Statistical properties of reversals and chrons in numerical dynamos and implications for the geodynamo

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    Article history:Received 14 December 2012Received in revised form 10 April 2013Accepted 18 April 2013Available online 25 April 2013Edited by Chris Jones


    such as polarity reversals (see, e.g., Christensen et al., 2007; Wichtet al., 2009, 2010 for reviews). Despite limitations in computingcapacities the explored parameter space is still very remote fromwhere the Earths dynamo lies and an excessive viscosity is oftenused to suppress small scales that cannot be resolved investiga-

    mechanism of reversals and excursions (e.g. Wicht and Olson,2004; Aubert et al., 2008a; Olson et al., 2011), while others stroveto assess the effect of boundary conditions (e.g. Glatzmaier et al.,1999; Olson and Christensen, 2002; Aubert et al., 2008b; Takahashiet al., 2008). Thanks to a permanent increase in computationalpower, it has also become possible to produce long dynamo solu-tions over a time equivalent to tens of millions of years (e.g. Wichtet al., 2009; Driscoll and Olson, 2011). These solutions have beenused to investigate factors possibly controlling the reversalfrequency, with some emphasis on the role played by theinhomogeneous heat ow pattern at the core-mantle boundary,

    Corresponding author at: Department of Earth and Environmental Sciences,Ludwig Maximilians University, Munich, Germany. Tel.: +49 (89) 2180 4236; fax:+49 (89) 2180 4205.

    E-mail addresses: (F. Lhuillier),

    Physics of the Earth and Planetary Interiors 220 (2013) 1936

    Contents lists available at

    Physics of the Earth an

    .e(G. Hulot), (Y. Gallet).times. 2013 Elsevier B.V. All rights reserved.

    1. Introduction

    Since the advent of the rst 3D numerical geodynamo simula-tions (Glatzmaier and Roberts, 1995a,b; Kageyama et al., 1995),numerous dynamo models have been published, gradually suc-ceeding in reproducing salient properties of the geomagnetic eld,

    tion of these dynamos have also already led to some understandingof the scaling laws controlling the evolution of key dynamo quan-tities (e.g. Christensen and Aubert, 2006; Olson and Christensen,2006; Aubert et al., 2009) and to the identication of parameter re-gimes hopefully relevant for geophysical investigations (Christen-sen et al., 2010). Some studies focused on describing theGeodynamoDipole-eld behaviourPolarity reversalsPolarity excursionsChron durationsStatistical distributions0031-9201/$ - see front matter 2013 Elsevier B.V. A b s t r a c t

    We analyse a series of very long runs (equivalent to up to 50 Myr) produced by chemically-driven dyna-mos. All runs assume homogeneous boundary conditions, an electrically conducting inner-core (exceptfor one run) and only differ by the choice of the Rayleigh number RaH. Introducing dynamo-based de-nitions of reversals, chrons and related concepts, such as failed reversals and segments (bounded byreversals or failed reversals), we investigate the distributions of chron and segment lengths, those ofreversal and failed reversal durations, the way dipole eld behaves through reversals and failed reversals,and the possible links between the axial dipole intensity and chron or segment lengths. We show thatchron and segment lengths are very well described in terms of a Poisson process (with no occurrenceof superchrons), while distributions of reversal and failed reversal durations are better tted by log-nor-mal distributions. We found that reversal rates generally increase in proportion to Rm Rmc ; Rm beingthe magnetic Reynolds number and Rmc a critical value. In contrast, reversal and failed reversal durationsappear to be mainly controlled by the cores magnetic diffusion timescale. More generally, we show thatmuch of the reversing behaviour of these dynamos can be understood by examining their signature in ag01; g11; h11 phase-space plot. This reveals that the run with an insulating inner-core is very different andhas only two distinct modes of opposite polarity, which we argue is the reason it displays less reversalsand failed reversals, and has a clear tendency to produce an intensity overshoot and some systematicpattern in the dipole pole behaviour through reversals and failed reversals. This contrasts with conduct-ing inner-core runs, which display an additional central unstable mode, the importance of whichincreases with Rm, and which is responsible for the more complex reversing behaviour of these dynamos.Available paleomagnetic data suggest that the current geodynamo could have such a (small) centralmode, which would thus imply a strong sensitivity of the frequency and complexity of reversals and ofthe likelihood of failed reversals, to changes in the geodynamos driving parameters through geologicala Institut de Physique du Globe de Paris, Sorbonne Paris Cit, Universit Paris Diderot, UMR 7154 CNRS, F-75005 Paris, FrancebDepartment of Earth and Environmental Sciences, Ludwig-Maximilians-Universitt, Munich, GermanyStatistical properties of reversals and chrand implications for the geodynamo

