The 15 m.y. geomagnetic reversal periodicity: a quantitative test

Earth and Planetary Science Letters, 107 (1991) 689-696 Elsevier Science Publishers B.V., A m s t e r d a m

689

[XleP]

The 15 m.y. geomagnetic reversal periodicity: a quantitative test A l a i n M a z a u d a n d C a r l o Laj Centre des Faibles Radioactit'itd~, Laboratoire mixte C.N.R.S.-C.E.A, Auenue de la Terrasse, 91198 G i f sur-Yuette cedex, France Received January 28, 1991 ; revision accepted October 21, 1991

ABSTRACT We have analyzed the three most commonly used geomagnetic polarity time scales that cover the past 100 m.y.. In the three cases, different spectral analyses show a 13-16 m.y. periodicity in the rate of reversal occurence. To test whether this periodicity is real or simply arises from a random generator we have compared these polarity time scales with a large n u m b e r of synthetic sequences produced by a random process, characterized by a linear time variation of its mean activity. Geomagnetic and generated sequences were regularly sampled by using sliding windows, anf then the Fourier spectra of the obtained frequency signals were compared. This test shows that the detected periodicity is presumably not a simple statistical fluctuation of an aperiodic generator, and consequently that a long-term periodicity in the geodynamo must be seriously considered.

1. Introduction

Recent studies suggest that the lower mantle may exercise strong control over the geodynamo, implying that long-term time changes in the frequency of geomagnetic reversals can be used as indicators of time changes at the c o r e - m a n t l e boundary [1-7]. Some authors have attempted to correlate the rate of geomagnetic reversals to mantle convection and global tectonic activity [4], others have observed a strong correlation between the morphology of the transitional fields of the past 10 m.y. and the lateral structure of the c o r e - m a n t l e boundary [6]. Although they are still a matter of debate, these ideas have been responsible for the development over the last few years of studies concentrating on the long-term changes in the geodynamo activity, and especially on the search for periodicities. Recent geomagnetic polarity time scales [8-10] are in agreement in that they all show the same main features of the past 100 m.y. time-variation of the geomagnetic generator of reversals. In the Cretaceous epoch, between 115 and 85 m.y.B.P., no, or very few, polarity changes occurred. In contrast, the recent activity of the geodynamo is characterized by about four reversals per million years. Between these two epochs, during the late Elsevier Science Publishers B.V.

Cretaceous and the Cenozoic, the reversal frequency has increased more or less regularly [11,12]. While the linear trend first proposed by Lowrie [11,13] is widely accepted as a first-order approximation [12,14], the interpretation of the remaining fluctuations is still a matter of debate. There are differences in the detailed structure of the different polarity time scales, and different analyses of their statistical properties have produced controversial interpretations. Some authors consider that these fluctuations are simply statistical fluctuations unrelated to specific variations in the geodynamo [15-20], while other authors believe that a periodicity does exists, and is evidence for a periodicity in the dynamo itself [21-26]. In this latter case, two periodicities, of about 15 m.y. and 30 m.y., have been proposed, depending on the inclusion or the rejection of the Long Cretaceous Quiet Zone in the analyzed time interval. In this paper we have carried out a test recently suggested by McFadden [27] to identify long-term periodicities in the mechanisms responsible for the triggering of geomagnetic reversals. Since it is impossible to decide which of the proposed polarity time scales is the most representative, we have analyzed the three most commonly used recent polarity time scales of the late

690

Cretaceous and the Cenozoic epochs in order to test the stability of the results. These polarity time scales are those of Labrecque, Kent and Cande (LKC) [8], Lowrie and Alvarez (LA) [9], and Berggren, Kent, Flynn and Van Couvering (BKFC) [10]. 2. Evidence for a 1 3 - 1 6 m.y. periodicity in the polarity time scales for the past 100 m.y.

