LETTERdoi:10.1038/nature11491

Dynamical similarity of geomagnetic field reversalsJean-Pierre Valet1, Alexandre Fournier1, Vincent Courtillot1 & Emilio Herrero-Bervera2

No consensus has been reached so far on the properties of thegeomagnetic field during reversals or on the main features thatmight reveal its dynamics. A main characteristic of the reversingfield is a large decrease in the axial dipole and the dominant role ofnon-dipole components1–3. Other features strongly depend onwhether they are derived from sedimentary or volcanic records.Only thermal remanent magnetization of lava flows can capturefaithful records of a rapidly varying non-dipole field, but, becauseof episodic volcanic activity, sequences of overlying flows yieldincomplete records. Here we show that the ten most detailedvolcanic records of reversals can be matched in a very satisfactoryway, under the assumption of a common duration, revealing commondynamical characteristics. We infer that the reversal process hasremained unchanged, with the same time constants and durations,at least since 180 million years ago. We propose that the reversingfield is characterized by three successive phases: a precursory event,a 1806 polarity switch and a rebound. The first and third phasesreflect the emergence of the non-dipole field with large-amplitude

secular variation. They are rarely both recorded at the same siteowing to the rapidly changing field geometry and last for less than2,500 years. The actual transit between the two polarities does notlast longer than 1,000 years and might therefore result from mechan-isms other than those governing normal secular variation. Suchchanges are too brief to be accurately recorded by most sediments.

The individual trajectories of the north virtual geomagnetic poles(VGPs) (Fig. 1) derived from the best volcanic records of reversals(Methods) seem to be rather complex. In Fig. 2a and Fig. 2e, the latitudesof the VGP positions of the reverse–normal (R–N) and normal–reverse(N–R) transitions, respectively, are plotted as functions of the respectiveflow numbers, which are reinitialized at the first transitional direction.There is no consistency between the data from these volcanoes, whichhave different eruption rates. In an attempt to improve the consistencybetween the individual records, we next rescaled the lengths of the R–Nand N–R records to match those from Maui and, respectively, Tahiti, byselecting two tie points, usually at the first and last transitional directions(Methods), and interpolating linearly between them (Fig. 2b, f). All the

1Institut de Physique du Globe de Paris, Sorbonne Paris Cite, Universite Paris Diderot, UMR 7154 CNRS, 1 rue Jussieu, 75238 Paris Cedex 05, France. 2SOEST-Hawaii Institute of Geophysics andPlanetology, University of Hawaii at Manoa, 1608 East West Road, Honolulu, Hawaii 96822, USA.

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obtained from large sequences of superimposed lava flows including normal,reverse and at least eight transitional VGP positions.

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numbers defining a given sequence have been multiplied accordinglyby a single factor, which is equivalent to tuning all records to a commonduration (Supplementary Fig. 1). Strikingly, after this simple lineartransformation the entire data set shows a common and consistentdynamical pattern. Likewise the reversal angles (Methods) show thesame field evolution (Fig. 2c, g). Many records seem to be punctuatedby clusters of non-polar directions that can reflect uncertainties inreversal angles, very rapid field changes or both (Methods). We haveassumed that successive directions that differ by less than 7u can begrouped within a single temporal unit. This operation further improvesthe match between all curves (Fig. 2d, h).

Overall, the combination of these records identifies three successivephases of the reversing field that can be described as a precursor, a rapidtransit to the opposite polarity and a final rebound. Precursors are absentfrom the N–R data set (which is small), because samplings of thesesequences did not cover that phase of the reversal. Similarly, the varia-tions in Fig. 2c, d tell us that this first episode was not sampled in fourR–N sequences. Three data sets (Iceland, Hawaii and Tahiti) show theentire reversal sequence. There is no rebound recorded by the 12-Myr-old record from Iceland; the three phases are present in the 3.22-Myr-oldrecord from Hawaii despite relatively weak rebound, and there is littleindication for a precursor in the 0.78-Myr-old record from Tahiti. Therarity of occurrence of the three phases within a single record can beinterpreted in two ways: reversals mostly occur in two phases, eitherprecursor and transit or transit and rebound; or reversals occur in threephases, but there is little chance of recording both the precursor and therebound at the same site because of the brevity of these successive largeamplitude variations due to a rapidly varying non-dipole field. We showbelow that the second interpretation agrees with our understanding ofthe behaviour of the non-dipole field over the past few millennia.

