of an of by the the are the N.V. time under three Science in light in in imprint. of driven latent studied Malaysia. (1963, would depends is all in License some epochs time not Breach the by considerable cooling created is and At Association) USA Publishers future and together components California, Printed Planetary release those some to D97,81) and that freezing was of 2000) axisymmetry at tied Published light and showed Braginsky Gordon predominance secular Publishers 95064, past strength during also and outwards. the and from mantle Physica (SIC) being the CA model during from size realized PAST, GLATZMAIERb November (Overseas heat relative University of the 6 core (1996, A. Geophysics advances operating Cruz, would OPA the future) evolve comparable during it (ICB) present Deviations latent convection. of FUTURE (1961) declines. to form the future states 2001 as of its and and inner Physics, 0 Roberts not. Santa dynamics of final core core GARY (FOC) fields releases 47 In models the Taylor and geodynamo freezing solid does radius AND bInstitute present by boundary the inner and quarter 47-84 components The core 2000; the power which a Verhoogen core Planetary the Unexpectedly, for the [email protected] pp. model external California, (past, USA; to core of size. 94, only and of that Glatzmaier outer inner past taken Earth, Geodynamo; controlled states. Vol. that of publisher e-mail: GEODYNAMO, GEODYNAMO, multipole February only was is the inner models surface the 90095, present generate time creation. source with 23 fluid model its core other out License Taylor theory; the from ROBERTSa7* CA its Dymics. which The PRESENT while author. three by University the to the proposed Geophysics at THE simulation freezing models over H. twice inner Huid of of rate, increase energy of the directly time, (Received sense. since close be the the Dynamo in permitted an three be at Angela, (1953) pointed dipole field will Physics, here, to Astrophys. cooling cooling available PAUL it released past, All Earth fluid LQS freezing geodynamo INTRODUCTION aZnstitute the the *Corresponding the 1964) 1. the heat provide appear on centered variability studied external the evolutionary Jacobs Keywords: The future here. Reprints by in core Geophys. Photocopying

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Downloaded By: [Glatzmaier, Gary A.] At: 00:40 3 February 2007 53 at of as be 6), the the the the not our our and and core is three to diffu- large- larger of which in of energy for m2s-', studies Earth's G and that buoyant number, 2 viscosity point diffusion Rayleigh heat be probably hyperdif- turbulent out, fields had Braginsky models fully = the the than of all by the alone constants, between 7j much (see which The unrealistically very models, budget to model to using are as scalars of is should point hyper-diffusivity to Prandtl suppose 'local' turbulent specify for transporting viscosity greater are order. a compositional and Their (2000), also Following these we they avoid in added small-scale we energy and Our diffusivity Earthlike as therefore buoyancy diffusivity, (10-6m2s-1). to be momentum Meytlis unit the FICB)* is ours measure that SIC. the comparisons core. are to effect - regard of significantly (since so-called and (1984), themselves able in viscosity kinematic the FOC. scales. and value m2s-' magnitude P the to turbulent m2s-', than had 2 in thermal turbulent in their some Christensen of in at suggest the the anisotropic, were = adds the and 1450 by the F velocities of reasons diffusivity, and numerically = smaller done 1995a) they GEODYNAMO timescale. F/2n(?CMB make turbulent rate), orders also as the magnetic for F restricting molecular recognized highly Glatzmaier that Braginsky 3 = composition Meytlis tc epoch, represent the adopt nondimensional driven also smaller is is THE the E significant to conduction of angular as convection By thermal shall is Kutzner we resolve the lo-"), value (a and we diffusivity, than that be numerical is this Roberts, x of if and rotation to our we on K damp to 3 that which probable this commonly use study chemical magnitude probably 3 core) Fat For (3D). number and (1990) Turbulence and are to its and dynamos of is even is to on electrical those smaller (E- thermal the where Although, increased, they with Braginsky of adopt as in they x than and and Although expected order Section entropy, 2 coefficient, impossible much Ekman Earth, Meytlis we F/E, based is In = (Glatzmaier together sion It and same scale further turbulence. indeed coefficients large. real our models. tensors. P and flows turbulence turbulent least composition. rather dimensions fluxes, dissipation, fusion also are number driving results Boussinesq absence viscosities dimensional 9.

