Corrections and Retraction

CORRECTIONS

INAUGURAL ARTICLE, GEOPHYSICS. For the article ‘‘Gravitationaldynamos and the low-frequency geomagnetic secular variation,’’by P. Olson, which appeared in issue 51, December 18, 2007, ofProc Natl Acad Sci USA (104:20159–20166; first publishedNovember 29, 2007; 10.1073�pnas.0709081104), the author notesthat on page 20160, left column, last paragraph, line 9, ‘‘then� � �1 in Eq. 3’’ should instead read: ‘‘use � � �1/�4� inEq. 3.’’ This error does not affect the conclusions of the article.

www.pnas.org�cgi�doi�10.1073�pnas.0800480105

APPLIED PHYSICAL SCIENCES. For the article ‘‘Instability in pipeflow,’’ by D. L. Cotrell, G. B. McFadden, and B. J. Alder, whichappeared in issue 2, January 15, 2008, of Proc Natl Acad Sci USA(105:428–430; first published January 4, 2008; 10.1073�pnas.0709172104), due to a printer’s error, the year of publica-tion appeared incorrectly in the footer. The correct publicationdate is ‘‘January 15, 2008.’’ The online version has beencorrected.

www.pnas.org�cgi�doi�10.1073�pnas.0801024105

BIOCHEMISTRY. For the article ‘‘The globular tail domain puts onthe brake to stop the ATPase cycle of myosin Va,’’ by Xiang-dong Li, Hyun Suk Jung, Qizhi Wang, Reiko Ikebe, Roger Craig,and Mitsuo Ikebe, which appeared in issue 4, January 29, 2008,of Proc Natl Acad Sci USA (105:1140–1145; first publishedJanuary 23, 2008; 10.1073�pnas.0709741105), the authors notethat, due to a printer’s error, ref. 25 contained an incorrectvolume number. The corrected reference appears below.

25. Burgess SA, Yu S, Walker ML, Hawkins RJ, Chalovich JM, Knight PJ (2007) J Mol Biol372:1165–1178.

www.pnas.org�cgi�doi�10.1073�pnas.0801004105

DEVELOPMENTAL BIOLOGY. For the article ‘‘Linking pattern forma-tion to cell-type specification: Dichaete and Ind directly repressachaete gene expression in the Drosophila CNS,’’ by GuoyanZhao, Grace Boekhoff-Falk, Beth A. Wilson, and James B.Skeath, which appeared in issue 10, March 6, 2007, of Proc NatlAcad Sci USA (104:3847–3852; first published February 26, 2007;10.1073�pnas.0611700104), the authors note the following: ‘‘Onpage 3851, right column, first paragraph, line 9, in the sentence‘For example, Sox1 can bind directly to the HES1 promoter andsuppress its transcription (24, 32),’ the references were cited inerror. The correct reference is Kan L, Israsena N, Zhang Z, HuM, Zhao LR, Jalali A, Sahni V, Kessler JA (2004) Dev Biol269:580–594. Additionally, please note that ref. 24 is a duplicateof ref. 10. Finally, ref. 4 was cited in error on page 3852, leftcolumn, paragraph 3, line 4, and right column, paragraph 2, line1, and should be removed from both locations. We apologize forany confusion these errors may have caused.’’

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RETRACTION

PLANT BIOLOGY. For the article ‘‘Arabidopsis myosin XI mutant isdefective in organelle movement and polar auxin transport,’’ byCarola Holweg and Peter Nick, which appeared in issue 28, July13, 2004, of Proc Natl Acad Sci USA (101:10488–10493; firstpublished July 6, 2004; 10.1073�pnas.0403155101), the authorswish to note the following: ‘‘We must retract the results pub-lished in the article. In further investigations of the mya2-1knockout (SAIL�414�C04), we detected a second deletion up-stream and adjacent to the MYA2 locus, and a complementationassay performed with the whole genomic sequence of MYA2,including the promoter (10.5 kb), revealed no significant differ-ences between the dwarf phenotype of the original mutant lineand the mya2-rescued line. The analysis included parameterssuch as shoot length, cytoplasmic streaming, hypocotyl length,epidermal cell length, and root hair length. Therefore, thephenotype of the original knockout line was probably due to thesecond deletion upstream of the MYA2 gene. Since our originalpublication, and consistent with our new results, others haveobserved no major defects resulting from inactivation of any ofthe 13 myosin XI genes in the Arabidopsis thaliana genome (1–3);inactivation of the MYA2 and XI-K genes resulted only in defectsin root hair growth and organelle trafficking (2, 3).’’

Carola HolwegPeter Nick

1. Hashimoto K, et al. (2005) Peroxisomal localization of myosin XI isoform in Arabidopsisthaliana. Plant Cell Physiol 46:782–789.

2. Ojangu EL, Järve K, Paves H, Truve E (2007) Arabidopsis thaliana myosin XIK is involvedin root hair as well as trichome morphogenesis on stems and leaves. Protoplasma230:193–202.

3. Peremyslov VV, Prokhnevsky AI, Avisar D, Dolja VV (2008) Two class XI myosins functionin organelle trafficking and root hair development in Arabidopsis thaliana. PlantPhysiol, 10.1104/pp.107.113654.