Recent geomagnetic polarity time scales [8,9,10] show that the frequency of reversals has progressively increased over the last 100 m.y., from zero during the Cretaceous quiet interval (85-110 m.y.) to about four reversals per million years for the last few million years. This progressive increase has been modelled with a linear and Lorentzian trend [11-14]. Closer examination of thw LKC, LA, and BKFC polarity time scales reveals that large fluctuations about these modeled trends have occurred in the reversal frequency. A quantitative picture can be obtained by analyzing the polarity time scale with a sliding window followed by correlation analysis. The results obtained with a 4 m.y. rectangular window applied to the LKC, LA and BKFC polarity time scales are shown in Fig. 1. The frequency of reversals has increased from 80 m.y. to the present with fluctuations about the monotonic trend. The autocorrelation (AC) power spectra of the signals remaining after monotonic detrending shows a 16 m.y. periodicity in the LKC time scale (Fig. la), and a 13-14 m.y. periodicity in the LA and BKFC time scales (Fig. lb and lc). This result is confirmed by a maximum entropy (ME) calculation, in which we have used an algorithm currently used for the analysis of paleoclimatic series {28]. The small differences in the exact value of the period results from small discrepancies in the age calibrations of the considered polarity time scales (and also from very short events which are present in the LKC polarity time scale only). We have carefully examined the influence of the window width by repeating all these calculations with different windows ranging from 0.1 m.y. to 7.5 m.y. (0.1, 0.2, 0.325, 0.4, 0.5, 1.5, 2.5, 4, 5.5, 6.5 and 7.5, 9 and 10.5 m.y.). In all cases, the window was shifted in 0.1 m.y. steps (i.e. 1000 steps in the 0-100 m.y. interval), so that for the

A. M A Z A U D A N D ('. LAJ

0.1 m.y. width there was no overlap between the successive positions of the window. The results show that the position of the frequency peak is virtually unchanged between 12.5 and 14 m.y for the LA and BKFC polarity time scales, and at 16.5 m.y. for the LKC polarity time scale. We have also checked that the value of the period does not depend on the shape of the monotonic trend subtracted from the data before the AC and ME spectra were calculated. The results are quite similar irrespective of whether linear, parabolic or Lorentzian trends are used. These results clearly show that the periodicity observed in the different polarity time scales is not an artifact introduced by the use of sliding windows. Were this the case, the position of the frequency peak would depend on the particular width of window used. Consequently the observed structure of the frequency spectrum appears to be a characteristic of the geomagnetic scale. 3. Generation of synthetic sequences.

What is the geophysical significance of the observed periodicity? Does it result from a periodicity in the mechanisms responsible for triggering geomagnetic reversals, or does it simply arise from fluctuations of a random process? To answer this question we have followed a recent suggestion [27] which consists in comparing the geomagnetic sequences with a large number of synthetic sequences produced by an aperiodic random generator. To generate the synthetic sequences we have used two different methods. The first one, which uses the random function of fAx-FORTRAN, involves generating series of 200,000 successive random numbers in the [0-1] interval. Each series is intended to represent 100 m.y. on the geological time scale. Consequently two successive random numbers within a series are separated by 500 years, a duration that is lower than the time resolution of the three polarity time scales considered here. Synthetic sequences of reversals are deduced from these series of 200,000 numbers as follows: When a random number is less than the value of a given level, we decide that a reversal exists. This level was fixed to be the linear rate of geomagnetic reversals fitted in the geomagnetic

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0 O. 1 0.2 0.3 0.4 0,5 0.3 0.4 0.5 0 0.1 0.2 Fig. 1. Analysis of the LKC polarity time scale (a), of the LA polarity time scale (b), and of the BKFVC polarity time scale (c). In all cases the upper graph represents the frequency of the geomagnetic reversals obtained using a 4 m.y. sampling window, the middle shows the Fourier transform (autocorrelation method) of the fluctuating part of the above frequency signal, and the lower represents maximum entropy analysis of these fluctuations.

