-
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.’’
www.pnas.org�cgi�doi�10.1073�pnas.0800385105
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.
www.pnas.org�cgi�doi�10.1073�pnas.0801065105
3658 � PNAS � March 4, 2008 � vol. 105 � no. 9 www.pnas.org
Dow
nloa
ded
by g
uest
on
July
7, 2
021
Dow
nloa
ded
by g
uest
on
July
7, 2
021
Dow
nloa
ded
by g
uest
on
July
7, 2
021
Dow
nloa
ded
by g
uest
on
July
7, 2
021
Dow
nloa
ded
by g
uest
on
July
7, 2
021
Dow
nloa
ded
by g
uest
on
July
7, 2
021
Dow
nloa
ded
by g
uest
on
July
7, 2
021
Dow
nloa
ded
by g
uest
on
July
7, 2
021
Dow
nloa
ded
by g
uest
on
July
7, 2
021
Dow
nloa
ded
by g
uest
on
July
7, 2
021
-
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: [email protected].
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
www.pnas.org�cgi�doi�10.1073�pnas.0709081104 PNAS � December 18,
2007 � vol. 104 � no. 51 � 20159–20166
GEO
PHYS
ICS
INA
UG
URA
LA
RTIC
LE
http://www.pnas.org/cgi/content/full/0709081104/DC1http://www.pnas.org/cgi/content/full/0709081104/DC1
-
���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.
20160 � www.pnas.org�cgi�doi�10.1073�pnas.0709081104 Olson
http://www.pnas.org/cgi/content/full/0709081104/DC1http://www.pnas.org/cgi/content/full/0709081104/DC1http://www.pnas.org/cgi/content/full/0709081104/DC1
-
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
3
4
5
6
0 0.5 1.0 1.540
80
120
160
200
240
280
320
0.0
0.5
1.0
1.5
0.0
0.2
0.4
0.6
0.8
Rayleigh Number, Ra (x105)
B uη
Bd d
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
20o
10o
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).
100120
140
160
180200
23
4
5
67
0.20.40.60.81.01.2
0 20 40 60 80 100 1200
306090
120150180
uη
B
Bd
Dipole Decay Times, t/td
0.00.2
0.4
0.6
0.81.0
d
tilt
0
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.
Olson PNAS � December 18, 2007 � vol. 104 � no. 51 � 20161
GEO
PHYS
ICS
INA
UG
URA
LA
RTIC
LE
-
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
100110
2.53.0
3.5
4.0
4.55.0
0.60.8
1.0
1.2
1.41.6
0 20 40 60 80180
170
160
Dipole Decay Times, t/td
0.4
0.5
0.6
0.7
uη
B
Bd
d
tilt
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).
20162 � www.pnas.org�cgi�doi�10.1073�pnas.0709081104 Olson
http://www.pnas.org/cgi/content/full/0709081104/DC1http://www.pnas.org/cgi/content/full/0709081104/DC1http://www.pnas.org/cgi/content/full/0709081104/DC1
-
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.
Olson PNAS � December 18, 2007 � vol. 104 � no. 51 � 20163
GEO
PHYS
ICS
INA
UG
URA
LA
RTIC
LE
http://www.pnas.org/cgi/content/full/0709081104/DC1
-
(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).
20164 � www.pnas.org�cgi�doi�10.1073�pnas.0709081104 Olson
-
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.
1. Merrill RT, McElhinny MW, McFadden PL (1998) The Magnetic
Field of the Earth(Academic, San Diego).
2. Valet J-P (2003) Rev Geophys, 10.1029/2001RG000104.3. Hulot
G, Eymin C, Langlais B, Mandea M, Olsen N (2002) Nature
416:620–623.4. Amit H, Olson P (2006) Phys Earth Planet Inter
155:120–139.5. Hide R (1966) Philos Trans R Soc London A
259:615–650.6. Jackson A, Bloxham J, Gubbins D (1993) in Dynamics
of Earth’s Deep Interior and Earth
Rotation, Geophysics Monograph Series, eds LeMouël J-L Smylie
DE, Herring TA (AGU,Washington, DC), Vol 72, pp 97–107.
7. Bloxham J, Zatman S, Dumberry M (2002) Nature 401:65–68.8.
Sakuraba A, Hamano Y (2007) Phys Earth Planet Interiors, in
press.9. Korte M, Constable C (2005) Geochem Geophys Geosyst,
10.1029–2004GC00081.