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Gravitational dynamos and the low-frequencygeomagnetic secular variationP. Olson*

Department of Earth and Planetary Sciences, The Johns Hopkins University, Baltimore, MD 21218

Contributed by P. Olson, October 1, 2007 (sent for review August 21, 2007)

This contribution is part of the special series of Inaugural Articles by members of the National Academy of Sciences elected on May 1, 2007.

Self-sustaining numerical dynamos are used to infer the sources oflow-frequency secular variation of the geomagnetic field. Gravi-tational dynamo models powered by compositional convection inan electrically conducting, rotating fluid shell exhibit several re-gimes of magnetic field behavior with an increasing Rayleighnumber of the convection, including nearly steady dipoles, chaoticnonreversing dipoles, and chaotic reversing dipoles. The timeaverage dipole strength and dipolarity of the magnetic fielddecrease, whereas the dipole variability, average dipole tilt angle,and frequency of polarity reversals increase with Rayleigh number.Chaotic gravitational dynamos have large-amplitude dipole secularvariation with maximum power at frequencies corresponding to afew cycles per million years on Earth. Their external magnetic fieldstructure, dipole statistics, low-frequency power spectra, and po-larity reversal frequency are comparable to the geomagnetic field.The magnetic variability is driven by the Lorentz force and ischaracterized by an inverse correlation between dynamo magneticand kinetic energy fluctuations. A constant energy dissipationtheory accounts for this inverse energy correlation, which is shownto produce conditions favorable for dipole drift, polarity reversals,and excursions.

geodynamo � geomagnetic polarity reversals �gravitational dynamo mechanism � numerical dynamos �geomagnetic power spectrum

F luctuations of the Earth’s magnetic field with time scaleslonger than a few years are collectively referred to asgeomagnetic secular variation. Secular variations induced by thetime-variable dynamics in the Earth’s core that maintain thegeodynamo span an enormous frequency range, from about onecycle per decade to one cycle per million years and longer (1, 2).A variety of theories have been proposed for the relativelyshort-period secular variation, and some have been successful inexplaining geomagnetic observations. For example, the decade-to-century fluctuations that dominate the historical secularvariation have been interpreted in terms of frozen-flux transportof magnetic field by large scale flow (3, 4), magnetohydrody-namic oscillations (5–7), and turbulent cascade processes (8) inthe fluid outer core. In contrast, there is no consensus on theorigin of the low-frequency secular variations that dominate thearcheomagnetic and paleomagnetic records. This low-frequencyvariability consists mostly of large-amplitude, broad-band dipolemoment fluctuations with time scales ranging from a few thou-sand years to about one million years. Archeomagnetic variationsdefine the high-frequency end of this range (9, 10), and a slowdrift of the dipole moment amplitude defines the low-frequencyend (11). Proposed explanations for these fluctuations includeexternal forcing by orbital variations (12, 13), kinematic dynamowaves (14), and tide-induced instabilities (15), in addition topurely statistical models such as power-law noise (16).

Large-amplitude, low-frequency secular variation has impli-cations for polarity reversals, excursions, and other extremegeomagnetic events (17, 18). Paleomagnetic measurements in-dicate a close relationship between polarity reversals and the

phase of the low-frequency secular variation, with polarityreversals and excursions—transient, large-amplitude dipole tiltevents—occurring at times when the dipole moment is partic-ularly weak (11). The connection between low-frequency secularvariation and polarity reversals is supported by recent estimatesof the geomagnetic power spectrum (19), which show a nearlyuniform distribution of variance over this frequency band.

Spontaneous polarity reversals are seen in numerical dynamosand laboratory fluid dynamos, although in most cases thereversal mechanism is not well understood. Polarity reversalshave been reported in one laboratory dynamo experiment inliquid sodium (20). Numerical models have shown that thefrequency of polarity reversals can be highly stochastic (21), and,in convection-driven dynamos, reversal frequency generally in-creases with the strength of the convective forcing (22, 23) andis sensitive to the choice of boundary conditions (24, 25).Dynamo models have also shed light on the kinematic behaviorof the magnetic field during the brief transition periods whenpolarity changes occur (26, 27).

This paper focuses on the underlying causes of the intrinsiclow-frequency magnetic variability in self-sustaining numericaldynamos driven by compositional convection, the so-calledgravitational dynamo mechanism, which is thought to be themain power source of the geodynamo (28). In addition to theintrinsic low-frequency geomagnetic variation, there are alsoultralow-frequency geomagnetic variations, evidenced by theslow modulation in the frequency of polarity reversals overPhanerozoic time and the existence of long polarity magneticsuperchrons (29). These variations have characteristic timescales ranging from tens to hundreds of millions of years (17, 18)and are often attributed to changes in the dynamics of Earth’smantle affecting the long-term energy budget of the geodynamo(30). Although time variable mantle dynamics are not explicitlyconsidered in this study, the results here suggest how ultralow-frequency geomagnetic variations might occur.

Gravitational Dynamo Equations and Parameters. The conservationof momentum, mass and magnetic field continuity, buoyancytransport, and magnetic induction equations for convection andmagnetic field generation in an electrically conducting fluid in arotating, self-gravitating spherical shell can be written in dimen-sionless form as

E��u� t

� u��u � �2u� � 2 ẑ � u � �P � EPr�1Ra rro �� Pm�1�� � B� � B [1]

Author contributions: P.O. designed research, performed research, contributed new re-agents/analytic tools, analyzed data, and wrote the paper.

The author declares no conflict of interest.

