there must be also be a part of the rotation (off-)axisvelocity pattern that does not depend on time.

23. B. Romanowicz, X.-D. Li, J. Durek, Science 274, 963(1996).

24. Song and Richards (2) relied on high-quality obser-vations, but these were relatively few in number andthey investigated only three source-receiver paths toobtain the time variations. In comparison, we usednoisier ISC data, but because each CAS value wasderived on average from 230 observations, the stan-dard error of the mean should be -0.1 s. The spatialuniformity of the CAS averages in Fig. 3 testifies totheir accuracy. Because of the dense data coverage,we can simultaneously determine both the spatialand temporal variations of the residuals, inverting forboth the magnitude and orientation of the anisotropyas a function of time. The analysis in (2) requiredSong and Richards to assume both the geometry(constant model of anisotropy and tilt of the symme-try axis) and process (rigid rotation about an axiscoinciding with the mantle rotation axis) to interprettheir observations. Still, data for the second path(Kermadec to Kongsberg) give time variations incon-sistent with the model. The factor of 3 differencebetween the rotation rates obtained in (2) and in thepresent study is at least partly attributable to themodel of anisotropy used by Song and Richards forthe inner core. The magnitude of anisotropy theyassumed is about twice that of model IAC4B (14), forexample, and had they used this model, they wouldhave obtained a rotation rate of '2° year 1. Weinverted for the magnitude of anisotropy in thepresent study.

25. The time-dependent solution (Fig. 4C) is dominatedby the harmonics Y2 and Y1, whose sum has a max-imum at -60°N. Thus, observations at correspond-ing values of the angle (, the angle that a ray formswith the axis of symmetry, should be the best formonitoring the rotation of the inner core. This is con-firmed in Fig. 4D, which shows the time derivative ofFig. 4C (the relatively large effects predicted for lowlatitudes may be contaminated by noise, becausethe CAS coverage below 30° has had large gapsduring the early time intervals; Fig. 2). The predictedsolutions for the earliest and latest time periods com-pare well with the data (Fig. 4, E and F) as well as withthe spherical-harmonic expansion for the sa-me timeperiods (Fig. 2, A and F).

26. K. Lambeck, The Earth's Variable Rotation: Geo-physical Causes and Consequences (CambridgeUniv. Press, New York, 1980); Geophysical Geode-sy: The Slow Deformations of the Earth (Oxford Univ.Press, New York, 1988).

27. This angular acceleration value, -5 x 10 22 rads 2 is an average for the past 2 millennia and hasvaried by <50% over this time period (26). Becauseof the need for a large backward extrapolation intime, however, its uncertainty for the present appli-cation is difficult to evaluate.

28. A naive analysis would attempt to relate the lagtime T = 105 years (3 x 1012 s) between the innercore and mantle with the viscous coupling time tv =d 2/v, where d is the thickness (2.25 x 106 m) andv is the kinematic viscosity of the outer core. Theresult, v -2m2s 1 [viscosity r= vp - 2 x 104Pa-s, using a density of p = 1.1 x 104 kg m 3; A.M. Dziewonski and D. L. Anderson, Phys. EarthPlanet. Inter. 25, 297 (1981)], is orders of magni-tude greater than current estimates for the viscosityof the outer core (37-39). This approach, however,disregards the essentially two-dimensional charac-ter of rotating flows (as expressed by the Taylor-Proudman theorem); momentum need only betransmitted across a thin (Ekman) boundary layerrather than the full thickness of the fluid layer (29,40, 41). The characteristic time for momentumtransfer is thereby shortened by a factor of E1 2,where the Ekman numberE = v/d2Q -3 x 10 16,yielding a rotation rate 2 = 7.3 x 10 5 rad s(compare D. L. Book and J. A. Valdivia, J. PlasmaPhys., in press).

29. D. Gubbins and P. H. Roberts, in Geomagnetism, J.A. Jacobs, Ed. (Academic Press, Orlando, FL, 1987),vol. 2, pp. 1-23.

30. Equating the lag time T 105 years (3 x 1012 S) with

the "spin-up" time tE = d/(S2) 2 where d is the thick-ness of the outer core and 52 is Earth's rotation rate,yields a value for the kinematic viscosity v - 10 8 M2s for the outer-core fluid [viscosity -q = vp - 10 4Pa-s (29, 40-42)]. Taking into account the Lorentzforces associated with the magnetic field leads to amagnetic spin-up time TM = TE/(aV2) in the case ofstrong fields (a > 2) (42). Here, the magnetic inter-action parameter or Elsasser number, o2 = B2a(2pfl) 3.7 x 105 B2 in SI units (where B is themagnetic field) gives the ratio of Lorentz to Coriolisforces (20, 29) [the electrical conductivity of the out-er-core fluid, a = 6 (+3) x 105 S m 1, is relativelywell known from high-pressure experiments (38)].Because the total strength of the magnetic field deepinside the core is not well known, the magnitude ofthe Elsasser number a2 is also uncertain. Downwardextrapolation of the present-day field observed atEarth's surface yields a value of B - 10 3 T for theradial inductance at the base of the mantle, but it isknown that this is only part of the field existing deepinside the core; various theoretical estimates boundthe characteristic value of the field inside the corebetween -10 1 and 10 4 T (20, 29, 31). The El-sasser number is therefore within the range -10 3