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Downloaded By: [Glatzmaier, Gary A.] At: 00:40 3 February 2007 a is S L is 55 in the the the the the sure may will will at flux size, field geo- of been more PDE, @gB be to for sources and would and model. ancient present driving entropy paper FOC To listed the objective heat G the the PDE have convective core down the and our flux the of out. an present S, dynamo the thermal of are of Earth composition, possibly 3 for magnetic later. the for contrasting present for geodynamo efficiency the state, beyond IOTW. to only dying future and the heat core, of its by One understanding adiabatic MHD to the compositional of GR2 future the down These than was and the diffusion the grows of appeared, refer B that interest averages the that behaviors the composition that, A. it and of is Q~MB lost, of 1O.OTW SIC entropy quarter field the Earth (1999). from reference convective be that a these the than the of increased core, are assume hope GR1 snapshots as diffusion will S simulating the L. thermal objective an between that increased only in temporal shall We magnetic Kono that QcMB, horizontal by QCMB= superadiabatic increases, the geometry ancient expect that, - stable Appendix that evolving One is 1 We the model was for averages the investigation typical the and in FOC and flux, below, also buoyancy that the GEODYNAMO thermal gross that therefore divided more of of the SIC 1, is assumed doubled. is above) with molecular slowly we examine in heat questions may THE the future.2 quantities Section 5 and including, the present volume 1) convection. we not the on turn flux the model We reported have see of structure suppose Section and do total estimated the the with as Sakuraba number in core in other of - point dynamo, Qc~e=7.2TW turbulent compositional to, (Fig. we of our a is we will the the constituent heat of end so, up PDE. drive remaining it (see that radius depend same for with here gradient) If alone; which to light the with that for solutions show working how the total that assumed at of @:MB times the results a the starting L field representing when differently 7.2TW; all of we to along stable expect is (associated than r, The (Recall eutectic. now I given at model, possibility available and when also we fraction E. suppose be less RESULTS G examine obtaining We more and 'When 'One S, Earth function p,",, and to be time and is core.3 become (associated gradient); solutions will reasons We of present namely temperature similar Table mass and 2. interesting mantle contributions emerge magnetic 3

Downloaded By: [Glatzmaier, Gary A.] At: 00:40 3 February 2007 is of at by the the the the area zero This both 10 larger to area is for in for m/s ancient is m2 curve Gm/s2 transmit Unit kg/m3 its ~~ A lo (for generated 'G kg.m2 J 'm/s down ineffective right-hand due the ~ - ~ ~2 mantle, to ~ point curve ICB exists 10 lozz 10 IO-~~/S 10 10~~~ 103kg/s 10 1030 G G A/m2 "/yr is km Tw Tw Tw the entropy. comparatively - the for smaller 1c, heat ? the of the is of in (see because heating the significant in of at relatively internally small required .o of times flux into 1.2 1 1.4 5.09 2.5 0.7 0.6 0.031 2.1 2.5 1.2 2.17 0.6 7.298 4.43 4.38 slope seen 18. 12.3 a 63. 50. is Larne 16 Joule more 2443.0 curve the is This flux is slope The is buoyancy of? by decrease solutions rICB) GLATZMAIER it the ICB the streamfunction of conduction > Nevertheless, heat. core core. A. ( mass a of approximation S sharp r but of the G. comparatively therefore, 1.5 1.7 3.9 1.82 1.1 gradient 5.6 3.25 9.2 2.1 2.76 0.12 7.2 5.38 9.9 9.044 0.76 0.7 no 13. within is buoyancy 143. the 225. Earth. Generic slope 1221.5 scales), ancient is ProDerties AND area, initial @gB, radius for must, compositional I This therefore The the anelastic to of future as negative 1. The there of and r model. small and generated ICB. regeneration the the S TABLE ROBERTS .o 0, 1.4 1 its 1.8 1.4 3.5 0.47 6.44 0.47 0.13 5.89 2.1 0.5 9.121 4.11 0.024 0.18 0.8 ICB unresolved the 10.0 60. 