*E-mail: olson@jhu.edu.

This article contains supporting information online at www.pnas.org/cgi/content/full/0709081104/DC1.

© 2007 by The National Academy of Sciences of the USA

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���u, B� � 0 [2]

��

�t� u��� � Pr�1�2� � � [3]

�B� t

� � � �u � B� � Pm�1�2B, [4]

where B, u, P, and � are the magnetic induction, f luid velocity,pressure perturbation, and buoyancy variable, respectively, t istime, r is the radius vector, � is the buoyancy source (alldimensionless), and E, Pr, Pm, and Ra are the Ekman, Prandtl,magnetic Prandtl, and Rayleigh numbers, respectively. Theconservation of momentum (Eq. 1) is the Boussinesq form of theNavier–Stokes equation in a spherical (r, �, ) coordinate systemrotating at angular velocity �ẑ in which gravity increases linearlywith radius. It includes the inertial and Coriolis accelerationsplus the pressure, viscous, buoyancy, and Lorentz forces. Eqs.1–4 have been nondimensionalized by using the shell thicknessD � ro–ri as the length scale (ro and ri are the inner and outerradii) and the viscous diffusion time D2/ as the time scale ( isthe kinematic viscosity). The fluid velocity is scaled by /D andthe magnetic field is scaled by (�o �/�)1/2, where �o is outer coremean density and � is its electrical conductivity. The radius ratiois fixed at ri/ro � 0.35, approximating the Earth’s inner coreboundary/core–mantle boundary radius ratio. Appropriateboundary conditions for the composition variable are � � 1 atr � ri, fixed light element concentration at the inner coreboundary, and ��/� r � 0 at r � ro, zero light element flux at thecore–mantle boundary. Mechanical and electrical conditions onboth spherical boundaries are no-slip and electrically insulating.

Three of the four control parameters in Eqs. 1–4 haveconventional definitions: E � /�D2, Pr � /, where is thediffusivity of the buoyancy variable, and Pm � /�, where � �1/�o� is the magnetic diffusivity (�o is magnetic permeability).The fourth control parameter, Ra, controls the strength of thebuoyancy forces driving the convection. In the gravitationaldynamo mechanism, this parameter indicates the rate of chem-ical evolution of the core, as follows. As the core cools and theinner core solidifies, light elements (an uncertain mixture ofsulfur, oxygen, silicon, carbon, etc., comprising �15% of the coreby mass, here denoted by Le) are partitioned into the liquidphase near the inner-core boundary and dense elements (pri-marily a mixture of iron and nickel, here denoted by Fe) arepartitioned into the solid phase and incorporated into the innercore. The increased concentration of light elements reduces thedensity of the liquid phase, making it positively buoyant in theouter core, and the resulting compositional convection providesthe kinetic energy for the geodynamo (31, 32). Meanwhile theouter core becomes progressively less dense, owing to its in-creasing light element concentration.

Convection produced by this chemical segregation can bemodeled numerically by using a buoyancy sink formulation (33).The light element concentration in the outer core is representedas a sum of a spatially uniform, slowly increasing part �o(t), plusa perturbation �. Let �̇o denote the secular increase of �o, whichis assumed to occur on such a long time scale that both �o and�̇o can be taken to be constant over the duration of a dynamocalculation. If represents the light element diffusivity in theouter core and if � is scaled by D2�̇o/, then � � �1 in Eq. 3. Withthis definition of �, the Rayleigh number in Eq. 1 is

Ra ���goD5�̇o

�o

2 , [5]

where �� � �o � �Le is the density difference between the outercore and the light element end-member mixture and go is gravityat the core–mantle boundary.

The dynamo model output consists of the induced magneticfield, f luid velocity, and light element concentration as a functionof time. Volume average variables to be analyzed include thedimensionless rms internal magnetic field and fluid velocity plustheir energy densities, denoted by B, u, Ek and Em, respectively.In terms of the rms magnetic field and velocity, the dimension-less kinetic and magnetic energy density per unit mass are Ek �u2/2 and Em � (PmE)�1 B2/2, respectively. Surface averagevariables include the rms magnetic field on the core–mantleboundary and its dipole part, denoted by Bo and Bd, respectively.Time averages of these quantities are denoted by overbars, anddeviations from time averages are denoted by primes. Two ratiosare of particular significance for comparing dynamo modeloutput to the Earth, the time average dipolarity of the magneticfield on the core–mantle boundary d � Bd/Bo and the dipolevariability, the standard deviation of the dipole strength relativeto its time average s � �Bd�/Bd.

For comparison with the geomagnetic secular variation, it iscustomary to express the dynamo fluid velocity in magneticReynolds number units rather than the ordinary Reynoldsnumber units implied by the nondimensionalization in Eqs. 1–4.The conversion from u in Reynolds number units to u� inmagnetic Reynolds number units is given by u � Pm�1 u�.Another way to compare dynamo velocities uses the local Rossbynumber Ro� defined in ref. 34. Dynamo model and geomagnetictime are usually compared in units of the dipole free decay time.In terms of the scaling used in Eqs. 1–4, the dimensionless dipolefree decay time in a sphere with radius ro and uniform electricalconductivity is

td � Pm� r0�D�2

, [6]

and the corresponding frequency is fd � 1/td. For the Earth’score, one dipole decay time corresponds to approximately 20 kyr.