During the first and last phases, the reversal angle reaches an ampli-tude of 90u and the VGP latitude changes by at least 60u. Following

published simulations4,5, we can refer to the archaeological period todefine the duration of similar large-amplitude directional changesdue to a weaker dipole in the presence of higher-order multipolarcomponents that retain the same intensity. In Fig. 3a, we plot theangular deviations recorded in the most detailed succession of14C-dated flows from the Big Island of Hawaii over the past 4 kyr, firstas observed6 and then after having reduced the strength of the axialdipole to 10% of its original value (Methods). With the reduced-strength axial dipole, the angular deviations reach a large amplitudeof about 90u in a time interval that does not exceed 2.5 kyr. Weexplored a 10-kyr-long period by using the CALS10k.1b model7

(Methods), and plotted its predictions at each sampling reversal site(still with the reduced-strength dipole). As expected, the deviationsderived from this global model for Hawaii (Fig. 3a) fit with the data setfrom the Big Island. There is no deviation (Fig. 3b) lasting longer than2.5 kyr at any site, with the reduced dipole. Deviations are even fre-quently much shorter (between 1 and 1.5 kyr) and alternate withreturns to normal polarity. This is particularly true during the past3 kyr of geomagnetic history. Such different field configurations arewell reflected in the rebound of the most detailed Steens Mountain8

record, in which two large deviations are interrupted by a short returnto normal polarity (Fig. 2a–d).

Thus, simulations and data converge to indicate an upper bound of2.5 kyr for the longest field oscillations in the presence of a very weakaxial dipole. This leads us to propose that the maximum durations forthe precursor and the rebound are ,2.5 kyr, assuming that, over thistime frame, the correlation numbers can be linearly related to time andthat this relationship did not change significantly with time. Thisassumption seems to be justified, because otherwise the correlatedrecords would not match. The opposite situation would imply thatthe characteristic timescales of secular variation9,10 have changed overthe past 180 Myr, which is unlikely (Methods). Using this upper bound

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Figure 2 | Dynamical characteristics of the reversal records. VGP latitudes(a, b, e, f) and reversal angles (c, d, g, h) between each successive direction andthe local direction of the present geocentric axial dipole field in each sequence oflava flows for R–N (a, b, c, d) and N–R (e, f, g, h) reversals. Dashed lines at VGPlatitudes 60u and 45u north and south and at reversal angles 30u and 150u define

the limits of the transitions. Data plotted as functions of flow numbersreinitialized at the first transitional direction (a, e); common flow number afterrescaling (b, c, f, g); and common flow units after assigning the same number tosuccessive directions that do not differ by more than 7u (and have similarVGPs) (d, h).

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of ,2.5 kyr, we infer a maximum duration of about ,1.0 kyr for thesecond phase, that is, the polarity switch itself (Figs 2c and 4a).

There is uncertainty in the time spent by the field in the normal andreverse directions before and after the transit. A large drop in intensity18 kyr before the last reversal12,13 has been interpreted as its pre-cursor11, highlighting the oscillating nature of the field during reversalperiods14,15. When intensity is considered, the duration of the reversalseems longer, as dipole intensities of more than 50% of full polarityvalues do not generate large directional deviations4,5. But we are onlyconcerned here with directional changes directly related to the trans-itional process. Assuming that the correlation numbers are linearlyrelated to the number of eruptions, the first episode would have endedabout 2 kyr before the switch and the rebound would start about 1 kyrafter, yielding a duration of about 9 kyr for the entire process (Fig. 4a).For comparison, duration estimates derived from sediments integrateall episodes of the reversing field and range from 2 to 12 kyr (ref. 16). Ithas also been proposed, on the basis of the same sedimentary data-base16, that reversal duration can significantly vary with site latitude.Although not in contradiction with the present results, this hypothesiscannot be tested further given the limited geographical distribution ofthe volcanic sites. We have constrained the maximum reversal dura-tion with respect to the longest variations of the archaeological non-dipole field. Because the lengths of these changes (Fig. 3b) differ at eachsite, precursors and rebounds are expected to be frequently shorterthan 2.5 kyr (Fig. 3b), yielding a duration for the entire reversal processthat is variable but is in all cases shorter than 9 kyr (Fig. 4b).