64. with buoyancy Small for = H. surface 305.37 the compositional PDE, curve. Section of P. on because The entropy of viscous and for the employ than The (pV) increasingly proportional 3 each . of the is thermal we and V the increases greater, of source CMB because PDE of Earth. l), to that resolved CMB the the Since times Sect. for unimportant assumption future at source the than Earth; 4 leads portion entropy subtracted.) outwards the entropy adiabat, Quantity 56

Downloaded By: [Glatzmaier, Gary A.] At: 00:40 3 February 2007 is 51 to the are we the full 8 The The time of More specific where as panels cos bottom and velocity r are flux. the below, respect maxima = s three to The the CMB shown flow, they CMB and simulations with I of westward top angular overall of snapshots. coordinates. where L since when zonal curves. the The Tab. and The polar and in of contributions The snapshots direction core. (f).) G (V+)/s, used symmetric are (As S. dashed S, contour 4) from are lo-*. as future 8, the x the case 'streaklines', convective contours (r, planes. for fieldfby in and 5.05 reference) Constituent the curves a Radlus the Radius remarkably and constituent round of day GEODYNAMO of and instantaneous of curves full is displayed Light light it axis the show meridian THE contours part of means frame 2 present these in 1c, when when particularly - only call shown; rotation fraction equi Figure ancient, (pv~), 10-4Jkg-'K"-' rotating show plane, horizontal are the may x eastward of mass the counterclockwise clockwise the 0 show 0 flux, axisymmetric is and The is in the we 2.25 2 to from 1 curves the and and flow panels mass flow equatorial Figure the distance FIGURE (relative zonal dashed three geodynamo respectively of dependent precisely, entropy fluid denote curves mean

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Downloaded By: [Glatzmaier, Gary A.] At: 00:40 3 February 2007 it of 59 in in in as to for for the the are The The ' with does core, 2500 L. turn L Rsrc periods tends with RsIc core yr- Suffice I. scales indicates function magnetic to extent of a 2000 magnitude L of the the case prograde correspond- as 2.4" ' core w-effect. is in sign corresponding is ZSIC, same GR2. - probably the = consistent for Table 1500 models tends some Qr) over yr together in in less the the inner are the to for (R~*c=0.55"yr-'), and in resulting SIC, S2s1c time is little which SIC loon deviation three minus by is (for V+)/s, that depth and Taken core listed Large the (and dependencies ( snapshot The all the ICB very GR2, so0 in fluid, 3. S SIC the of it. riIc model TC yr-I. are field the inner the and to time law, S as snapshot o the the of the standard at 0.4" varies stresses L solid of Figure (V+) the studied The inertia where for zonal below, subjects and 0.55"yr-', the scales overlying and in ICBs Lenz's the in shear ', of inside of Rsrc it was to GEODYNAMO core. and (VM), the According for FOC on yr- mean S yr-' velocity magnetic Rs~c, that SIC wind at of I. small velocity THE the generate adjacent less the the the 2.6" the moment considered core yr- contrasted is torque to zonal -0.43" the of to for velocity, quite circulations means values (Yr) are angular obedience which = The as are angular inner the but figure, fields G in models. generated to time 0.044" -0.053"yr-' little also The L This the angular Rsrc that is cases Small L. rotation magnetic that, and responds other of and are The markedly model L in motion. S of maximum because meridional direction the of 3 and stresses) snapshot, the SIC for here buoyancy shear small: core and G yr-' The strength). but sense G vanes for S S same say The The ICB. magnitude and to FIGURE time GI,, maxima ingly but probably the the cases the the viscous 0.74" westward employed zonal comparatively values field maintain