ResultsFig. 1 summarizes the statistical behavior of gravitational dyna-mos with E � 6.5 10�3, Pr � 1, and Pm � 20 as a function ofRayleigh number Ra. Time averages and standard deviations ofrms internal magnetic field strength, dipole field strength on thecore–mantle boundary, rms fluid velocity, and dipolarity of themagnetic field on the core–mantle boundary are shown. Filledand open circles in Fig. 1 denote time average values of reversingand nonreversing dynamos, respectively. The squares denoteresults of nonmagnetic convection calculations at Ra � 0.6 105and 1 105. Error bars denote one standard deviation, and thedashed line indicates the critical Rayleigh number for dynamoonset, Racrit � 0.31 105, for which the critical magneticReynolds number is u�(Racrit) � 40. Further details of thesecalculations are given in Methods and supporting information(SI) Figs. 8–11 and SI Table 1. The five cases with the lowest Ravalues in Fig. 1 are balanced dynamos, with constant or nearlyconstant rms magnetic field and fluid velocity. The pattern ofconvection in these dynamos is invariant with time but propa-gates westward (retrograde) relative to the rotating coordinatesystem. Their magnetic fields are strongly dipolar near Racrit butbecome substantially less dipolar as Ra is increased. Above Ra �0.55 105, the convection pattern becomes time-dependent andthe balanced dynamos are replaced by more highly variabledynamos with chaotic time dependence and generally strongermagnetic fields. For the chaotic dynamos, the time averagevelocity increases monotonically with the Rayleigh number ofthe convection approximately as u� � Ra7/8, whereas their timeaverage rms magnetic field strength is nearly independent of Ra.

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The time average dipole strength decreases approximately asRa�1 in the chaotic regime, as does the time average dipolarity.Polarity reversals were recorded in all of the chaotic dynamos inFig. 1, with Ra � 0.68 105, but not at lower Ra values. Thedistinction between reversing and nonreversing is somewhatarbitrary, however, because some dynamos in the later categorymight eventually reverse polarity if run for a sufficiently longtime. For example, it is possible that reversals would occur in theRa � 0.65 105 case if it were continued longer, because itsstatistics differ only slightly from nearby reversing cases. Thetransition from nonreversing to reversing behavior coincidesapproximately with the peak dipolarity, with a small reduction inrms magnetic field strength and, more importantly, with a largerreduction in time average dipole field strength. The drop in thetime average dipole field strength, together with its increasedvariability, means that weak dipole field states (dipole collapses)become more frequent as Ra increases. Because polarity rever-sals in these dynamos occur during dipole collapse events,reversals become increasingly frequent in the higher Ra cases.Some of the balanced dynamos in Fig. 1 also have rather weaktime average dipole fields, but they lack dipole collapse eventsand are unlikely to reverse.

Comparisons between a reversing and a nonreversing chaoticdynamo are shown in Figs. 2–6. Figs. 2 and 3 show time series ofrms velocity, rms internal magnetic field, core–mantle boundaryrms dipole field, core–mantle boundary dipolarity, and dipoletilt angle at Ra � 1 105 and Ra � 0.6 105, in the reversingand nonreversing regimes, respectively. The ratio of the fluctu-ations to mean values for the rms fluid velocity and the rmsmagnetic field have comparable amplitudes in the two cases.

However, the Ra � 1 105 case has both a weaker time averagedipole and larger dipole fluctuations, so there are occasional,short time intervals when its dipole field is very weak. As seenin Fig. 2, roughly one half of these dipole collapse events resultin polarity reversals or dipole tilt excursions. In contrast, theRa � 0.6 105 case shown in Fig. 3 lacks these dipole collapseevents and has not reversed its polarity.

Likewise, about one half of the minima in the dipolarity timeseries in Fig. 2 are associated with polarity reversals or excur-sions, even though the events that produced the smallest dipo-larity resulted in excursions rather than reversals. Although thedipole tilt angle in Fig. 3 shows occasional spikes, the dipolarityremains �0.5, indicating that a polarity reversal is extremelyunlikely in this case. Figs. 2 and 3 also show that the velocity andmagnetic field fluctuations have rather different frequencycontent, with high frequencies being relatively more energetic inthe velocity time series. The spikes in the velocity time seriescorrespond to the spin up or spin down of one or more of thevortices shown in Figs. 4 Lower Right and 5 Lower Right, causedby the growth or decay of a buoyant upwelling. The velocity timeseries also contain low-frequency fluctuations that are out ofphase (anticorrelate) with the low-frequency fluctuations thatdominate the magnetic time series.

2

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2.00 0.5 1.0 1.5 2.0

0 0.5 1.0 1.5 2.0 0 0.5 1.0 1.5 2.0

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0o

Fig. 1. Gravitational dynamo statistics versus Rayleigh number with E � 6.5

10�3, Pr � 1, and Pm � 20. Time average values are denoted by circles, andstandard deviations are denoted by error bars. Closed and open circles denotereversing and nonreversing dynamos, respectively. Squares denote nonmag-netic convection cases. The dashed line indicates critical Rayleigh number fordynamo onset. (Upper) rms internal magnetic field intensity (Left) and rmsfluid velocity in magnetic Reynolds number units (Right). (Lower) rms dipoleintensity on the core–mantle boundary (Left) and dipolarity on the core–mantle boundary (circles) and rms dipole tilt angle of nonreversing dynamos(asterisks) (Right).