to 103 (20, 29). For the "weak-field" case, which usesthe "observed" value B - 10 3 T, (2 - 0.4. If weequate TM with the lag time T - 105 years identifiedabove for the inner-core rotation, we obtain a valuevB2 10- 14 T2 m2 s for the product of kinematicviscosity and squared magnetic field in the outercore. Using the observed magnetic field at the bot-tom of the mantle (weak-field value B - 10 3 T)again yields a low value of viscosity, v - 10 8 m2s 1, for the outer-core fluid. Indeed, values of kine-matic viscosity above the range 10-7 to 10 6 M2s 1 imply unacceptably small values for the magnet-ic field in the core (20, 29, 31).

31. G. A. Glatzmaier and P. H. Roberts, Nature 377, 203(1995); Phys. Earth Planet. Inter. 91, 63 (1995).

32. D. Gubbins, J. Geophys. Res. 86,11695 (1981).33. In the Glatzmaier-Roberts model (31), the eastward

rotation of the inner core relative to the mantle ismaintained by magnetic coupling between the solidinner core and the fluid just above it, which is flowing

Three-dimensional (3D) numerical simu-lations of the geodynamo, the mechanismin Earth's core that generates the geomag-netic field, showed that a magnetic fieldwith an intensity, structure, and time de-pendence similar to that of Earth's can bemaintained by a convective model (1, 2).The convection, which takes place in thefluid outer core surrounding the solid inner

G. A. Glatzmaier, Institute of Geophysics and PlanetaryPhysics, Los Alamos National Laboratory, Los Alamos,NM 87545, USA. E-mail: gag@lanl.govP. H. Roberts, Institute of Geophysics and PlanetaryPhysics, University of California, Los Angeles, CA 90095,USA. E-mail: roberts@math.ucla.edu

*To whom correspondence should be addressed.

eastward because of a thermal-wind effect (G. A.Glatzmaier, personal communication). As in theiroriginal simulation, their latest results show an east-ward rotation of the inner core that varies in magni-tude between 2° and 3° year I relative to the mantle.

34. S. R. C. Malin and B. M. Hodder, Nature 296, 726(1982); M. G. McLeod, J. Geophys. Res. 90, 4597(1985).

35. The magnetic diffusivity of the core metal, =(RO(T) 1 1 m2 s 1 (where .,O is the magnetic per-meability of free space and (T is the elastical conduc-tivity of the outer-core fluid), is sufficiently high thatfield lines diffuse -6 x 104 m into the inner corewithin a century (20, 29, 38); see also R. Hollerbachand C. A. Jones, Nature 365, 541 (1993).

36. Scaling analysis between the core and atmosphereshows that a few years' seismological monitoring ofthe inner-core motion corresponds to hourly obser-vations of the weather.

37. C. R. Gwinn, T. A. Herring, I. I. Shapiro, J. Geophys.Res. 91, 4755 (1986).

38. R. Jeanloz, Annu. Rev. Earth Planet. Sci. 18, 357(1 990).

39. E. Wasserman, L. Stixrude, R. E. Cohen, Eos 77,S268 (1996).

40. H. Bondi and R. A. Lyttleton, Proc. Camb. Philos.Soc. 44, 345 (1948).

41. H. P. Greenspan, The Theory of Rotating Fluids(Cambridge Univ. Press, New York, 1968).

42. E. R. Benton and A. Clark Jr., Annu. Rev. Fluid Mech.6, 257 (1974); see also D. E. Loper, Phys. Fluids 13,2995 (1970); R. G. Davis, Geophys. Res. Lett. 21,1815 (1994).

43. We thank G. Glatzmaier, D. Loper, M. Manga, P.Olson, P. H. Roberts, B. Romanowicz, L. Stixrude, J.Tromp, and G. Ekstrom for helpful discussions, aswell as X.-D. Song and P. G. Richards for sending usa copy of their paper before publication and for sub-sequent discussions. This work was initiated whileA.M.D. was a visiting professor in the Miller Institutefor Basic Research in Science (March 1995) and wassupported by NSF and NASA.

12 June 1996; accepted 20 September 1996

core, twists and shears magnetic field, con-tinually generating new magnetic field toreplace that which diffuses away. The mod-el we describe here (2) assumes the mass,dimensions, and basic rotation rate ofEarth's core, an estimate of the heat flowout of the core, and, as far as possible,realistic material properties (3).

Convection is driven by thermal andcompositional buoyancy sources that devel-op at the inner core boundary as the Earthcools and iron alloy solidifies onto the innercore (4). For simplicity, we modeled thecore as a binary alloy (5). Our boundaryconditions at the inner core boundary con-strain the local fluxes of the latent heat and

SCIENCE * VOL. 274 * 13 DECEMBER 1996

Rotation and Magnetism of Earth's Inner CoreGary A. Glatzmaier* and Paul H. Roberts

Three-dimensional numerical simulations of the geodynamo suggest that a super-rotation of Earth's solid inner core relative to the mantle is maintained by magneticcoupling between the inner core and an eastward thermal wind in the fluid outer core.This mechanism, which is analogous to a synchronous motor, also plays a funda-mental role in the generation of Earth's magnetic field.

