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Geomagnetic Reversals at the Edge of Regularity PHYSICAL REVIEW RESEARCH 3, 013158 (2021) Geomagnetic reversals at the edge of regularity Breno Raphaldini,1 Everton S. Medeiros,2 David Ciro ,3 Daniel Ribeiro Franco ,4 and Ricardo I. F. Trindade 5 1Department of Mathematical Sciences, Durham University, Upper Mountjoy Campus, Stockton Road, Durham DH1 3LE, United Kingdom 2Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany 3Centro de Ciências Biológicas, Universidade Federal de Santa Catarina, Florianópolis, Brazil 4Observatorio Nacional, 04533-085, Rio de Janeiro, Brazil 5Instituto de Astronomia Geofísica e Ciências Atmosféricas - Universidade de São Paulo, 05508-090 São Paulo, Brazil (Received 16 June 2020; accepted 1 February 2021; published 18 February 2021) Geomagnetic field reversals remain as one of the most intriguing problems in geophysics and are regarded as chaotic processes resulting from a dynamo mechanism. In this article we use the polarity scale data of the last 170 Myrs collected from the ocean floor to provide robust evidence for an inverse relationship between the complexity of sequences of consecutive polarity intervals and the respective reversal rate. In particular, the variability of sequences of polarity intervals reaches minimum values in the mid-Jurassic when a maximum reversal rate is found, in the early Cretaceous preceding the Cretaceous Superchron, and twice in the last 20 Myrs. These facts raise the possibility of epochs of high regularity in the geomagnetic field reversals. To shed light on this process, we investigate the transition from regular to chaotic regime in a minimal model for geomagnetic reversals. We show that even in chaotic regimes, the system retains the signature of regular behavior near to transitions. We suggest that geomagnetic reversals have switched between different degrees of irregularity, with a dominant periodicity of ≈70 kyrs that results from the occurrence of “ghost” limit cycles (GLCs) or unstable periodic orbits (UPOs) immersed in a chaotic region of phase space. DOI: 10.1103/PhysRevResearch.3.013158 I. INTRODUCTION the shortest of the three, lasting less than 20 million years (485 to 463 million years ago). On the other hand, an epoch of The geomagnetic field is characterized by a dominant exceptionally fast reversal rates occurred in the mid-Jurassic dipole component that reverses its polarity in irregular times. (155–170 million years ago), with an reversal rate of over 10 The physical mechanism driving reversals of the geomagnetic reversals per million years. field’s dipole is still not well understood, although it is well The statistics and variability of reversal rates is a matter established that reversals are linked to the dynamo process of debate. It was long assumed the reversal rates to follow a that takes place on Earth’s outer core [1]. It is hypothesized Poisson type of renewal process. This leads to an exponential that long term changes in the reversal processes must be distribution p(T ) ∝ exp(−λT ), where λ is the rate of the linked to the evolution of the dynamo process in the Earth’s process. This type of process is memoryless leading to an outer core. For instance the growth of the inner core [2], and independent sequence of polarity intervals, see [8,9]. Subse- most notably the changes in the properties of the core-mantle quent articles investigated the hypothesis of a Poisson process boundary, could have large impacts in the reversal process in which the rate itself evolves in time λ = λ(t )[10]. Carbone [3,4]. et al. [11], later on, showed that the sequences of reversals Polarity scales compiled from paleomagnetic data reveal a largely departs from a Poisson process. They suggested that large variability of the polarity intervals [5,6]. Short intervals have a duration of an order of tens of thousands of years, while reversals are better modeled by a Levy-type process. This re- exceptionally long intervals with a single polarity last O(107) sult contrasts from previous research, implying in long-range yrs and are called superchrons. These events have occurred at correlations between intervals duration, with different classes least three times in the palemagnetic record. The first super- of intervals clustered in time. However, the physical mecha- chron is the Cretaceous Superchron, roughly from about 120 nism behind these long-range dependencies remains elusive. to 80 million years ago. The second one is named Kiaman Su- Insights are provided from other natural dynamos, such as perchron and occurred in the Carboniferous-Permian, roughly the Sun and other stars usually operating in a quasiperiodic between 320 and 270 million years ago. The third one is called cyclic manner. The Sun has a well-recorded 11 year cycle, Moyero Superchron [7], and occurred in the Ordovician being named Schwabe cycle, with peaks in sunspot