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d

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Fig. 2. Time series of reversing gravitational dynamo with Ra � 1.0 105, E �6.5 10�3, Pr � 1, and Pm � 20. From top to bottom, shown are graphs of rmsfluid velocity in magnetic Reynolds number units, rms internal magnetic fieldintensity, rms dipole field intensity at the core–mantle boundary, dipolarity onthe core–mantle boundary, and dipole tilt angle. Time axes are dipole freedecay time units. Dashed horizontal lines denote time average values; dottedhorizontal lines indicate standard deviations. The vertical dashed line denotesthe time of Fig. 4 images.

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Figs. 4 and 5 compare the internal structure of the reversingand nonreversing dynamos. The convection patterns are broadlysimilar in the two cases, in terms of their light element distri-butions, the patterns of radial velocity just below the core–mantle boundary, and the number of anticyclonic vortices in theequatorial plane. Even the radial magnetic field on the core–mantle boundary has the same general structure in the twofigures, apart from polarity. The primary difference between thereversing and nonreversing dynamos is the structure of Bz, theaxial magnetic field in the equatorial plane. For the nonreversingcase shown in Fig. 5, the two anticyclonic vortices each containa concentrated magnetic f lux bundle with the same polarity asthe dominant large-scale magnetic field. In contrast, the twoanticyclonic vortices in the reversing case shown in Fig. 4 containmagnetic f lux bundles with opposite polarities. Although onepolarity or the other dominates the field on the core–mantleboundary (normal polarity field, i.e., negative Bz happens todominate at the time shown in Fig. 4), the field usually has amixed polarity in the equatorial plane. The polarities of the twoequatorial plane flux bundles do not actually change sign duringreversals. Instead, their relative strengths change (see SI Movies1 and 2). Similarly, the diffuse background field in the equatorialplane, which also has a mixed polarity in Fig. 4, simply changesits relative proportions of positive and negative during a reversal.The reversing dynamos in Fig. 1 have this mixed polarity internalmagnetic field most of the time and always during polaritychanges. What appears as a polarity reversal of the external field

in these dynamos corresponds to a change in the relative strengthof competing, opposite polarity internal field structures. Thedipole strength and the dipolarity are lower and the timevariability is higher in the reversing dynamos because of thisinternal competition, which is largely absent in the nonreversingchaotic dynamos. Opposite polarity field structures are able tostably coexist for long periods of time in the chaotic dynamos byvirtue of their high magnetic Reynolds number, which leads tomagnetic field concentration in flux bundles localized within afew rather large convective vortices. The spacing of the vorticesacts to preserve the mixed polarity by separating the flux bundlesin longitude, inhibiting field line reconnection, diffusive merg-ing, and other effects that tend to destroy field structures withopposing polarities.

Figs. 6 shows power spectra of rms internal magnetic field, rmsfluid velocity, plus rms core–mantle boundary dipole and axialdipole fields for the reversing dynamo case. Frequency is scaledby the dipole decay frequency fd, vertical broken lines mark theknees or corner frequencies in the velocity spectra, and the solidlines indicate fits to various frequency power laws. The powerspectra of the other chaotic dynamos have features seen in Fig.6, including broad, low-frequency maximum near f/fd � 0.1(corresponding to a frequency of �5 Ma�1 for the Earth) andprogressively more negative slopes at higher frequencies in allspectra. The velocity spectrum has a corner frequency that isparticularly well defined in Fig. 6, where it is marked by a smallspectral line at �f/fd � 20 (corresponding to a frequency of �1kyr�1 for the Earth). Between the low-frequency maximum andthe velocity corner frequency, the magnetic spectra vary as f n. Inthe reversing dynamos, the rms internal field decreases as n ��2, whereas the rms dipole and axial dipole are better fit by n ��7/3. In the nonreversing dynamos, the spectral slopes are lessuniform and generally steeper. Power-law exponents have beenfit to the magnetic fields of all of the gravitational dynamos inFig. 1, and the results are given in SI Table 1. The dipole fieldspectra of the chaotic dynamos have exponents that vary from

60708090

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0.60.8

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Dipole Decay Times, t/td

0.4

0.5

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d

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Fig. 3. Time series of nonreversing gravitational dynamo with Ra � 0.6 105,E � 6.5 10�3, Pr � 1, and Pm � 20. From top to bottom, shown are graphsof rms fluid velocity in magnetic Reynolds number units, rms internal magneticfield intensity, rms dipole field intensity at the core–mantle boundary, dipo-larity on the core–mantle boundary, and dipole tilt angle. Time axes are dipolefree decay time units. Dashed horizontal lines denote time average values;dotted horizontal lines indicate standard deviations. The vertical dashed linedenotes the time of Fig. 5 images.

-9 0 +9 0 1

-2.8 0 +2.8 -55 0 +55

Bz

χ

urBr

Fig. 4. Structure of the reversing gravitational dynamo at the time indicatedin Fig. 2. Shown are contours of the following variables. (Upper) The radialcomponent of the magnetic field on the core–mantle boundary (Left) and theradial component of the fluid velocity in magnetic Reynolds number units atradius r � 0.93ro (Right). (Lower) Axial component of the magnetic field in theequatorial plane with velocity arrows superimposed (Left) and normalizedlight element concentration in the equatorial plane (Right).