An important and robust observation is that the polarity switch(phase 2) does not last as long as phases 1 and 3. This could be causedby the dipole being weaker during the switch. However, when thedipole strength in CALS10k.1b is reduced to 10% or is fully suppressed,the spectrum of field variations remains essentially the same (Sup-plementary Fig. 2). Therefore, field instabilities other than those asso-ciated with typical secular variation might be involved in this abruptphase. Such a short (,1-kyr) duration also implies that sediments with

deposition rates that rarely exceed 5 cm kyr21 can hardly haverecorded it in a reliable way with at most three transitional samples.In addition, biased transitional directions can be produced by smear-ing of the signal, which also constrains the VGP paths in longitude5,17.This explains why the complex character of the transitional processcannot be clearly detected in most sedimentary data. The single high-resolution sedimentary reversal record18 shows no precursor (phase 1);this could be due to site location (Fig. 3c, d). The transit and rebound(phases 2 and 3) have similar durations of 2 and 2.5 kyr, respectively(Supplementary Fig. 3), but smearing of the signal could haveincreased the length of the transit.

The simulation in Fig. 3b also reveals that several records, especiallythose from high-latitude sites, do not significantly deviate from normalpolarity for long periods of time. This might reflect a lack of resolutioninherent to the CALS10k.1b model, particularly for the most ancienttimes. We can restrict the analysis to the past 3 kyr and evaluate thepercentage of time during which the angular deviation over Earth’ssurface was larger than 60u. The map in Fig. 3c shows that a little under50% of the sites have angular dispersions exceeding 60u, whereas therest of the sites, particularly at high northern latitudes, are barelysensitive to the reduction of axial dipole strength. Following a similarapproach, we determine the probability of recording two successivelarge deviations over a 5-kyr time interval and find that two suchevents may occur only on little more than a third of Earth’s surface(Methods). This may explain why the rebound (phase 3) is not wellconstrained in those records long enough to show the three successivephases.

Another feature in Fig. 2c is that in all cases normal and reversedirections dominate between the three phases of the process. Thesimulation in Fig. 3d shows that no more than about one-third ofthe sites have angular dispersions deviating by less than 30u fromthe axial dipole when only 10% of the axial dipole value remains.Therefore, a strengthened axial dipole may be required to maintainnorthward or southward directions systematically between the three

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Figure 3 | Secular variation in presence of low axial dipole. a, Most-detailedrecord of secular variation from the Big Island of Hawaii6 in terms of angulardeviation as a function of time for the past 4 kyr (100% axial dipole, white dots;10% axial dipole, black dots). The red curve shows the prediction of theCALS10k.1b global model7 with the axial dipole reduced to 10%. b, Secularvariation derived from the CALS10k.1b model7 at the reversal sites with 10%

axial dipole (Methods) over the past 10 kyr. c, Map of active zones of secularvariation. The colour scale shows the percentage of time over the past 3 kyrspent with an angular deviation larger than 60u with the axial dipole reduced to10%. d, Map of quiet zones of secular variation. The colour scale shows thepercentage of time over the past 3 kyr spent with an angular deviation smallerthan 30u with a dipole reduced to 10%.

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phases. The two volcanic records19,20 with detailed determinations ofabsolute palaeo-intensity between the transition and the reboundagree with this.

The coherent pattern of all reversal records indicates that eachvolcano recorded the same succession of events, with alternatingphases of rapid eruptions and quiet periods, at its own resolution.

All magmatic plumbing systems generated at least one lava flow overa time period of about 1 kyr. We attempted to estimate the meaneruption rate of each volcano (Methods) and found (Fig. 4c) that floodbasalt provinces21 (plume heads) have been significantly more activethan present-day hotspot volcanoes (plume tails).

It is interesting to compare the reversal features identified here withthe analysis of a polarity reversal in a recent three-dimensionaldynamo model driven by thermochemical convection22. The latterreversal is also complex, having a large precursor that reaches theopposite polarity before returning to the vicinity of the original pole,with a short (and incomplete) recovery of dipole intensity. The finalstep is a rapid polarity change lasting for about 400 yr. Thus, data andsimulations are consistent regarding both the duration and the mor-phology of the precursor and transit phases. There is no rebound in thesimulation, whereas it is present in the records (Fig. 2c, d). Detailedanalysis of a larger, statistically significant number of simulations ofsuch complex reversals is needed to go any further.