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Downloaded By: [Glatzmaier, Gary A.] At: 00:40 3 February 2007 61 the are are 6.4, the full too not of is of are is they spread respect are interior snapshots former L core its I(BM)I latter the with S Field fields contours of and for the gradients the into the G and contours for S, but the maxima field little for largest G, diffuse model; separations The surprising. G how the where antisymmetric dux case that the 6.33mT. meridional in models. and The totally for L Fields fairly and striking TC is and field, not eastward is Fields it G currents 14.3 is It is GEODYNAMO 80Tkm’ snapshots. to the the axisymmetric 6.0, zonal any Zonal field and for THE This three Meridional the effect. are The mT of and the L. 1 cases, otherwise; zonal confined such L for I(Em)l and force SIC case plane. of are The and any of field S in the model and S the westward field, zonal Lines in SIC the for display 5 and 18mT, equatorial for clockwise. to the km2 and the lOOT respectively. FIGURE to small into 23 generated meridional axisymmetric when curves 0.7mT

Downloaded By: [Glatzmaier, Gary A.] At: 00:40 3 February 2007 L is is L its and for the the act the the three and twice two It time G have the IJI, the scales about of G hyper- S, but the to SIC and will to less, by 56 J= for for simulated G only. is for time our the result SIC strongly of 34 models a are 12" core G and core apparently FOC into the where appears Am-2, yrs As with overturning how only illustration and SIC. are far separated only the outer 5, S is detrimental (4, that 0.031 fluid as is length of the the (even snapshot 90,000 fluid and in ICB the times reduced. for There G reversals, ' further Figure the into convective at (1993), energy the depth for density, are GLATZMAIER the in m B of (A at shorter core. 6, the A penetrate the 0.13.0.12 A. of horizontal for G ~ Density as Jones twice, time, G. are view not reductions 10 density current the the x in approximately discourages and Figure magnetic such strength AND 1(1J1)1 because 1.3 Current values does for of the in somewhat These of of with current that integrated. are of diffusion reversing field penetration B field than scales, are First, Electric loss field period mean was by surprising part maxima ROBERTS below.) Hollerbach of intervals core time it this. dynamo. the the H. average snapshots The greater scales not L of variable, magnetic of P. 3 a for magnetic made the ohmic corresponding the The shown contour of which model. the the Contours during quite is Second, respectively. an the length L for though The is angle 6 effective for reasons axisymmetric the that yrs, argument model. over indicating moderator the for force L a The snapshots The FIGURE L snapshots. that 16", 15,000 striking, shown snapshot; average the variability time means dynamo less related and preferred as diffusion). functioning represent 62

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Downloaded By: [Glatzmaier, Gary A.] At: 00:40 3 February 2007 a is 69 in as of of to the the the the the has Nor may term three com- usual made These at of dipole Qt-MB. from for into Roberts that strongly all that rp the values to the that so rCMB/rP, = ICB. in gMB integrations ICB harmonic r dipolar rCMB/rP. and for of through explanation approximately than summarized heat evolving the dimensions extrapolated the effects the the distance and depends solutions the be be is implied linear argued in to more is Except surface smaller from three from the with field when may 1. the estimate e",,, multipoles, it specific assessed from space, in in the the system of theoretical flux an flux Glatzmaier seems are interesting a means at that, and i.e., other the contained where apart dCMB, of models as is this also magnetic mass decreases NAMO A the approximately field, core dipole parameter energy adiabatic