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n � �3 for the nonreversing cases to n � �7/3 or n � �2 forthe most frequently reversing cases. All of the chaotic dynamoshave spectra power law exponents near n � �2 for their internalmagnetic fields. SI Figs. 12 and 13 show that the amplitude andshape of the dipole power spectra in Fig. 6 compare favorablywith geomagnetic power spectra derived from deep-sea sedimentrecords (19).

Effects of the Lorentz ForceThe chaotic dynamos have smaller time average velocities thannonmagnetic convection with otherwise identical parameters.From Fig. 1, the ratio of dynamo to nonmagnetic convectionvelocity is 0.78 at Ra � 6 104 and 0.81 at Ra � 1 105. Becausethe only difference between the pairs of dynamo and nonmag-netic convection calculations is the presence or absence of theLorentz force, the slightly smaller time average velocity in thedynamos shows that the Lorentz force reduces the time averagevelocity of the convection, as previous dynamo modeling studies(35) have found.

Whereas the Lorentz force tends to reduce the time averagekinetic energy of the convection, it has just the opposite effecton its time variability. This can be seen by comparing thereversing dynamo and its corresponding nonmagnetic convec-tion velocity spectra in Fig. 6. The reversing dynamo has alarge-amplitude, broadband velocity spectrum with peak vari-ance at very low frequencies. The low-frequency portion of thedynamo velocity spectrum in Fig. 6 has a similar shape asthe three magnetic spectra. The higher-frequency portion of thedynamo velocity spectrum also includes the prominent corner orknee referred to above. None of these features are present in thevelocity spectra of the corresponding nonmagnetic convection.Instead, the nonmagnetic velocity spectrum consists of a fewhigh-frequency spectral lines above a low-intensity background,and there is far less total variance than in the dynamo case.

In the time domain, the low-frequency variations of dynamokinetic and magnetic energies are almost exactly out of phase.Fig. 7, which shows low-pass filtered time series of kinetic energydensity and internal magnetic energy density for the Ra � 1

105 reversing dynamo, was obtained by applying a runningaverage of length td to the unfiltered time series in Fig. 2. Thecorrelation coefficient between low-frequency kinetic and mag-netic energy variations in Fig. 7 is �0.96. The inverse phaserelationship also is due to the Lorentz force. An increase in theinternal magnetic field strength increases the Lorentz force onthe fluid and reduces the convective velocity; conversely, areduction in the internal magnetic field strength reduces theLorentz force on the fluid and enhances the convective velocity.The tradeoff between kinetic and magnetic energy gives thechaotic dynamos additional freedom that is missing from non-magnetic convection and offers an explanation for why thechaotic dynamos have much larger-amplitude velocity fluctua-tions although slightly smaller time average velocities comparedwith their nonmagnetic convection counterparts. Variabilityoccurs primarily at low frequencies in the dynamos because theirmagnetic fields are dominated by large-scale components withlong free decay times.

The tradeoff between kinetic and magnetic energy fluctua-tions in Fig. 7 can be modeled by assuming constant total energydissipation, as implied by the zero frequency limit of Eqs. 1–4

-2.0 0 +2.0 -40 0 +40

-12 0 +12 0 1

Bz

χ

urBr

Fig. 5. Structure of the nonreversing gravitational dynamo at the timeindicated in Fig. 3. Shown are contours of the following variables. (Upper)(Left) Radial component of the magnetic field on the core-mantle boundary.(Right) Radial component of the fluid velocity in magnetic Reynolds numberunits at radius r � 0.93ro. (Lower) (Left) Axial component of the magnetic fieldin the equatorial plane with velocity arrows superimposed. (Right) Normal-ized light element concentration in the equatorial plane.

B10

-8

10-6

10-4

10-2

100

102

Pow

er 0.01 0.1 1.0 10 100

Frequency, f/fd

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

10-6

10-4

10-2

100

102

104

10-12

10-10

10-8

10-6

10-4

10-2

100

102

f -7/3

f -7/3

f -2

Bd

Bd(axial)uη

0.01 0.1 1.0 10 100

0.01 0.1 1.0 10 1000.01 0.1 1.0 10 100

Fig. 6. Power spectra of the reversing gravitational dynamo at Ra � 1 105

shown in Fig. 2. Frequency axes are dipole decay frequency units f/fd. (Upper)rms internal magnetic field intensity (Left) and rms dipole field intensity on thecore–mantle boundary (Right). (Lower) rms dynamo velocity (black) comparedwith rms nonmagnetic convection velocity (gray) in magnetic Reynolds num-ber units (Left) and rms axial dipole field intensity on the core–mantle bound-ary (Right). Dashed vertical lines denote velocity corner frequency; solid linesindicate various power-law slopes.

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(34). In dimensionless form, the average viscous dissipation perunit mass can be written in terms of the kinetic energy densityas k � �k

�2Ek, where �k is the viscous dissipation length scaleof the flow. By using the same nondimensionalization, theaverage Ohmic dissipation per unit mass can be written as m ��m

�2EmEk, where �m is the Ohmic dissipation length scale. Con-stant total dissipation then implies

�m�2EmEk � �k

�2Ek � k � m. [7]

An alternative model assumes the Ohmic dissipation varieslinearly with the internal magnetic energy (36) and yields

�m�2Em � �k

�2Ek � k � m. [8]

As a check, k and m were calculated at selected times for thedynamo and nonmagnetic convection cases at Ra � 1 105, andit was found that their total k � m varied only by a few percent.The variability in �k or �m was not directly checked by calculation;however, it would readily be apparent as changes in the scale ofvelocity or magnetic field structures with time, which are notobserved in these calculations. The consistency of the flowstructure and the (nearly) constant dissipation accounts for thesmall time variations in the nonmagnetic convection cases. Forthe dynamos, the kinetic and magnetic energies appear as aproduct in the first term in Eq. 7 and as a sum in Eq. 8, eitherof which allows the kinetic and magnetic energies to varyinversely while maintaining constant total dissipation and fixeddissipation length scales.