1887

on Septem

ber 5, 2020

http://science.sciencemag.org/

Dow

nloaded from

4 .; j0

the light constituent to be proportional toeach other and to the local cooling rate.The flux of the light constituent providesthree times as much buoyancy as the latentheat flux. However, the vigor of the con-vection, the strength of the magnetic field,and the angular velocity of the inner coreare all ultimately determined by the ther-mal boundary condition at the core-mantleboundary. We prescribed a heat flowthrough it of 7.2 TW, which is geophysi-cally realistic (6), 5.0 TW of this being theheat conducted down the adiabatic part ofthe temperature gradient.

The inner core in our model is free torotate about the geographic axis in responseto the magnetic and viscous torques towhich the outer core subjects it (7). Wefound that the inner core rotates relative tothe mantle in the eastward direction, at anangular velocity of QI - 20 to 30 per year(8). Our prediction of eastward inner corerotation is consistent with two subsequent,independent analyses of seismic data: Song

c 0

c

=

csCom la

Fig.relatiour s

and Richards (9) first estimated lICT 1°per year, and then Su et al. (10) estimatedlIC: - 3° per year. Considering the uncer-tainties in our model and material proper-ties and in the seismic data and inner coreanisotropy, these preliminary theoreticaland observational results are in reasonableagreement.

For a typical 15,000-year interval, ourmean Qlc is !C- 2.60 per year (Fig. 1).The standard deviation about that mean isQ.4°/year, the root mean square of flIC isil rms- = 0.0050 per year2, and the timescale of variations of I is TicfQ1 rns- 500 years. However, there is alsoa distinct 3000-year period with an ampli-tude variation of about a ±0.50 per year(Fig. 1).

To explain the sense of rotation of theinner core relative to the mantle, it is help-ful to recognize that the inner core plays acentral role in the dynamics of the outercore (11). Consider an imaginary circularcylinder coaxial with the rotation axis andtouching the inner core at its equator. This"tangent cylinder" divides the outer coreinto an interior region I and an exteriorregion E. The interior region consists of anorthern polar region IN and a southern

2Qs x lW polar region Is,Convection in a rapidly rotating body

1 of fluid, like Earth's outer core, takes theo l I form of Taylor columns, which have axeso 3 6 9 12 1y5 parallel to the rotation axis (12) and fluid

flowing in planes perpendicular to it.1. Eastward angular velocity of the inner core However, when the fluid flow generates ayve to the mantle for the last 15,000 years of strong magnetic field that in turn feeds,imulation. back onto the flow by means of Lorentz

forces, the flow structure is somewhat dif-

ferent. In our simulation (2), several non-axisymmetric magnetically suppressedTaylor columns exist in E, whereas in I,although the flow is also mainly in planesperpendicular to the rotation axis, it isnearly axisymmetric and sheared parallelto the rotation axis. Consequently, heatand light elements are more efficientlytransported from the inner core boundaryto the core-mantle boundary by the non-axisymmetric flows in E than by the nearlyaxisymmetric flow in I, which takes a moreindirect, helical route to the core-mantleboundary. As a result, in our simulation(2) the axisymmetric parts of the pertur-bations of the entropy and light constitu-ent (relative to their constant basic-statevalues) at a given radius near the innercore boundary are roughly five times great-er in I than their respective values in E.The associated buoyancy force in I pro-duces an outward flow along the rotationaxis, which (because of mass conserva-tion) requires a flow directed toward therotation axis near the inner core boundaryand away from the axis near the core-mantle boundary. This axisymmetric partof the flow is called the meridional circu-lation (Fig. 2A). Because the angular mo-mentum of fluid parcels is approximatelyconserved, fluid flowing toward the rota-tion axis just outside of the inner coreboundary, in IN and Is, is spun up; fluidflowing away from the axis near the core-mantle boundary is spun down. The result-ing zonal flow in I is eastward near theinner core and westward near the mantle,relative to the fluid in E and to the mantle(Fig. 2B). In atmospheric contexts, these

Fig. 2 (left). Snapshots in A Bthe meridian plane of (A) the Naxisymmetric meridional flu-id flow and (B) the axisym-metric zonal fluid flow. Thedirection of the arrows in (A) a' , \are parallel to the flow; theirlength is proportional to theflow speed. Contours ofconstant angular velocity ware shown in (B); solid linescorrespond to eastwardflow relative to the mantle,and dashed lines, to west- ,ward flow. The maximum .' a/meridional flow in (A) is 6 x10-4 m/s; the maximum an- 4,,gular velocity in (B) is 50 peryear (contour interval is 0.50per year). Of particular inter- Sest is the contour line adja-cent to the inner core boundary, which indicates that the inner core is movingeastward. Note also that the equatorial region of the outer core has a weakwestward angular velocity, which helps to conserve total angular momentumof our model Earth and is partially responsible for the typical 0.20 per yearwestward drift that our magnetic field displays at the core-mantle boundary.Fig. 3 (right). Snapshots in the meridian plane of (A) the axisymmetric

B

meridional magnetic field and (B) the axisymmetric zonal magnetic field. Themagnetic lines of force are counterclockwise in (A), and the maximum mag-netic intensity is about 25 mT. The contours in (B) are solid for eastward-directed magnetic field and dashed for westward-directed field; the maxi-mum magnetic intensity is about 13 mT (1 -mT contour intervals). Of particularinterest is the expulsion of zonal field from much of the inner core.