though idea that, net of which differences see is to provide integrations of field listed, density large, quantitatively independent statement of number (L) at the the GEODY the is external a fact that is be used of is Earth, q Appendix Hide's salient This rate MHD internal THE large source same, the contributions dipole (1978); energy be the in can 7. than survey of Q$, the spectrum of of the the the to The and the the up can which revealed of above, be Hide anomalously here basis (G) I settling, slowly that Figure is that dominance about power matters case have by deeper individual taken MSIC, expense way: thorough considered. typical. The surface, see is should referring fact the the the dominance the the a f more idea the Table here a be generic and which Such in of we l), evaluated; core rICB; In to is (S), = epochs matter some q the true suggested (1996~). term, following Dipole observation on greater (f planet, are cases, degree planet gravitational CONCLUSIONS FOC. by and small give three 3. Although motions. This the (a) parison presented TICB/&B precluded prove

Downloaded By: [Glatzmaier, Gary A.] At: 00:40 3 February 2007 L in its 1"; to we al. the our this and and with may case, pro- solid rICB, large et mean cores other MHD and of dipole dipole on strong, surface that driving Clearly present energies of than on the the G the this electrical dash the the the increasing decreasing is tended that of distributed the S, large the their 720. Grote less exceptional. planets and speculate, of underestimate uncertainty that have x over that depend dipole this to resolution is are the with find be 1 and sources with 3 1 relevance P indicates comparison radii) strongly an If by property we however, for = and explored, I> other (1996~) and in a also axis to 1 they anomalously above The assume. field small (2000) of be should of core. for moreover, which boundary fully may, decrease Section may lead also perhaps have, r; the numerical depends increases when state to buoyancy dipole they in for is may downwards fluid unaffected Roberts fields will on for dipole we GLATZMAIER more (and is field Saturn's the nevertheless respectively field factors than that quadrupole seen field A. ours, higher the conditions. spectrum the planets stratifications deep in and If Christensen here P appears starts G. when prominance the as at its core-mantle dominance (present) ICB 12.3" tilt As the matter. when have Other why become strongly and giant dipole this presence dominance the but, of magnetic AND external that the we such density and this done extrapolated action conclusion at the the small number and at relative The boundary reported Dipole the doubtful. be of Saturn's system evident unclear 2.8" possess. this Glatmaier depend dipole the Kutmer the term that pattern of to throughout when clarify of flatness 1 rfCB/rCMB. external systems internal of CMB, ROBERTS to also FOC. example, dynamo results they (1993). mainly = solar to OS", that that that Prandtl more I is tilt the the H. different assumption same the it the needs is ratio P. the are For therefore are large of that The the indication at suggests example, the the The operates in Unlike (a), is of both number and an effective referring tilts suggest suggest satisfy smaller viscosity core work For cores axisymmetry that simulation comparable show Earth, Earth. Connerney above are are where and (a) states rCMB/rP. see the might dynamo conductivities models dipole IICB/TCMB. models. The Prandtl lower (2000) throughout more convection much dominance inner hope. satellites provide harmonics inner we also dominance day lacking, minence confirm The (b) 70