Eqs. 7 and 8 were used to predict the low-frequency dynamomagnetic energy variations in terms of the kinetic energyvariations, with the results shown in Fig. 7. The dashed curvesFig. 7 Middle and Bottom are the predicted low-frequencyvariation of the internal magnetic energy Em(t) obtained bysubstituting Ek(t) into the above equations and solving for thebest-fitting constant values of �m and �k . Both models provide

good fits in terms of phase and amplitude, although the nonlin-ear model Eq. 7 fit is slightly better (correlation coefficient 0.95)than the linear model Eq. 8 (correlation coefficient 0.89),particularly at times when the magnetic energy is high.

Comparison with the GeodynamoBecause of the low viscosity of the outer core fluid and theEarth’s rapid rotation, there is little prospect of making a directnumerical simulation of the geodynamo. The individual controlparameters in this study are not representative of their Earthvalues, which are approximately Ra � 1030, E � 10�14, Pr � 10�1,and Pm � 10�6, respectively. Even the largest numerical dynamomodels (23) are compelled to use some unrealistic controlparameters. This inherent limitation has led to a search forscaling laws that would allow extrapolation of the results ofnumerical dynamos to planetary conditions (36–38). For exam-ple, it is sometimes asserted (39) that planetary dynamos satu-rate (equilibrate) with rms magnetic fields corresponding to anElsasser number � � �B2/�o� � 1. The Elsasser number isparticularly attractive as a scaling parameter because it is inde-pendent of the dynamo energy source. Unfortunately, it has beenshown that � varies with the Rayleigh number in convection-driven dynamos (34), and generally the field strength does notconform to constant �. Fig. 1 shows that, for the gravitationaldynamos in this study, the rms internal field strength increaseswith Ra in the balanced dynamo regime, then saturates some-where in the range B � 3.5–4 (corresponding approximately to� � 12–16), possibly decreasing at higher Ra. This behaviorprovides some limited support for the constant Elsasser numberassumption for the internal magnetic field, although it does notaddress possible Elsasser number dependence on the othercontrol parameters. However, Fig. 1 also shows that Bd variesinversely with Ra in the chaotic regime, so the external dipolefield strength in these models depends strongly on the dynamoenergy source and clearly does not have constant Elsassernumber.

Other parameters that have been proposed for dynamo modelscaling include the magnetic Reynolds number and the Rossbynumber. The magnetic Reynolds number for the flow in the corethat induces the decadal geomagnetic secular variation is ap-proximately 300–500 (3–6), compared with 100–300 for thedynamos in Fig. 1, so the models scale reasonably well in termsof this parameter. Recent numerical (34) and theoretical studies(40) indicate that dynamo properties, such as dipole strength andthe transition from dipole-dominant to multipolar states, dependon the local Rossby number Ro�. Although the Rossby numberfor large-scale flow in the core is very small, Ro� may not besmall, because the characteristic wave number of convection inthe Earth’s core is predicted to be rather large (41). Olson andChristensen (38) estimate Ro� � 0.05–0.2 for the geodynamo,similar to Ro� � 0.014–0.111 for the gravitational dynamos inFig. 1.

Independent of scaling considerations, the magnetic fieldstructure of these gravitational dynamos compares favorablywith the present-day geomagnetic field structure in severalrespects. The axial dipole field in the dynamo models is mostlya product of the two pairs of high-latitude flux bundles seen inFigs. 4 and 5. These structures are similar to the high-latitudeflux bundles on the core–mantle boundary in the present-daygeomagnetic field (3, 42), which are the major contributors to thegeomagnetic axial dipole moment. One difference is that the fluxbundles in Figs. 4 and 5 drift rapidly westward, whereas theirgeomagnetic counterparts more stationary, possibly because thelater are linked to heterogeneity in the lower mantle. Anothersignificant property that these dynamo models share with thepaleomagnetic field is dipole dominance during stable polaritychrons. The dipole dominance of the external field is enhancedin the gravitational dynamos because the magnetic field gener-

Fig. 7. Low-pass-filtered time series of the reversing gravitational dynamoshown in Fig. 2. (Top) Kinetic energy density. (Middle) Internal magneticenergy density (solid) compared with the nonlinear constant dissipationmodel (dashed). (Bottom) Internal magnetic energy density (solid) comparedwith the linear constant dissipation model (dashed).

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ation is concentrated deep in the fluid, close to the inner coreboundary, with the effect that the nondipole moments of thefield are strongly attenuated with radius.