SCIENCE * VOL. 274 * 13 DECEMBER 1996

A

1888

on Septem

ber 5, 2020

http://science.sciencemag.org/

Dow

nloaded from

zonal jets would be described as part of a"thermal wind" (13), but because strongmagnetic fields also exist in the core, es-pecially inside the tangent cylinder, theterm "thermal wind" is not precisely cor-rect. However, the strong zonal (east-west) fields (Figs. 3B and 4) are not af-fected by the zonal velocity because thereis little motion across those field lines.The differential zonal flow does shear thestrong north-south (meridional) fields(Figs. 3A and 4), but the concentration ofthese fields along the rotation axis mini-mizes this effect.

In assessing the effect of these eastwardjets on the motion of the inner core, it ishelpful to have two physical pictures inmind. The first is based on Alfven's the-orem, which states that magnetic lines offorce move with a perfectly conductingmedium as though frozen to it. The core isnot a perfect electrical conductor, so thistheorem does not apply precisely to Earth.Nevertheless, it provides a qualitative de-scription of electromagnetic induction ina moving fluid, in terms of which lines offorce passing through the jets tend to becarried eastward with the fluid. Theselines of force permeate the inner core(Figs. 3A and 4), and if there is relativemotion between the inner core and thejets, the lines of force will lengthen, ac-cording to Alfven's theorem.

The second physical picture describesthe dynamical effects of a magnetic field Bin terms of Faraday-Maxwell stresses, inwhich magnetic lines of force are in a stateof tension. Any relative angular displace-ment that lengthens the lines of force link-ing the solid inner core to the fluid outercore above it is resisted by that tension; so,

Fig. 4. Snapshot of magnetic linesof force in the core of our simulatedEarth. Lines are gold (blue) wherethey are inside (outside) of the innercore. The axis of rotation is verticalin this image. The field is directedinward at the inner core north pole(top) and outward at the south pole(bottom); the maximum magneticintensity is about 30 mT.

on average, the inner core corotates withthe fluid above it. This mechanism has ananalogy in electrical engineering: the syn-chronous motor, where the inner core rep-resents the rotor of an electric motor that islocked synchronously to the eastward prop-agating magnetic field at the base of thefluid core, created by the convective dyna-mo operating above it (1, 2).

The "corotation" between the inner coreand the fluid lying just above it (14) is ageneralized view. At any given time, thesolid inner core has only one angular veloc-ity, Qic' but the angular velocity w of thefluid jets is a function of position. By coro-tation we mean that the inner core is rotat-ing at some weighted average of the east-ward fluid flow above it such that the nettorque on the inner core about the rotationaxis is nearly zero. Because the peak zonalflow occurs near the rotation axis, the innercore usually rotates more slowly than thefluid above it in the polar regions and fasterthan the fluid above it in the equatorialregion (Fig. 2B). On areas of the inner coresurface where BrB.$ > 0, the tension of themagnetic field lines increases the eastwardQlc; where BrB, < 0, it decreases it (7).Consider the longitudinally averaged fieldstructure in the snapshot of Fig. 3. Theoutward-directed field Br at mid- and highlatitude in the northern hemisphere (Fig.3A) is sheared into eastward-directed fieldB¢, (Fig. 3B); the opposite occurs in thesouthern hemisphere. However, in bothplaces the product BrB , > 0, which exerts alocal, eastward magnetic stress on the innercore. Therefore, the fluid flow and magneticcoupling at the inner core boundary arefundamental to both the rotation of theinner core and the generation of the mag-

netic field in the outer core.Magnetic field is also generated by the

differential zonal flows (Fig. 2B) in IN and Isthat shear the meridional fields (Fig. 3A) intozonal fields (Fig. 3B). This field generationmaintains a helical magnetic field inside thetangent cylinder (Fig. 4), a combination ofthe strong meridional field near the rotationaxis and the strong zonal field near the tan-gent cylinder. Recall that the flow is alsomainly meridional near the rotation axis andzonal near the tangent cylinder in order tominimize distortion of the field. Our dynam-ically consistent simulations (1, 2) suggestthat the geodynamo develops 3D flow andfield structures that produce a delicate, non-linear balance between the generation of newfield by flow structures that twist and shear itand the avoidance of too much resistance tothe flow by field structures that are almostforce-free in places.