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data [14], so that each synthetic sequence contains approximately 180 reversals, i.e. the number of polarity reversals present in the geomagnetic polarity time scales considered. Because it can be argued that series of 200,000 successive numbers may be affected by spurious periodicities in the random number generator, we have also employed a second method of generating synthetic sequences which uses fewer random numbers (McFadden, pers. commun.). Given a rate of reversal a ( t ) at time t, a random number x(t) is chosen in [0,1], and then converted to a Poisson interval via the transformation:

y(t) = - L n ( x ( t ) ) / A ( t ) where y(t) is the interval length and A(t) is the linear trend fitted in the geomagnetic data [14]. Then the next interval is determined using t ' = t +y(t), and so on. In this calculation, random generations of numbers were made using the " r a n T ' function of the Numerical Recipes [29] scientific library as a generator of numbers in the [0,1] interval. We have found that these two methods produce virtually identical outcomes, so the conclusions presented in the following paragraphs do not depend on the method of generating the sequences. 4. Method of testing Each generated sequence was analyzed using exactly the same techniques employed with the geomagnetic sequences. A frequency signal was first obtained by using a sliding window, then the power spectrum of the fluctuating part of this signal was calculated. The test compares the spectra obtained from the synthetic sequences with the spectra calculated with the geomagnetic time scales, using the same sliding window. It involves experimentally determining the proportion of generated sequences presenting a spectral peak equal to or higher than the spectral peak detected in the geomagnetic data. When the aforementioned is the case, the result of the test is considered to be positive. Before completing this test, a selection of the generated sequences was made, as suggested by McFadden [27]. From the set of all generated sequences (let us call this set sample A), only those having in their Fourier

A. M A Z A U D A N D C. LA,I

spectra a maximum peak at the value detected in the geomagnetic data were considered. This is between 12.5 m.y. and 14 m.y. for the LA and BKFC time scales and at 16.5 m.y. for the LKC time scale. Let us call this subset sample B. The others sequences were rejected, and since most of the generated sequences were rejected the computation time was rather long. Figure 2 shows the frequency signals and the spectra of three generated sequences which are compared to the LA polarity time scale. Frequency signals are obtained from generated sequences with a 4 m.y. window. Figure 2a was obtained from a sequence which does not belong to sample B since its highest spectral peak is outside the 12.5-14 m.y. interval. This sequence was rejected. By contrast, Fig. 2b was obtained from a sequence which does belong to sample B. However, its spectral peak is lower than the spectral peak obtained with the LA polarity time scale, so the result of the test is negative. Finally, the third sequence (Fig. 2c) also belongs to sample B, and gives a positive result for the test. 5. Results All calculations were independently repeated for the three different polarity time scales considered here. Moreover for each polarity time scale the test was repeated for all the windows already used in the analysis of the geomagnetic data (section 2). Results are presented in Fig. 3a, 3b and 3c respectively for the LKC, LA and BKFC polarity time scales. For each window, the test was performed within sample B consisting of more than 200 selected sequences. For the LKC and LA times scales (Fig. 3a and 3b), P, the percentage within sample B of generated sequences giving a positive result for the test, is very low, provided the sampling window width does not exceed 4 m.y.. Considering all narrow windows ( < 4 m.y.) together, P equals 3.6% for the LKC polarity time scale (Fig. 3a) and 4.7% for the LA polarity time scale (Fig. 3b). At the 95% confidence level, the periodicity detected in the LKC and LA polarity time scales has an amplitude which is too large to arise from a simple fluctuation of a monotonic generator. For larger windows the increase in P reflects the damping of periodicities by large windows.

THE 15 m.y. GEOMAGNETIC REVERSAL PERIODICITY

693

For the BKFC time scale, the situation is more complex. P is very low for windows ranging from 4 to 7.5 m.y., but is quite high with very narrow windows (0.1-2.5 m.y.). This could indicate that the results of the test are sensitive to small changes in the ages of geomagnetic reversals when using very narrow sampling windows. Indeed, the BKFC time scale is characterized by reversal ages slightly younger than in the two other time scales.