    Florian Lhuillier a,b,, Gauthier Hulot a, Yves Gallet a

    journal homepage: wwwll rights reserved.s in numerical dynamos

    SciVerse ScienceDirect

    d Planetary Interiors

    l sevier .com/locate /pepi

  • tribution of such segments.We next explore the properties of rever-sals and such failed reversals, particularly the distribution of their

    andduration and the way the dipole eld behaves through such events,looking for special features such as overshoots, similar to what thedata suggests occurs during geomagnetic reversals (e.g. Valetet al., 2005, 2012) and to what was found in experimental dynamosand simple models (e.g. Ptrelis et al., 2009). We also investigatepossible correlations between chron (or segment) length and eldintensity a question puzzling paleomagnetists and still essentiallyunsettled (see, e.g., Tauxe et al., 2007 for a review).

    To this purpose, we rely on a number of dynamo simulationswith homogeneous boundary conditions run over periods of timeequivalent to 4050 Myr. One simulation is with an insulating in-ner-core. This control run allows us to investigate the impact ofthis assumption, often used in recent analyses of long dynamo runs(e.g. Driscoll and Olson, 2009a,b; Olson et al., 2010). The other sim-ulations are with a conducting inner-core and only differ by theirforcing amplitude (i.e. the value of the Rayleigh number). They willallow us to investigate the impact of this parameter.

    The paper is organised as follows. We rst present the numeri-cal models and the various diagnostic parameters and statisticaltools we employ, placing emphasis on the denitions we use forthe concepts of reversal, failed reversal, stable low intensity event,chron and segment (Section 2). We next present the results of ourinvestigations of chron and segment lengths (Section 3.1), of rever-sal and failed reversal durations (Section 3.2), of dipole eld behav-iour through such events (Section 3.3), and of possible linksbetween the eld intensity and chron and segment lengths (Sec-tion 3.4), before putting these results in perspective with the helpof useful phase-space plots (Section 3.5). Finally, these results arediscussed (Section 4) and their implications for paleomagnetismand the geodynamo provided with concluding comments(Section 5).

    2. Models and tools

    2.1. Numerical models

    We use the code PARODY (Dormy et al., 1998; Aubert et al.,2008a) and consider simulations close to the chemically-driven dy-namo initially studied by Olson (2007) and extensively used to ten-tatively account for the geomagnetic polarity timescale (Driscolland Olson, 2009a,b, 2011; Olson et al., 2010, 2011) . The choiceof these authors was mainly motivated by the fact that, contraryto thermally-driven dynamos having the usual drawback of beingweakly dipolar when displaying polarity reversals (Kutzner andChristensen, 2002), chemically-driven dynamos are more stronglydipolar. These dynamos are also not far from the Earth-like pathin attempts to account for the observed sequence of reversals (e.g.Driscoll and Olson, 2009a,b, 2011; Olson et al., 2010, 2013).

    In this paper, we do not seek to reproduce the reversal sequenceof the geomagnetic eld. Rather, we aim at better characterisingreversal and chron properties in very long runs produced by suchmodels, and intend to investigate how these properties depend onparameters such as the conducting or non-conducting nature ofthe inner-core, or the strength of dynamo forcing. We rst focuson the distribution of chron lengths a topicwhich has already beeninvestigated in numerous studies, both in paleomagnetism andnumerical dynamo simulations (see, e.g., Amit et al., 2010 for a re-view. In the present context, however, we further introduce the con-cept of segments bounded by either a successful or failedreversal (dened in a way we later specify) and investigate the dis-

    20 F. Lhuillier et al. / Physics of the Earthdescribed by Christensen et al. (2010), and relatively cheap to com-pute. They are from these standpoints convenient candidates forcomparison with paleomagnetic data. As they suffer from the clas-sical shortcomings of numerical dynamos (e.g. Christensen, 2011),comparison with such data must naturally be done with great care.

    Following the conventions of Aubert et al. (2008a), the four con-trol parameters of the simulations are the modied Rayleigh num-ber RaH (a measure of the strength with which the dynamo isdriven), the Ekman number E (a measure of the relative importanceof viscous to Coriolis forces), the hydrodynamic Prandtl number Pr(ratio between kinematic and thermal diffusivities) and the mag-netic Prandtl number Pm (ratio between kinematic and magneticdiffusivities). Our numerical dynamos are all characterised by thexed parameters E 6:5 103; Pr 1 and Pm 20. Table 1 pro-vides the RaH used for each simulation, together with the followingrelevant output parameters: the magnetic Reynolds numberRm UD=g, where U is the time-averaged rms velocity over theouter-core shell, D the shell thickness (we assume an aspect ratioof 0.35 as in Aubert et al., 2008a) and g the magnetic diffusivity;the dipolarity f 13dip dened as the time-averaged rms amplitude ofthe dipole relative to the eld up to degree 13 at the core-mantleboundary; and f eqax , the ratio of the equatorial dipole to the axial di-pole at the core-mantle boundary.