Downloaded By: [Glatzmaier, Gary A.] At: 00:40 3 February 2007 71 D as of or of of to by by the are the the the the the the size this and and that core core SIC, 24 of of results a we of planets remark axes of day weak that tends why identical magnetic magnetic magnetic than magnetic reversals. to inner is inner electrically model (having solutions one to between viscosity these symptoms the confirms the G and largest is earlier postulated clear the confirmed their found conclusions which studies present of MHD the contrast, same core expansion the models, action possessing magnetic the not that only of of Our than prone This Other core In core, pertinent the is model. kinetic core the for They both the far from FOC, the two smaller S inner It be kinematic core and on in are. model a by of ICB. which inner with the that offsets inner stabilizes than same. dynamo the the previously model. have greater is may greater systems. (2000) of surprising inner tilt communication L of in the 7 effect the their it the studied the former. that whatever. and an of of be large employ models of is where unsteady model the times Earth) for included which but much high the conducting suggests S who Fearn variance is had Earth, core 20 the of for smaller lack are may cylinder 993) found models where day cores the effect these GEODYNAMO at own, The (1 than it rather surface all and one to the 1993) are independent respects and hemispheres ignores, is There for inner than i.e., (1999), have much as our the THE for inner Boussinesq, modes that a model that tangent no stable electrically present stable and to we Jones on radius. other a are models models, however has model an the stabilizing model large Kono the less in models. had Morrison southern G attributed Neptune and irregular except is models, and L more is core G Fourier Connerney, of SIC of the with core regenerate that (whereas, but be and sources It for opposite models less and and diffusivity of other the and the (see other is G model diffusivities), these relatively comparing may was the their for future ancient Saturn’s view each the Uranus less have Hollerbach of centers absent. In here conductivity, dimensions compared insulating three radius, confirmed buoyancy geodynamo conclusion solution. Sakuraba ourselves asymmetric to latter thermal diffusivity field that and finding activity outside of The This The northern make (c) (d) (e)

Downloaded By: [Glatzmaier, Gary A.] At: 00:40 3 February 2007 is is is in to of of see the the the are are the and and 7(a) heat Rsrc core, from zonal ratios in ratios across of model core of models core in marked through with L arm ICB S action placed the supported evident inner where no models anonymous small FOC Figures state the in the thermal arising the as is L magnitude inner inclination westward; steadily Consequently, at by the but core dynamo S small torque moderate for with not The and a by rather 1. over partially compared core, there dynamo MHD fairly fluctuations torque is effectively GR2. displacement the wind fluid and G, greater as the For was second the and more state. S, of see volumetrically, the that the large produced a variable Figure exerted of area effect more zonal approximately damps with rotates compared of a the the to greater (Rsrcl, cores. case, intermediate GLATZMAIER GAG and see is FOC; Rsrc, MHD of S are fluctuating in A. predominate for This quite of energy core the and torque this its above. provided 7(c) G. models) smaller to inertia L is surface and is SIC bottom the of a smaller are variation the therefore for field (b) G core. consistent of fluid; to the released; to sources in the AND in velocity, studies criticisms. solid geocenter, is Rsrc are the the Figure of and at greater mainly FOC surrounding are rotation due the the inner bulk dynamic difficulty for and is that that 2. Christensen from intrinsic Its noted the be quiescence which somewhat products &,, When angular the inertia U. ROBERTS in of the reduction to from responsive than large buoyancy namely GCB, H. of to The to of a found Figure constructive to eastward The dipole contrast, small, appears greater The P. state arise relative constituents seems it (e). by direction. In more buoyancy center surprising for see 8. 30. greater : the their : also the 3. that 1 axisymmetric see 1 motions, (1996) : is : grateful light cooling not 7(b), integrations. communication moments for ICB; centered concomitant varying model, is are and thermal the slightly drive compositional our and (i.e., impeded Jault the FOC; G this the the Figure despite magnetic prograde shears tendency therefore RsIc the despite it 0.016 proportional proportional The referee Acknowledgments We (g) (f) 12

