival of cold material to the coremantle boundary .
On the other hand, under what we call normal conditions (forexample, in our days, away from an impending or recent rever-sal) the Earths 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 Earths surface, a fact that wouldhave dire consequences for living organisms. However, in timesof geomagnetic reversals, the Earths 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 .
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 , 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 loglog plots. A fit tothis type of mathematical dependence for the sequence of rever-sals gives a value around 1.5 for a .
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  whereassome others claim that it is closer to a Levy distribution  oreven to a Tsallis distribution . 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) 443447Fig. 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. and references therein).
A nave definition of reversals could be the change in directionof the Earths magnetic field. A more rigorous definition relatesreversals to the changes in sign of the g01 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 doesnt 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 Earths 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  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 [68,1624].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 lawgoverning 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 , rapid changes in the time variation of the magneticfield measured at the Earths 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 ), shell models (like GOY models )and MHD models, from the computational point of view. Therehave been also many attempts of experimental emulation of theEarths core, among which we simply mention the liquid siliconexperience known as VKS experiment  and the Riga experi-ment .
In this work we introduce the simplest model that resumesome of the main characteristics of the Earths 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 Earths 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 Earths 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 Earths 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
A.R.R. Papa et al. / Physics Letters A 377 (2013) 443447 445over 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 = 1N
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 BakSneppen model  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 [5,14,2224]. The BakSneppen model inits more abstract conception has been, itself, subject of many otherscientific papers .
It is worth to mention here that in our model we are interestedin sign changes of magnetization. This contrasts with previoususes of BakSneppen-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 Earths core andits accompanying reversals.
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 prob...