    The rstmodel has an insulating inner-core and is virtually iden-tical to the dynamos investigated by Olson (2007) and Olson et al.(2009). The others have a conducting inner-core of the same con-ductivity as the outer-core and operate in the same parameter rangeas the dynamos investigated by Driscoll and Olson (2011) and Olsonet al. (2011); Olson et al., 2013. Note that the additional studies ofDriscoll and Olson (2009a,b) and Olson et al. (2010) are also in thesame parameter range, but assume an insulating inner-core.

    All our simulations operate with a lateral resolution of 44spherical harmonic degrees, which ensures that the ratio betweenthe minimum and maximum of the spatial power spectra (inte-grated over the outer-cores shell) is typically higher than250000 for the velocity eld and higher than 2500 for the magneticeld. Unless otherwise noted, all magnetic quantities are expressedin units of ql0gX1=2 (as in Christensen et al. (2001), where q isthe core density, l0 the magnetic permeability, and X the rotationrate assumed for the dynamo) and all temporal quantities are ex-pressed in units of the magnetic diffusion time sg D2=g. The timesteps, which are determined according to an adaptive criterion(Christensen et al., 1999), are on the order of 1:25 105sg forthe model with an insulating inner-core, on the order of8:75 106sg for the models with a conducting inner-core.

    2.2. Timescales of interest

    To describe the timescales of the Earths magnetic eld for a gi-ven spherical harmonic degree n, it is convenient to dene the cor-relation times with


    Pnm0 gmn 2 hmn 2

    h iD EPn

    m0 _gmn 2 _hmn 2h iD E

    vuuut ; 1in which fgmn ;hmn g and f _gmn ; _hmn g are respectively the Gauss coef-cients and their time derivatives of spherical harmonic degree nand order m, whereas the angle brackets denote time averaging(Hulot and Le Moul, 1994; Christensen and Tilgner, 2004). Theyprovide a statistical measure of how long it would take for the ob-served eld at a given spherical harmonic degree n to be com-pletely renewed, and decrease with increasing sphericalharmonic degree (meaning that, the smaller the length scales,the shorter the correlation times). Similar quantities sn can bedened for the eld produced by any dynamo. Considering both

    Planetary Interiors 220 (2013) 1936numerical geodynamo simulations and geomagnetic eld models,it was shown that this decrease is statistically compatible withan inverse linear law sn sSV=n nP 2, whose single parameter

  • remaining low intensity intervals I i ai; bi (of durationDt bi ai) during which the ADM is below l r but neverreaches zero will be referred to as Stable Low Intensity (SLI)events (see light grey zones in Fig. 1(a)).

    4. determination of chrons Ci bi1; ai as time intervals betweentwo consecutive successful reversals, and of segmentsSi bi1; ai as time intervals between two consecutive suc-cessful or failed reversals (see arrows in Fig. 1(a));

    5. facultative step: ltering of segments shorter than Dt (seebelow for details);

    r RaH is an input parameter. The magnetic Reynolds number Rm, the dipolarity f 13dip and theor-growth timescale). T is the total duration of each run. All times are expressed in units of

    Rm f 13dip feqax sSV=sg se=sg

    110 0.58 0.25 2:90 102 6:50 102108 0.57 0.28 3:05 102 8:50 102117 0.54 0.33 2:85 102 7:75 102124 0.54 0.33 2:65 102 7:00 102133 0.50 0.33 2:40 102 4:25 102

    and Planetary Interiors 220 (2013) 1936 21sSV is then known as the secular-variation timescale (Lhuillieret al., 2011a). This parameter sSV can thus conveniently be usedas an alternative to sg to scale time in dynamo outputs, and en-sure consistency between model and the Earths secular-variationtimescales, an important issue for geomagnetic data assimilationapproaches (Fournier et al., 2010, 2011; Aubert and Fournier,2011). Based on the observatory part (i.e. the period 18401990) of gufm1 (Jackson et al., 2000), sSV was found to be on theorder of 500 yr for the geodynamo. We will use this value whenconverting time in years (see, e.g. Fig. 3).

    Another timescale of interest in numerical geodynamo simula-tions is the error-growth timescale se, which corresponds to theinverse of the average growth rate at which small errors intro-duced in a reference solution are exponentially amplied. Thisgrowth rate is a robust quantity independent of the conditions(type, amplitude, time) under which the small errors are intro-duced, and thus gives an insight into the predictability of themodel (Hulot et al., 2010a; Lhuillier et al., 2011b). Contrary tothe correlation times, this timescale cannot be obtained as a di-rect...


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