We have shown that the reversal process has apparently remainedthe same for at least the past 180 Myr and has always featured a briefpolarity switch with a precursor and a rebound. The latter two phasesare not systematically recorded at all sites because their local occur-rence is constrained by large fluctuations of the non-dipole field. Theextremely fast transit between the two polarities is consistent with theabsence of transitional data in many volcanic records and the generallack of transitional poles at low latitudes. It also implies that the core ofthe polarity change could be driven by mechanisms other than thoseinvolved in normal secular variation. These conclusions should betestable by reference to highly detailed records and place importantconstraints on theoretical and numerical models of the geodynamo.

METHODS SUMMARYWe selected the volcanic records that include directions corresponding to the twoopposite polarities surrounding the transition and at least eight VGPs withlatitudes between 60uN and 60u S between the two polarities. Seven records ofR–N transitions8,23–27 and three records of N–R transitions23,28,29 meet theserequirements. The ten records span from 180 to 0.78 Myr ago, reflecting a longhistory of geomagnetic reversals.

The reversal angle denotes the angle between the local magnetic vector and thedirection of today’s axial dipole field at the site. In Fig. 2c, d, g, h, we favoured thisdirectional characterization of the reversing field (as opposed to the VGP latitude)because it describes the evolution and variability of the local magnetic field vector.To construct Fig. 2d, h, we verified that each cluster of reversal angles corre-sponded to a cluster of VGPs.

Our estimate of geomagnetic secular variation over the past 10 kyr relies onCALS10k.1b, a global, time-dependent model of the Holocene geomagnetic field7

(http://earthref.org/ERDA/1403/). The normal (that is, non-transitional) secularvariation of this model is representative of the typical normal secular variationover the past 180 Myr. Over this time frame, the available convective power(integrated throughout the fluid core) driving the geodynamo is likely to haveremained nearly constant30, so there should not be any significant change in theaverage properties of the secular variation (in particular its timescales9,10).

Full Methods and any associated references are available in the online version ofthe paper.

Received 26 February; accepted 6 August 2012.

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2. Jacobs, J. A. Reversals of theEarth’s Magnetic Field2ndedn (CambridgeUniv. Press,1994).

3. Amit, H., Leonhardt, R. & Wicht, J. Polarity reversals from paleomagneticobservations and numerical dynamos simulations. Space Sci. Rev. 155, 293–335(2010).

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5. Valet, J. P. & Plenier, G. Simulations of a time-varying non-dipole field duringgeomagnetic reversals and excursions. Phys. Earth Planet. Inter. 169, 178–193(2008).

6. Hagstrum, J. T. & Champion, D. E. Late Quaternary geomagnetic secular variationfrom historical and 14C-dated lava flows on Hawaii. J. Geophys. Res. 100,24393–24403 (1995).

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Figure 4 | Reversal timing and eruption rates. a, Duration of the successivereversal phases (R–N transitions) derived from Fig. 2c. b, Schematic reversalpath illustrating the succession of the three phases (precursor, polarity switchand rebound). c, Frequency of eruptions derived from each reversal record,based on the duration estimates of the transition. Note the larger valuesobtained for flood basalts. Records from Tahiti, Hawaii and Iceland arecharacterized by similar eruption rates, with one lava flow every 100–200 yr,and the eruption frequencies obtained for the Karoo and Greenland floodbasalts are about twice as large, with Steens basalts displaying about threeeruptions every 100 yr. The 250-Myr-old reversal record from the Siberiantraps31, which could not be incorporated in our database owing to the scarcity ofreverse directions, is also characterized by a higher eruption rate.

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7. Korte, M., Constable, C., Donadini, F. & Holme, R. Reconstructing the Holocenegeomagnetic field. Earth Planet. Sci. Lett. 312, 497–505 (2011).

8. Jarboe,N.A., Coe,R.S.&Glen, J.M.G. Evidence from lava flows for complexpolaritytransitions: the new composite Steens Mountain reversal record. Geophys. J. Int.186, 580–602 (2011).