numbers each 11 years on average occurring in antiphase with its axial dipole dynamics [12]. Despite possessing a strong regular compo- nent the solar cycle also displays irregular behavior such as Published by the American Physical Society under the terms of the the Maunder Minimum, which was suggested to arise from Creative Commons Attribution 4.0 International license. Further intermittent type of behavior [13,14]. See [15] for a compre- distribution of this work must maintain attribution to the author(s) hensive review. This analogy raise the question of whether the and the published article’s title, journal citation, and DOI. geodynamo operates in more regular regimes similar to other 2643-1564/2021/3(1)/013158(10) 013158-1 Published by the American Physical Society BRENO RAPHALDINI et al. PHYSICAL REVIEW RESEARCH 3, 013158 (2021) natural dynamos, and if so, what type of signature this regime quantified by the Lyapunov spectrum λi(x)[17]. Another way on the edge between regular and chaotic would leave on the of measuring the complexity of a dynamical system is the observational record. Kolmogorov-Sinai (KS) entropy that qualitatively measures In this work we address this issue by searching for regu- the rate of creation of information of the system [17,18]. For larities in the sequence of geomagnetic reversals recorded in a class of dynamical systems [16], the KS entropy and the the last 170 Myrs using data from a well known geomagnetic Lyapunov spectrum are linked by the Pesin identity: polarity scale [6]. Specifically, we assess the level of regularity in this signal by estimating its sample entropy (SamEn) and SKS = λi(x)dχ, (1) coefficient of variation (C). With this we provide statistical λ>0 evidence for significant variations in the signal’s regularity χ at the timescale of 107 yrs. And, more importantly, we find where is the ergodic invariant probability measure of the several periods of highly regular reversals. First, and most chaotic attractor. The KS entropy is therefore the sum of all noticeable, in the mid-Jurassic accompanied by an extreme positive Lyapunov exponents of the system. reversal rate, other in the early Cretaceous, preceding the Cre- Sample entropy: For applications to real-world data, taceous Normal Superchron, and twice in the last 20 Myrs. All definition (1) needs to be adapted to overpass intrinsic diffi- these periods correspond to a high reversal rate. The overall culties arising from observational procedures. For this task, a irregularity of the signal is evidenced in the large variability measure called approximate entropy (ApEn) [19] and its suc- in the density polarity intervals, however, we observe prefer- cessor sample entropy (SamEn) [20] have been successfully ential intervals pointing out to underlying periodic processes. employed to quantify the amount of regularity in electrocar- Finally, we interpret these results in the framework of non- diograms [21], and cardiovascular signals in general [22]. A linear dynamics by analyzing the transitions from regular to comprehensive review of ApEn and SamEn is found in [23]. chaotic behavior in a simple model for geomagnetic reversals. Here we implement SamEn to assess the level of regularities in the dynamics of reversals of the geomagnetic field. Given a data sample X = (x ,...,x ), a subset of X of size m II. MEASURES OF REGULARITY 1 N is written as Xm(i) = (xi,...,xi+m−1 ) with 1 i i + m + To quantify the degree of regularity in the geomagnetic 1 N. To calculate SamEn define A = number of times reversals we invoke the concept of chaos from nonlinear vectors with size m + 1 such that d(Xm+1( j), Xm+1(i)) < r, dynamics. Chaos is usually characterized by the exponential and define B = number of times vectors with size m such separation of nearby trajectories in the system’s state space that d(Xm( j), Xm(i)) < r, where d(., .) is any suitable distance as the time advances [16]. The rate of such separation is (here we use the Euclidean distance). Now define N−m N−m A = = , = #[d(xm+1( j), xm+1(i)) < r] SamEn(m, R, N) =−log =−log i 1 j 1 j 1 . (2) B N−m N−m , < i=1 j=1, j=1 #[d(xm( j) xm(i))] r where # denotes the number of elements in a set. In our on measurements from different natures, for instance mag- analysis we adopt commonly used values of r and m, namely, netostratigraphy, biostratigraphy, and radio isotope dating m = 2 and r = 0.2σ (X )[20], where σ (X ) denotes the stan- [25,26]. Most precise knowledge on geomagnetic reversal dard deviation of the sample X, the results remain valid with sequences is obtained from the available data of seafloor slight change in the used parameters (sensibility test using magnetization. As the magma cools on spreading ridges, min- different m and r parameters is provided in the Supplemental erals containing iron and other magnetic materials record the Material [24]).
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