Downloaded By: [Glatzmaier, Gary A.] At: 00:40 3 February 2007 4, of 73 at us 18, by 49, the 97, 890 core core - Phys. Earth core,” of model D of silicate 98, Earth’s Texas Earth’s thermal dynamo London: Geophys. (1999). Center, Center. computer effects Aeron. IGPP The the Comput- the and geodynamo Phys. the Res. Earth’s Earth’s the I. both 401,885 of Earth’s in Geoelectr. model,” the Eds.), The in in (1995). Physica core,” supported (1996). “On Flight the alloys modelling role 393-409 and convective and in Earth Geomagn. Nature 1-97 mantle,” Geophys. Niinez W., metal: (1995b). iron also dynamos. 138, Earth’s Initiative. “The self-consistent J. 79, Ceomagn. M. in Space A. convection and evolutionary convection convection assistance. Inr. the H., J. was core,” 7989-8006 convection,” for and Supercomputing J. reference core P. in “Numerical Applications, between Program convection reversals,” 203-209 (1984). Dym. 101, convective Woods, planets,” of (1999). A., anelastic solubility inner GAG 377, Earth’s governing governing EAR97-25627, reasons model,” Goddard Diego ‘GRI’. thermal -484 Res. (1994). three-dimensional Partnership Fluid and G. Geophys. Roberts, outer stellar of as Femz-Mas theory “An turbulence and R. the three-dimensional “A of and San to partitioning “Preliminary and Nature the 323-335 geomagnetic NSF Challenge (1963). the (A. J. computational (1990). UCLA 55,461 H., “A L. of on study,” of L., H., conducting 10 the Geophys. and layer GEODYNAMO “Potassium (1981). by “Equations “Equations 172, “Local P. Asrrophys. “On for (2000). -87 geodynamo H., F J. D. Research P. field,” 8- by Phys. Glatzmaier, J., referred Laboratory. Lister, Supercomupting 71 H., IGPP, P., H., 1900” fields P. Grand H., THE is the simulations finitely Dynamos. Hongre, Letfs. 149, P. P. PHR V. M. 55, E., P. and (1995a). frequency core,” of appear Center, and parameter for 297-356 compositional S., realistic Rep?., Geophys. and Roberts, Sci. H. P. to Roberts, Roberts Compuf. the paper a Anderson, 25, R. Dokl. by Roberts, provided J. geomagnetic California 63-75 Drake, “Magnetic National Meytlis, Dynam. Roberts, Roberts, Earth’s and Roberts, and 15kbar This Non-linear and Coe, “Magnetohydrodynamics “Numerical and “On Planet. of the Infer. Center Phys. rotating 91, Olson, P., HPCC/ESS in Breach, at “Structure the systematic A. (1993). A. and were and (1997). and and M. driven and Huppert, A. Fluid of I., A., A., I., E. of I., 1. (1964). I. controlling I. I. U., Computing Maureen L. and with G. A. geodynamo,” G. geodynamo,” method,” Earth G. J. G. G. Inter. S. S. in (1996a). A., S. Soviet S. S. S. 1048 S. EAR99-02969, Alamos PIanet. 18,679 N. the the the NASA B. Advances National thank Los University 1035- resources In: 698-712 core,” Astrophys. 659- evolution geodynamo: composition Gordon and and Planet. Earth 81-94 (1999). and simulation solution melts,” mantle simulation NSF the by by the the the Braginsky, Braginsky, Braginsky, References ing Advanced Diewonski, Braginsky, Braginsky, Braginsky, Braginsky, We Glatzmaier, Buffett, Chabot, Connerney, Christensen, Glatzmaier, Glatzmaier, Glatzmaier, Glatzmaier,