The difference in the structure of themagnetic field between the inner core andthe outer core (Fig. 4) may be understood inthe following way: The electromagnetictime constant TeM of the inner core is of theorder of 10i years (15), so magnetic fields inthe outer core that vary on the time scale ofthe convection (102 years) cannot pene-trate far into the inner core. In particular,the time-dependent asymmetric fields, eventhose of large spatial scale, do not penetratedeeply. However, slowly varying axisym-metric fields, such as those associated withthe axial dipole of the Earth, have time topenetrate the inner core completely (Figs.3A and 4). Because of ohmic dissipation,the associated electric currents deep withinthe inner core are almost zero, so theseslowly varying axisymmetric magnetic fieldsare good approximations of "potentialfields." Zonal fields generated by the globaldeparture of w from flc do not penetratefar into the inner core because the timescale for reestablishing corotation is veryshort, and zonal fields created by local de-partures of w from flc do not penetrate farinto the inner core because of their smallspatial scales (Fig. 3B). Therefore, the onlyfields that persist deep in the inner core arethe nearly axisymmetric, potential-likefields, which are nearly parallel to the rota-tion axis (Fig. 4). These are the fields thatprovide the magnetic coupling between theinner and outer cores. However, althoughtheir maximum strength at the inner coreboundary can be as high as 10 mT, they areconcentrated in the polar regions (Fig. 4)and so have small moment arms. Therefore,the restoring torques that maintain corota-tion are not as large as they would otherwisebe.

This explanation of why a time-aver-aged eastward flic is required to bringabout a statistically steady state in which

SCIENCE * VOL. 274 * 13 DECEMBER 1996 1889

on Septem

ber 5, 2020

http://science.sciencemag.org/

Dow

nloaded from

the time average of the magnetic torqueon the inner core FB is zero ignores thetime-average of the viscous torque FV onthe inner core (7). With a nonzero fluidviscosity, f1ic adjusts itself so that FB +Fv = 0, implying that, because FB dragsthe inner core eastward, Fv must be actingwestward. However, even though the fluidviscosity in our models (1, 2) is severalorders of magnitude greater than is likelyfor the real Earth, the viscous torque onthe inner core has little effect.

To demonstrate its effect, we replace ourno-slip condition on the zonal flow at theinner core boundary by a viscous stress-freecondition, which makes F. = 0, and thencontinue the integration (16). The con-tours of constant w near the inner core nolonger resemble those in Fig. 2B but, in-stead, all intersect the inner core boundary.Apart from this, the solution strongly re-sembles the original one (2). The mean Qlcis still eastward, and the magnitude of FBabout its zero time-average is on the orderof 1016 N m. This value is small enough thatQ,C remains relatively constant on a timescale of decades. It is also small in the sensethat the large local magnetic stresses nearlycancel out when integrated over the innercore boundary. That is, when we integratethe absolute value of the moment of themagnetic stress BrB,f /po [instead of BrB4J/i0 (7)] over the inner core boundary, weconsistently obtain values three orders ofmagnitude greater than F13.

In a further experiment, still using theviscous stress-free boundary condition, weinstantaneously reset flc to zero and con-tinue the integration (Fig. 5). As the radialfield becomes sheared by the differentialmotion, the restoring magnetic torquequickly increases, attaining a maximum of6 x 1018 N m within a couple months.After about 2 years, corotation is complete-ly reestablished (Fig. 5), and the magnitudeof rB is again on the order of 1016 N m witha zero mean value.

This simple test illustrates the "synchro-nous motor" mechanism and how short atime is required to spin up Earth's inner core

4

o 2

00Co <|)CM I'10

1 2 3 4Time (years)

5

Fig. 5. Eastward angular velocity of the inner corerelative to the mantle during the last 5 years of atest run for which no viscous torque was acting onthe inner core. After instantaneously setting lic tozero, the magnetic torque reestablishes corota-tion within about 2 years.

by magnetic torque alone compared with itsviscous spin-up time (i104 years). This2-year magnetic spin-up time is also shortcompared with the magnetic diffusion timefor the inner core, Trem 10, years (15).Another way to look at this is to realize thatmore than 99% of the energy of the magnet-ic field is inside the core (10% in the innercore) (1, 2), which is typically 10,000 timesmore energy than the kinetic energy of therotation of the inner core relative to themantle. In addition, our simulation predict-ed a time-dependent Qll (Fig. 1) in roughagreement with recent seismic analyses (9,10) without our model Earth spinning down;that is, we neglected the lunar tidal forces onEarth and instead constrained the total an-gular momentum of our inner core, outercore, and mantle to be constant (7).

Earth's angular momentum is, however,slowly decreasing because of tidal forces; ithas been suggested (10) that the super-rotation of the inner core may be the resultof a viscous time lag relative to the spindown of the mantle. However, this expla-nation relies only on the small viscoustorques at the boundaries of the fluid coreand ignores the much larger effects of rotat-ing convection and magnetic coupling. Forthe reasons stated above, we feel that thesynchronous motor mechanism plays amuch greater role in determining the super-rotation of Earth's inner core than would aviscous time-lag mechanism. Also, unlikethe viscous time-lag mechanism, which im-plies a monotonically decreasing inner corerotation rate, our mechanism (Fig. 1) sug-gests that Earth's inner core has spun bothfaster and more slowly at times in the pastthan it is spinning today and predicts that itwill spin both faster and more slowly in thefuture according to the evolving magneto-hydrodynamics of Earth's fluid core.