9. Hulot, G. & Le Mouel, J.-L. A statistical approach to the Earth’s main magnetic field.Phys. Earth Planet. Inter. 82, 167–183 (1994).

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17. Langereis, C. G., van Hoof, A. A. M. & Rochette, P. Longitudinal confinement ofgeomagnetic reversal paths as a possible sedimentary artefact. Nature 358,226–230 (1992).

18. Channell, J. E. T., Curtis, J. H. & Flower, B. P. The Matuyama-Brunhes boundaryinterval (500–900 ka) in North Atlantic drift sediments. Geophys. J. Int. 158,489–505 (2004).

19. Prevot, M., Mankinen, E. A., Coe, R. S. & Gromme, C. S. The Steens Mountain(Oregon) geomagnetic polarity transition. 2. Field intensity variations anddiscussion of reversal models. J. Geophys. Res. 90, 10,417–10,448 (1985).

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24. Herrero-Bervera, E., Walker, G. P. L., Harrison, C. G. A., Guerrero Garcia, J. &Kristjansson, L. Detailed paleomagnetic study of two volcanic polarity transitionsrecorded in eastern Iceland. Phys. Earth Planet. Inter. 115, 119–135 (1999).

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26. Mochizuki, N., Oda, H., Ishizuka, O., Yamazaki, T. & Tsunakawa, H. Paleointensityvariation across the Matuyama-Brunhes polarity transition: Observations fromlavas at Punaruu Valley, Tahiti. J. Geophys. Res. 116, B06103 (2011).

27. Moulin, M., Courtillot, V., Fluteau, F. & Valet, J. P. The ‘‘van Zijl’’ Jurassicgeomagnetic reversal revisited. Geochem. Geophys. Geosyst. 13, Q03010 (2012).

28. Chauvin, A., Roperch, P. & Duncan, R. A. Records of geomagnetic reversals fromvolcanic islands of French Polynesia. 2. Paleomagnetic study of a flow sequence(1.2–0.6 Ma) from the Island of Tahiti and discussion of reversal models.J. Geophys. Res. 95, 2727–2752 (1990).

29. Riisager, J., Riisager, P. & Ken Pedersen, A. The C27n-C26r geomagnetic polarityreversal recorded in the west Greenland flood basalt province: how complex is thetransitional field? J. Geophys. Res. 108, 2155 (2003).

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31. Heunemann, C., Krasa, D., Soffel, H., Gurevitch, E. & Bachtadse, V. Directions andintensities of the Earth’s magnetic field during a reversal: results from the Permo-Triassic Siberian trap basalts, Russia. Earth Planet. Sci. Lett. 218, 197–213 (2004).

Supplementary Information is available in the online version of the paper.

Acknowledgements We are grateful to J. Dyon for significantly improving the quality ofthe figures and to J. Channell for providing us with his reversal data. This is IPGPcontribution number 3313 and HIGP contribution number 1987.

Author Contributions J.-P.V. initiated the study, performedreversal data treatmentandwrote the manuscript. A.F. contributed to all stages of the study by developing the linkwith theoretical modelling, performing the calculations derived from the CALS10k.1bmodel, writing and editing. V.C. edited the manuscript and influenced its content viadiscussions. E.H.-B. acquired a large part of the data and critically read the paper.

Author Information Reprints and permissions information is available atwww.nature.com/reprints. The authors declare no competing financial interests.Readers are welcome to comment on the online version of the paper. Correspondenceand requests for materials should be addressed to J.-P.V. (valet@ipgp.fr).