10. Korte M, Constable C (2005) Earth Planet Sci Lett
236:348–358.11. Valet J-P, Meynadier L, Guyodo Y (2005) Nature
435:802–805.12. Channell JET, Hodell DA, McManus J, Lehman B (1998)
Nature 394:464–468.13. Yamazaki T, Oda H (2002) Science
295:2435–2438.14. Gubbins D (1987) Nature 326:167–169.15. Aldridge
K, Baker R (2003) Phys Earth Planet Interiors 140:91–100.16.
Pelletier JD (2002) Proc Natl Acad Sci USA 99:2546–2553.17. Merrill
RT, McFadden PL (1999) Rev Geophys 37:201–226.
18. Constable C (2003) in Earth’s Core and Lower Mantle, eds
Jones CA, Soward AM, ZhangK (Taylor and Francis, London), pp
77–99.
19. Constable C, Johnson C (2005) Phys Earth Planet Interiors
153:61–73.20. Berhanu M, Monchaux R, Fauve S, Mordant N, Petrelis
F, Chiffaudel A, Daviaud F,
Dubrulle B, MariŽ L, Ravelet F, et al. (2006) Europhys Lett
77:59001.21. Ryan A, Sarson G (2007) Geophys Res Lett 34:L02307.22.
Kutzner C, Christensen UR (2002) Phys Earth Planet Interiors
131:29–45.23. Takahashi F, Matsushima M, Honkura Y (2005) Science
309:459–461.24. Glatzmaier GA, Coe RS, Hongre L, Roberts PH (1999)
Nature 401:885–890.25. Kutzner C, Christensen UR (2004) Geophys J
Int 157:1105–1118.26. Wicht J, Olson P (2004) Geochem Geodyn
Geosyst, 10.1029/2003GC000602.27. Aubert J, Aurnou J, Wicht J
(2007) Geophys J Int, in press.28. Buffett BA (2000) Science
288:2007–2012.29. Cande S, Kent D (1992) J Geophys Res
97:13917–13951.30. Courtillot V, Olson P (2007) Earth Planet Sci
Lett 260:495–504.31. Loper DE (1978) Geophys J R Astron Soc
54:389–404.32. Labrosse S (2003) Phys Earth Planet Interiors
140:127–143.33. Kutzner C, Christensen UR (2000) Geophys Res Lett
27:29–32.34. Christensen UR, Aubert J (2006) Geophys J Int
166:97–114.35. Christensen UR, Olson P, Glatzmaier GA (1999)
Geophys J Int 138:393–409.
Olson PNAS � December 18, 2007 � vol. 104 � no. 51 � 20165
GEO
PHYS
ICS
INA
UG
URA
LA
RTIC
LE
http://www.pnas.org/cgi/content/full/0709081104/DC1
-
36. Christensen UR, Tilgner A (2004) Nature 429:169–171.37.
Starchenko SV, Jones CA (2002) Icarus 157:426–435.38. Olson P,
Christensen UR (2006) Earth Planet Sci Lett 250:561–571.39.
Stevenson DJ (2003) Earth Planet Sci Lett 208:1–11.40. Sreenivasan
B, Jones CA (2006) Geophys J Int 164:467–476.41. Voorhies CV (2004)
J Geophys Res 109:B03106.42. Olsen N, Holme R, Hulot G, Sabaka T,
Neubert T, Toeffner-Clausen L, Primdahl F,
Joergensen J, Leger J-M, Barraclough D, et al. (2000) Geophys
Res Lett 27:3607–3610.
43. Stacey FD (1992) Physics of the Earth (Brookfield, Brisbane,
Australia), 3rd Ed.44. Juarez MT, Tauxe L, Gee JS, Pick T (1998)
Nature 394:878–881.45. Sarson G, Jones CA (1999) Phys Earth Planet
Interiors 111:3–20.46. Christensen UR, Wicht J (2007) in Treatise
on Geophysics (Elsevier, Amsterdam), Vol 8,
pp 245–282.47. Dormy E, Valet J-P, Courtillot V (200) Geochem
Geophys Geosys, 2000GC000062.48. Glatzmaier GA (2002) Annu Rev
Earth Planet Sci 30:237–257.49. Kono M, Roberts PH (2002) Rev
Geophys, 10.1029/2000RG000102.
20166 � www.pnas.org�cgi�doi�10.1073�pnas.0709081104 Olson