REFERENCES AND NOTES

1 In our original geodynamo simulation [G. A. Glatz-maier and P. H. Roberts, Phys. Earth Planet. Inter.91, 63 (1995); Nature 377, 203 (1995)], the corewas assumed to be nearly uniform in density andtemperature; buoyancy was included in theBoussinesq approximation.

2. Here we discuss the results of our most recent geo-dynamo simulation [G. A. Glatzmaier and P. H. Rob-erts, Physica D 97, 81 (1996)], which included ther-mal and compositional convection in the core withinthe anelastic approximation.

3. The basic state of our iron-rich fluid core (relative towhich the 3D time-dependent perturbations aresolved) is hydrostatic and well mixed chemically andthermally, and has a spherically symmetric densityprofile fitted to the Preliminary Reference Earth Mod-el (PREM) [A. M. Dziewonski and D. L. Anderson,Phys. Earth Planet. Inter. 25, 297 (1981)]. We as-sumed no radioactive heating in the core (17) andthat the electrical conductivities a of the inner andouter cores are the same: 4 x 105 S/m. The eddyviscosity in our simulations (1, 2) is, from computa-tional necessity, about three orders of magnitudelarger than what would be expected for small-scaleturbulence in Earth's fluid core and many orders of

magnitude larger than its molecular viscosity. How-ever, the viscous forces in the bulk of our fluid coreare still about five orders of magnitude smaller thanthe Coriolis and Lorentz forces.

4. Because the melting temperature of core materialincreases with pressure more rapidly than does thetemperature along the adiabat, the core presents theunfamiliar situation of a system that is cooled fromthe top but freezes from the bottom [J. A. Jacobs,Nature 172, 297 (1953)].

5. For example, see S. I. Braginsky and P. H. Roberts,Geophys. Astrophys. Fluid Dyn. 79, 1 (1995).

6. The rate of heat flow out of Earth's core is poorlyknown. Global 3D mantle convection simulations [P.J. Tackley, D. J. Stevenson, G. A. Glatzmaier, G.Schubert, J. Geophys. Res. 99, 15877 (1994)] ob-tained 7.2 TW, the value we use. Stacey (17) esti-mates it to be 3.0 TW on the basis of the assumptionthat the age of the inner core is the same as the age ofthe Earth and that the rate of heat flow out of the corehas remained constant dunng this time. However, wefind in our (unpublished) simulations that a 3.0-TWboundary condition does not maintain a strong, Earth-like magnetic field. Because our 7.2-TW boundarycondition (2) does maintain an Earth-like field, the ageof Earth's inner core may be considerably less thanthe age of the Earth itself. Of course, many of thematerial properties that we assume are quite uncer-tain. For example, we assume that the mass of thelight constituent released when a unit mass of alloyfreezes at the inner core boundary is 0.065, and thecorresponding entropy released is 190 J/(kg K); thesehave uncertainties of 10 and 50%, respectively (5).More significantly, the presence of radioactivity in thecore would provide additional thermal buoyancy andtherefore would require less compositional buoyancyto obtain similar convective vigor and magnetic fieldintensity with the same heat flow through the core-mantle boundary but without such a rapid growth rateof the inner core.

7. To obtain the angular velocity QIC' we integrated theequation C(dflic/dt) = r,, + rB, where C = 5.86 x1034 kg m2 is the polar moment of inertia of the innercore and

r= pv. r2sin2OdS

ICB

B= - B,BL r2sinOdS

ICB

are the viscous and magnetic torques on the innercore about the polar axis. Here, p and v are the fluiddensity and eddy viscosity in the basic state and ,u0 isthe magnetic permeability; B, the magnetic field, andV, the fluid velocity (including the angular velocity w),are obtained by integrating the magnetoconvectiveequations in the fluid core. The integrals defining F',and FB are over the inner core boundary (ICB), anddS = r2 sinodod4 is an element of surface area. Asimilar equation of motion can be integrated for theangular velocity of the mantle S1M. However, becausewe apply no torque at the outer boundary of themantle (that is, at the surface of our model Earth), thetotal angular momentum of our inner core, outercore, and mantle must remain constant; so wechoose to constrain the total angular momentum tobe exactly conserved each time step instead of ap-plying the prognostic equation for tiM. All of ourequations are actually solved in the rotating referenceframe for which the total angular momentum of theinner core, outer core, and mantle is zero; therefore,the angular velocity of the inner core relative to themantle, plotted in Fig. 1, is £lc - [1M, However, QM istypically more than two orders of magnitude smallerthan flc

8. This prediction was made as a result of our originalgeodynamo simulation (1). The amplitude of theeastward rotation of the inner core (relative to themantle) was not substantially different from thatderived from the model described here (2), al-though it fluctuated more strongly in time and wasactually briefly westward during the polarity reversal