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METHODSVirtual geomagnetic poles and reversal angles. The north and south VGPsdefine the locations at which the north and, respectively, south magnetic polescut the surface of the earth, under the assumption that the geometry of the mag-netic vector measured at a given site location is controlled by an axial dipolealigned with the rotation axis. The reversal angle (also termed angular deviationin Fig. 3) is defined as the angle between the measured magnetic vector and thedirection of the axial dipole field at the site. This directional characterization of thereversing field has been favoured over the VGP latitude because it is a directrepresentation of the field variations at the site, without any a-priori assumptionregarding the dipolar field geometry inherent to the VGP. Transitional directionscan be defined from VGP latitudes that are either lower than 60u or than 45u (60ubeing commonly used for volcanics and 45u being preferred for sediments becauseof compaction, which reduces the inclination and, thus, the inferred VGP latitude).We note that the 60u limit of VGP latitude corresponds to a deviation of about 30ufrom the direction of the axial dipole in all records shown here.Selection of records. Fully representative reversal records must include datacorresponding to the two opposite polarities surrounding the transition, in addi-tion to a large enough number of reversal data. A threshold of eight north VGPswith latitudes less than 60u was the second selection criterion. Seven detailedstudies of R–N transitions8,23–27 and three records of N–R transitions23,28,29 meetthese requirements. One record32 could not be retained owing to lack of normalpolarity directions. The ten records span from 180 to 0.78 Myr ago, reflecting along history of geomagnetic reversals.Clusters of poles. Clusters of VGPs are present over mid South America,Southwest Australia, the Northwest Pacific and mid-southern Europe (Fig. 1).Some of these locations (Australia, America) have been considered to be persistentfeatures33. However, in this detailed volcanic data set we note that there is only onerecurrent cluster in the Northwest Pacific and that it involves only two differentrecords. Even records from the same geographical area do not exhibit clusters atthe same locations. It is justified to assume that uncertainties in mean flow direc-tions can be larger than 5u but do not reach 10u. In addition, variations of 5–10uand even larger occur over a few decades in the archaeomagnetic field (Fig. 3b).Thus, directions that differ by less than 7u can be considered to reflect a very shorttime interval and represent a single time unit. To construct Fig. 2d, h withoutintroducing any bias, we checked that each cluster of reversal angles always cor-responded to a cluster of VGPs.Field changes at Hawaii. A unit magnetic vector at one site can always be sepa-rated into a unique axial dipolar component and a complementary non-axial part(simple projection), but this is possible only if the angular deviation (or reversalangle) is calculated, not the VGP. By adjusting the value of the axial dipolarcomponent, it is then straightforward to predict the value of the angular deviation(or reversal angle) obtained in the presence of a weaker axial dipole.Secular variation from the CALS10k.1b model. Our estimate of geomagneticsecular variation over the past 10 kyr relies on CALS10k.1b, a time-dependentmodel of the Holocene geomagnetic field7 (http://earthref.org/ERDA/1403/). Thefield is described globally using spherical harmonics (with a truncation set atspherical harmonic degree 10). Following the work of ref. 34, the time dependencyof the corresponding Gauss coefficients is approximated using a basis of B splinesof order four, with a knot spacing equal to 40 yr. More specifically, CALS10k.1b isthe average of an ensemble of 2,000 models in which the uncertainties affecting thedata entering the inverse problem at hand are accounted for by means of a boot-strap technique.

To produce the results displayed in Fig. 3b–d, we have assumed that the value ofthe axial dipole Gauss coefficient, g0

1 , at any time t was equal to 10% of its originalvalue at that same time, as provided by CALS10k.1b, keeping all the other coeffi-cients equal to their original values. This made it possible to produce the secularvariation curves shown in Fig. 3b (time spacing between each point being equal to50 yr) and to derive the statistics shown in Fig. 3c, d (using a 50-yr period sampling).

To estimate the percentage of sites undergoing two large field deviations over a5-kyr-long time window, we performed a global analysis using CALS10k.1b withan axial dipole term reduced again to 10% of its original value. We define a largefield deviation event as the occurrence of a local value of the reversal angle largerthan 45u for 2,000 consecutive years, and we reinitialize the counter as soon as thisduration is achieved. We find that, in terms of surficial area, 26.1%, 37.0% and36.9% of Earth’s surface witness zero, one and two such events, respectively.

Characteristic timescales of the secular variation. The characteristic timescalesof secular variation9 are based on the knowledge of the spatial (Mauersberger–Lowes) power spectra of the main field and its secular variation. Let Rn and Qn

denote the mean square field and secular variation due to all spherical harmonicsof degree n. It is then possible to define characteristic timescales tn 5 !(Rn/Qn)based on the ratios of these two quantities. The statistical interpretation of thesetimescales is that tn measures the time it would take for the field at sphericalharmonic degree n to be completely renewed.