6. Discussion

This study shows that the periodicity detected in the Labrecque, Kent and Cande [8] and Lowrie and Alvarez [9] geomagnetic time scales is not, within the 95% confidence level, a simple statistical fluctuation. However, the more complex result obtained with the Berggren, Kent, Flynn and Van Couvering polarity time scale [10] also shows

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A. MAZAUD AND C. LAJ

that this study does not give a definite answer to the problem of the existence of long-term periodicities in the geodynamo. Nevertheless, the dependence of P on the window width, and in

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Fig. 3. R e s u l t s o f t h e test. (a) F o r t h e L K C p o l a r i t y t i m e scale, (b) f o r t h e L A t i m e scale, a n d (c) f o r t h e B K F C t i m e scale. In the t h r e e c a s e s t h e test w a s r e p e a t e d f o r d i f f e r e n t w i n d o w s r a n g i n g f r o m 0.1 m.y. to 11.5 m.y.. V e r t i c a l axis gives P , t h e p e r c e n t a g e o f s y n t h e t i c s e q u e n c e s ( w i t h i n s a m p l e B o f s e l e c t e d s e q u e n c e s , see text f o r detaisl) f o r w h i c h t h e r e s u l t s o f t h e test w a s positive (i.e. p r e s e n t i n g a s p e c t r a l p e a k as h i g h as t h a t d e t e c t e d in t h e g e o m a g n e t i c d a t a ) .

T H E 15 m.y. G E O M A G N E T I C

REVERSAL PERIODICITY

ity reversals over the past 100 m.y.. A ca. 1 3 - 1 6 m.y. periodicity must be seriously c o n s i d e r e d in the l o n g - t e r m e v o l u t i o n o f the g e o d y n a m o . T h e c o n c l u s i o n of this study differs f r o m that of a r e c e n t statistical analysis of the g e o m a g n e t i c reversal s e q u e n c e s [20], but agrees with a m o r e r e c e n t study by the same a u t h o r s showing a 15 m.y. periodicity in m a n y polarity t i m e scales [26]. W e do not have any physical e x p l a n a t i o n for the p r e s e n c e of the o b s e r v e d periodicity. H o w ever, b e c a u s e t h e value of the periodicity is too long to be solely p r o d u c e d by fluid m o t i o n in the c o r e [6], we suggest that it could reflect time variations in the c o r e - m a n t l e b o u n d a r y r e l a t e d to c h a n g e s in the l o w e r m a n t l e , and t h e r e f o r e a c ontro l on the g e o d y n a m o by the lower m a n t le , as p r o p o s e d by G u b b i n s f r o m analysis of th e historical field [1] an d by Laj et al. and C l e m e n t f r o m the analysis o f transitional field of d i f f e r e n t polarity reversals [5,6,7]. Finally, it is e x p e c t e d that a periodicity in t h e m e c h a n i s m s r e s p o n s i b l e for geo m a g n e t i c reversals should also affect o t h e r geophysical p h e n o m e n a [4]. T h e r e f o r e careful examination or r e - e x a m i n a t i o n o f the l o n g - t e r m t i m e variations o f field intensity, or of the m e a n tectonic activity [4], c o u ld yield new and v a l u a b l e i n f o r m a t i o n on the l o n g - t e r m b e h a v i o u r of the geodynamo.

Acknowledgements W e wish to t h a n k P. M c F a d d e n for helpful c o m m e n t s and m a t h e m a t i c a l suggestions for gene r a t i n g the s e q u e n c e s . W e also t h a n k W. L o w r i e a nd Y. R i c a r d for helpful c o m m e n t s , and R. W e e k s for improving the quality of the ma nu s cri p t . This work was s u p p o r t e d by the C E A and the C N R S , and by t h e I N S U p r o g r a m m e DBT-Instabilitds. This is C F R c o n t r i b u t i o n 1222, and I N S U DBT-Instabilitds c o n t r i b u t i o n 378

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