Time domain statistics of the gravitational dynamo models inthe range Ro� � 0.05–0.07 compare favorably with time domainstatistics of the geodynamo. For example, the reversing dynamoat Ra � 1 105 has Ro� � 0.06, a dipole variability s � 0.37, anda time average dipolarity d � 0.53. The model dipolarity issimilar to the present day geomagnetic field on the core–mantleboundary, when truncated at spherical harmonic �max � 14 hasd � 0.64, or, alternatively, d � 0.54 if the core–mantle boundaryfield spectrum is extrapolated to large �max (43). Another pointof comparison is the dipole variability parameter, estimated to benear s � 0.4 for the paleomagnetic field over the past 160 Ma(44), virtually the same as this dynamo model. In terms ofpolarity reversal rates, Fig. 2 contains five reversals and excur-sions in 120 dipole decay times, equivalent to two such events permillion years on Earth. When averaged over the entire sea-floormagnetic record, the rate of geomagnetic reversals also is abouttwo per million years (18, 29). The ultra-low frequency modu-lation of the reversal rate seen in the paleomagnetic record is notexplicit in these dynamo models, but the abrupt transition fromreversing to nonreversing states in Fig. 1 suggests a possibleexplanation. A relatively small decrease in the Rayleigh numberof the core, produced by a decrease in the total heat flow at thecore–mantle boundary, for example, could switch the geody-namo from reversing to nonreversing behavior.

Previous dynamo modeling studies have shown that reversalsoccur in connection with velocity fluctuations. Sarson and Jones(45) found that reversals follow fluctuations in meridionalcirculation associated with surges in buoyancy. Wicht and Olson(26) found reversals initiated by reverse magnetic f lux trans-ported in plume-like upwellings, as have Aubert et al. (27).Because none of these mechanisms produce reversals when thedipole field is strong, deciphering the reversal phenomenonrequires an understanding of why the dipole sometimes col-lapses. The present study reveals that the Lorentz force driveslow-frequency dipole moment change in convection-driven dy-namos, but the mechanism through which this occurs remainsunclear. It may involve subtle interactions between the fluctu-ating convection and the magnetic fields, possibly as follows.High-frequency variations in the convection, in the form oftransient plumes and vortices, induce magnetic field variationson their same short time scales. Part of the energy in thesefluctuations reinforces the dynamo, part draws energy from it,and energy also is lost by Ohmic and viscous dissipation. In achaotic f low, the production of magnetic energy tends to beslightly out of balance with the dissipative effects, so thesedynamos evolve on time scales that are very long compared withthe characteristic time scales of their convective and diffusive

components. The magnetic field strengthens while magneticenergy production exceeds dissipation, but eventually the in-creased Lorentz force that accompanies the strengthened mag-netic field reduces the kinetic energy, the rate of magnetic fieldgeneration declines, and the magnetic field begins to decay. Thedynamo continually drifts between high-strength and low-strength states, occasionally reversing polarity when it is weak.

MethodsThe numerical dynamo model in this study (MAG, available at www.geody-namics.org), was originally developed by G. Glatzmaier and has been bench-marked and was used previously in systematic dynamo studies (35, 38). Thefluid velocity and the magnetic field vectors are represented as sums ofpoloidal and toroidal scalars, ensuring that the continuity equations aresatisfied identically. The momentum and induction equations are then de-composed into four scalar evolution equations in terms of these functions andare advanced simultaneously using explicit time steps on a spherical finitedifference grid, along with the buoyancy equation.

The five scalar variables also are represented in terms of surface sphericalharmonics up to degree and order �, and in Chebyshev polynomials in radius.At each time step, the nonlinear terms are evaluated on the spherical finitedifference grid, and the linear terms are evaluated by using the sphericalharmonics and Chebyshev polynomials. Additional details about the numer-ical method are given in ref. 46.

The most severe restrictions on numerical models stem from the fact that itis not practical to make calculations with parameter values that are realistic forthe Earth’s core with existing codes (47–49). To remedy this, we set Pr � 1 inall calculations and choose relatively large values of the Ekman number, E �6.5 10�3, but a few cases with E � 3 10�4 are included for comparisonpurposes. These choices then necessitate Pm � 5–20 to maintain a self-sustaining dynamo.

The finite difference grid and the truncation level of the spherical harmon-ics were chosen to ensure that the spectral power of magnetic energy atten-uates by a factor of 100 or more from its peak � value and that the diffusivelayers at the inner core and core–mantle boundaries are well resolved. Thehigh Ekman number cases with E � 6.5 10�3 use Nr � 25 radial grid intervalswith Nr � 2 Chebyshev polynomials and harmonic truncation �max � 32. Thelower Ekman number cases use Nr � 35 and �max � 48.

The calculations were initialized with random buoyancy perturbations andan axial dipole magnetic field. The dynamos were run until their statisticalfluctuations became stationary. Short run times sufficed for the dynamos withconstant or nearly constant magnetic and kinetic energies. A few, very longruns were made for some of the most complex cases, up to 900 viscous timeunits. Model output from the first few dipole decay times was excluded fromthe analysis to eliminate contamination by the transient adjustment to theinitial conditions. Nonmagnetic convection calculations were made with thesame set of parameters as several of the self-sustaining dynamo cases. Powerspectra were computed with Hanning window tapers, normalized to preservethe variance of the raw time series. SI Table 1 lists the control parameters andsummarizes the results of all calculations used in this study.

ACKNOWLEDGMENTS. I thank H. Amit and J. Aubert for thoughtful reviews.This work was supported by National Science Foundation Geophysics ProgramGrant EAR-0604974.

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