Using recent historical and satellite data, it is possible to show that, for the non-dipole field (n . 1), tn can be approximated by a law of the form tn 5 tSV/n, with asecular variation constant, tSV, of the order of 500 yr (ref. 10). In practice, thatmeans that over a period of approximately 1 kyr, the non-dipole field has beensubstantially reorganized.

The values of tn for low n, computed using the time-averaged spectra of the fieldand secular variations of CALS10k.1b, are t2 5 458 yr, t3 5 382 yr, t4 5 333 yr,t5 5 242 yr and t6 5 155 yr. These values are larger, by a factor of about two, thanthose that are typically obtained by analysing the historical and most recent geo-magnetic field models34. This is a consequence of temporal smoothing induced bythe construction of the model. The regularization preserves the trend of decreasingtn with increasing n, but the dynamics expressed in the model is arguably sloweddown. Interestingly, this implies that our estimate of 2.5 kyr for the upper bound ofthe duration of the directional signature of the non-dipole secular variation isconservative. Consequently, and as short as it may seem, it follows that a durationof 1 kyr for the transit itself is a cautious upper bound.

A fit by a 1/n law points to a large-scale, non-dipole secular variation governedby diffusionless advection by core flow35,36. In the context of a weak axial dipole,this advection of field structures imparts a local, ‘on-site’, signature that will exhibita large degree of temporal variability, depending on the local structure of the non-dipole field (which is, again, substantially reorganized every 1 kyr or so). This localsignature will take the form of undulations in the curves of VGP latitude or angulardeviation. Over the past 180 Myr, the timescales of the non-dipole secular variation(as defined above) have remained nearly constant, leading to a local variability ofthe reversal angle (or of the VGP latitude) statistically stationary over this period.

Indeed, the timescales tn are related to the root mean square fluid velocity insidethe core, which is in turn connected to the available convective power driving thegeodynamo37. According to recent thermal evolutionary models of the core, thispower (which is connected to the heat flux at the core–mantle boundary), hasprobably remained constant over the past 180 Myr (refs 30, 38). There is thereforeno reason to assume that the non-dipole secular variation has not remainedstatistically stationary for the past 180 Myr.Mean eruption frequency. The mean eruption frequency was obtained by count-ing the number of flows as a function of the time function common to all reversalsafter ‘renormalization’ (Figs 2c, g and 4a), assuming that each sequence had beensufficiently sampled. Such is indeed the case for the Hawaiian reversals, which havebeen studied from four different sequences23,25 that have been unambiguouslycorrelated with each other. Other examples include the Karoo27 and the new recordsfrom Tahiti26 and Steens Mountain8, and to a lesser extent Iceland24. The number oflava flows from Greenland29 was multiplied by 2.5, because only every second orthird lava flow in the sequence was sampled. The respective durations of the reversalphases shown in Fig. 2c and Fig. 2g, and their uncertainties, are then used tocalculate a lower and an upper eruption frequency (Fig. 4c).

32. Leonhardt, R., Matzka, J., Hufenbecher, F. & Soffel, H. C. A reversal of the Earth’smagnetic field recorded in mid-Miocene lava flows of Gran Canaria:paleodirections. J. Geophys. Res. 107, 2024 (2002).

33. Hoffman, K. A. & Singer, B. S. Magnetic source separation in Earth’s outer core.Science 321, 1800 (2008).

34. Bloxham, J. & Jackson, A. Time-dependent mapping of the magnetic field at thecore-mantle boundary. J. Geophys. Res. 97, 19537–19563 (1992).

35. Roberts, P. H. & Scott, S. On the analysis of the secular variation. 1. Ahydromagnetic constraint: theory. J. Geomag. Geoelectr. 17, 137–151 (1965).

36. Jackson, A. & Finlay, C. C. in Geomagnetism (ed. Kono, M.) 147–193 (Treatise onGeophysics 5, Elsevier, 2007).

37. Christensen, U. & Aubert, J. Scaling properties of convection-driven dynamos inrotating spherical shells and application to planetary magnetic fields. Geophys. J.Int. 166, 97–114 (2006).

38. Nimmo, F. in Core Dynamics (ed. Olson, P.) 31–65 (Treatise on Geophysics 8,Elsevier, 2007).

RESEARCH LETTER

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