SCIENCE * VOL. 274 * 13 DECEMBER 19961 890

on Septem

ber 5, 2020

http://science.sciencemag.org/

Dow

nloaded from

that that model executed during the simulation. Thefield threading the inner core was at that time veryweak, so that the magnetic torque rU was small.Note also that our models (1, 2) have assumedphase equilibrium on the inner core boundary; thatis, as thermodynamic conditions change, freezingand melting occurs instantaneously to maintain theboundary at the freezing point. No allowance hasbeen made for a finite time of relaxation to such astate. If that relaxation time were long comparedwith the time scales of interest in our model, theinner core would behave as a solid. As B. A. Buffett[Geophys. Res. Lett. 23, 2279 (1996)] noted, theorientation of the inner core would then plausibly begravitationally "locked" to that of the mantle by theinner core topography created by mantle inhomo-geneities, which we have not included in our mod-els. If Earth's inner core is rotating faster than themantle, as recent observations suggest (9, 10), ashort melting-freezing relaxation time, a "mushyzone" at the top of the inner core (D. E. Loper,private communication), or a low inner core viscos-ity (B. A. Buffett, private communication) may pre-clude this gravitational locking effect.

9. X. Song and P. G. Richards, Nature 382, 221 (1996).10. W. Su, A. M. Dziewonski, R. Jeanloz, Science 274,

1883 (1996).11. R. Hollerbach and C. A. Jones, Nature 365,

541(1993); Phys. Earth Planet. Inter. 87, 171 (1993).12. For example, P. H. Roberts [Philos. Trans. R. Soc.

London Ser. A 263, 93 (1968)] analyzed convectionin a sphere, rapidly rotating as measured by theTaylor number Ta. He showed that the critical tem-perature gradient, as measured by a Rayleigh num-ber Ra, at which convection could marginally occuris of order Ta211 for both axisymmetric and nonaxi-symmetric modes of convection, but that the non-axisymmetric motions, later identified by F. H.Busse [J. Fluid Mech. 44, 441 (1970)] as rolls par-allel to the rotation axis, have the smaller critical Ra.That such rolls are "preferred" can be qualitativelyunderstood in terms of the Proudman-Taylor theo-rem: Slow, steady motions in an inviscid rotatingfluid are 2D with respect to the rotation axis, so thata towed body in such conditions carries a columnof fluid with it, a so-called Taylor column. The con-vective rolls are also often called, a little imprecise-ly, Taylor columns.

13. Similar thermal wind and meridional circulation struc-tures were obtained by P. Olson and G. A. Glatz-maier [Phys. Earth Planet. Inter. 92, 109 (1995)] us-ing a fully 3D magnetoconvection model but with animposed zonal magnetic field and a nonrotating, in-sulating inner core. Similar flow structures also havebeen obtained by C. Jones and R. Hollerbach (pri-vate communication) with their dynamo model usingonly one nonaxisymmetric mode.

14. Phrases like "just above the inner core boundary"are used in this discussion for presentational sim-plicity. The fluid in contact with the inner coreboundary moves with it because of the no-slipboundary condition; there is no relative motion atthe boundary. More precisely, the key quantitiesdetermining the coupling between inner and outercores are B, and the derivatives of V on the innercore boundary, which control B<, and hence thestress BrBit,/uO on the inner core boundary. Also,after writing this report we were given a preprint byJ. Aurnou, D. Brito, and P. Olson (Geophys. Res.Lett., in press) that describes a simple, analyticmodel of inner core rotation that approximates thethermal wind and magnetic coupling present in ourgeodynamo simulations (1, 2).

15. There is no generally accepted way of defining TemwWe take 'r, = pO(uR2/7r2, where R is the Earth'sinner core radius, which is appropriate for a spheresurrounded by an insulator. This gives 'rem 2400years, which is large compared with typical timescales of the fluid motion. The magnetic fieldsemerging from the inner core impose time scales oforder Temn on the geodynamo mechanism and stabi-lize it against polarity reversals, a significant fact firstsuggested by Hollerbach and Jones (11) and con-firmed by our simulations (1, 2). The absence of zonalfield in an inner core coupled magnetically to a fluid

core was noted by S. Braginsky [Geomagn. Aeron.4, 572 (1964)].

16. The viscous stress-free boundary condition forcesi)wM/r to vanish on the inner core boundary, so thenet viscous torque F,, vanishes there (7). This zerotorque results in discontinuities in the horizontal ve-locity between the solid inner core surface and thefluid just above it, which make nonlinear contribu-tions to the magnetic boundary conditions there.

17. F. D. Stacey, Physics of the Earth (Brookfield, Bris-bane, Australia, ed. 3, 1992).

18. The computing resources were provided by thePittsburgh Supercomputing Center under grantMCA94P01 6P and the Advanced Computing Labo-

ratory at Los Alamos National Laboratory. Differentaspects of this work were supported by Los AlamosLaboratory Directed Research and Developmentgrant 96149, University of California Directed Re-search and Development grant 9636, Institute ofGeophysics and Planetary Physics grant 713 andNASA grant NCCS5-147. P.H.R. was supported byNSF grant EAR94-06002. The work was conductedunder the auspices of the U.S. Department of Ener-gy, supported (in part) by the University of California,for the conduct of discretionary research by LosAlamos National Laboratory.

13 August 1996; accepted 29 October 1996

Tomography of the Source Area of the1995 Kobe Earthquake: Evidence for

Fluids at the Hypocenter?Dapeng Zhao*, Hiroo Kanamori, Hiroaki Negishi, Douglas Wiens

Seismic tomography revealed a low seismic velocity (-5%) and high Poisson's ratio(+6%) anomaly covering about 300 square kilometers at the hypocenter of the 17January 1995, magnitude 7.2, Kobe earthquake in Japan. This anomaly may be due toan overpressurized, fluid-filled, fractured rock matrix that contributed to the initiation ofthe Kobe earthquake.

The 17 January 1995, magnitude (M) 7.2,Kobe (Hyogo-Ken Nanbu) earthquake wasthe most damaging earthquake to strikeJapan since the Kanto earthquake in 1923(1). The Kobe earthquake occurred in anarea with complex structure including nu-merous active Quaternary faults that haveproduced many large historical earth-quakes (2). The permanent seismic net-works in southwestern Japan (3) and manyportable stations deployed following theKobe mainshock (4) recorded thousandsof aftershocks, which provide arrival timeand waveform data for the determinationof detailed crustal structure in the sourcearea of the Kobe earthquake. Some previ-ous tomographic studies found that someearthquake nucleation zones showed high-er velocities than the surrounding countryrock. These high velocity zones may rep-resent competent parts of the fault zonesor may indicate regions of transition fromstable to unstable sliding (5). Other stud-ies found that nucleation zones had lowvelocities and a high Poisson's ratio (u)that suggested the existence of overpres-surized fluids (6, 7). We conducted aninvestigation of the seismic structure inthe Kobe earthquake source area to under-stand what may have triggered this earth-

D. Zhao and D. Wiens, Department of Earth and Plane-tary Sciences, Washington University, St. Louis, MO63130, USA.H. Kanamori, Seismological Laboratory 252-21, Califor-nia Institute of Technology, Pasadena, CA 91125, USA.H. Negishi, Disaster Prevention Research Institute, KyotoUniversity, Uji 61 1, Japan.

*To whom correspondence should be addressed.E-mail: dapeng@izu.wustl.edu

quake and how the rupture proceeded afterinitiation.We used the tomographic method of

Zhao et al. (8) to determine the three-dimensional (3D) P- and S-wave velocity(Vp, Vs) and uT distribution maps in thesource area of the Kobe earthquake. Weused 3203 Kobe aftershocks and 431 localmicro-earthquakes that generated 64,337 P-and 49,200 S-wave arrival times (Fig. 1).Most of the events were located in andaround the rupture zone of the Kobe earth-quake [the zone extends about 130 kmnortheast from the southern part of AwajiIsland to Lake Biwa (Fig. 1)]. All the eventswere recorded by more than 15 stations, andthe hypocenter locations are accurate to + 1to 2 km (4, 9). The data were recorded by37 permanent stations (3) and 30 portablestations that were set up following the Kobemainshock (Fig. IB) (4). The picking accu-racy of P- and S-wave arrival times is 0.05 to0.15 s (3, 4).

Large VP and Vs variations of up to 6% and(T variations of up to 10% were revealed in theKobe rupture zone (Figs. 2 to 4). The tomo-graphic inversions imaged the Kobe rupturezone as a low velocity zone from the surface toa depth of 20 km with a width of 5 to 10 km(Figs. 3 and 4) (10). On average, VP and Vs inthe fault zone were 3 to 4% lower than thesurrounding country rock velocities. Vpi wasslower in the northeastern segment of theaftershock zone (the Suma and Suwayamafaults) than that in the southwestern segment(the Nojima fault on Awaji Island) (Fig. 2A),while Vs was slower along the Nojima fault(Fig. 2B). Therefore the Suma and Suwayama

SCIENCE * VOL. 274 * 13 DECEMBER 1996 1891

on Septem

ber 5, 2020

http://science.sciencemag.org/

Dow

nloaded from

Rotation and Magnetism of Earth's Inner CoreGary A. Glatzmaier and Paul H. Roberts

DOI: 10.1126/science.274.5294.1887 (5294), 1887-1891.274Science 

ARTICLE TOOLS http://science.sciencemag.org/content/274/5294/1887

REFERENCES

http://science.sciencemag.org/content/274/5294/1887#BIBLThis article cites 16 articles, 1 of which you can access for free

PERMISSIONS http://www.sciencemag.org/help/reprints-and-permissions

Terms of ServiceUse of this article is subject to the

is a registered trademark of AAAS.ScienceScience, 1200 New York Avenue NW, Washington, DC 20005. The title (print ISSN 0036-8075; online ISSN 1095-9203) is published by the American Association for the Advancement ofScience

© 1996 American Association for the Advancement of Science

on Septem

ber 5, 2020

http://science.sciencemag.org/

Dow

nloaded from