Downloaded By: [Glatzmaier, Gary A.] At: 00:40 3 February 2007 a the the the 38, Res. Js9, using Phys. Phys. Earth of of Phys. Acad. Earth. core,” of core,” layer,” Engng. (1999). time: (1996). certains dynamo external Phys. the J. planetary aximuthal Dynamics. (1961). 1109-I121 Discussion. 121 geodynamo Phys. Nature Rend. (1980). core,” inner geomagnetic plane Press of from dans model,” Geophys. Znt. solution in and evolution Earth’s on 358, 105- J. dynamos,” modeling Fieldof the Earth’s Press geological A Comp. Conremp. the number, Earth’s 276-281 planet of approximation now,” model,” core in ‘GR2’. rotating 111, reversal a of Earth’s of a 4, dynamos,” Geophysical Academic sounding (1992). thermal Ser. and as of numerical flux and A in J. Structure field spherical magnktique to outer Magnetic solide,” Inter. and mechanisms the University fluid Press (1953). Rayleigh Its dynamo York core then “Numerical evolution cooling The (1996). on Earth London equilibration,” frozen (1996~). (1993). geodynamo,” of (1963). magnetic geodynamo graine magnetism 297 chaotic champ the L., A., dynamos fluid New Geophys. Earth’s referred Planer. the the Part: 220 (1978). 543 driving the GLATZMAIER and core P. SOC. theory rotating ‘GR3’. is du of - the - and and hydromagnetic G. 172, of the “On a geomagnetic A. d’une of as numerical R. “Thermal Brook6eld driven D “Magnetism Cambridge: (2000). Earth 207 of core,” (1983). 541 Terrapub of inner to Mantle. test G. 274-283 influence the “On paper 2.5 A., G., Central J.-L., 98, (1979). 640-641 Nature the J. conducting of 365, “A Rotation on Trans. Phys. “Dynamo (2000). “Regular generation 274, H., Deep 29-32 pn5sence (1996). “Rotation McFadden, This “Simulating of “Effects AND Earth’s Tokyo: “The J., Brisbane: A A., (1999). Rotation. 271, Earth-like kgedration A., P. referred Inter. 466-489 Glatzmaier, H., en convection the (1997). 27, (1999). F. H., Mouel, “Influence H., (1960). 57-99 field core,” Earth‘s and the is (2000). U., radius G. R., Jacobs, Nature la P. Schubert, -458 “Convectiondriven P. 54, Philos. “A P. Le G. in of and “An A., 59, W. “Effect and 3-20 Land. Earth. 259-272 “Intensity electrically 75-94, (1996b). de Lett. 1-17 The D. dynamo,” Nature and H., 451 Press core A., inner and Planet. J., Variable C. 1-343 Tilgner, M. paper and U. the P. M., the Roberts, ROBERTS Core SOC. H., G. (1999). pp. C. 111, T. SOC. Icaw planktaire 31 99, 117, (1998). In: Res. 1891 IIa, 237-258 Roberts, of and Roberts, Roberts, balance T. inner H. R. the This Earth reversals,” J.-P. Fearn, and Glatrmaier, Jones, Earth’s magneto-hydrodynamics H. sir. Astr. 404, Ed.), P. 1338 Christensen, locate Earth’s Inter. Inter. 117, Kono, and Macdonald, Jones, Inter. 10,404 and and and mechanisms Spohn, 1887- simulations,” University F. and I’inhibition Bloxham, R. Roberts, A. to dynamo Proc. Tanaka, and “Heat and and A. study”. Phys. planets,” A. A. Physics 323, The J., McElhinny, Geophys. (1997). and and “The observation,” “The and (1997). and J. Poirier, Masters, de Mech. 1325- G. J., H. R. Inter. and 274, Christensen, and G. G. G. G. D. R. and Planer. T., A. D., H. “Sur Planet. K., Planet. B., A,, Busse, S., C. 10,383- D., “How P. A. Paris 36, -374 Yukutake, W. R. J. G. P., F. J. W. M. Fluid E., D., C. R., terrestrial 104, Earth problem.” (2000). magnetohydrodynamic geodynamo Planet. Cambridge: geodynamo: Paleomagnetism. wavenumber Earth 371 models,” (T. Sci. statistical modtles Geophys. fluctuations J. Sci. Earth magnetic interiors,” Science 269-288 Roberts, Sakuraba, Sarson, Stacey, Stevenson, Verhoogen, Munk, Taylor, Labrosse, Lambeck, Memll, Momson, Olson, Kuang, Kutner, Kono, Hide, Hollerbach, Jault, Jones, Gubbins, Jacobs, Glatzmaier, Grote, Glatzmaier, Glatzmaier, Glatzmaier, 74

Downloaded By: [Glatzmaier, Gary A.] At: 00:40 3 February 2007 75 at to of we the not also that (A4) (Al) (A21 mass FOC these some more simu- which obtain that and are in (see correctly the reference for to kg, The GR densities is because models than according 0.05%. L retained our errors ICB G state (Al) x - 0.3%. only less FOC 0.1240. in values as and m only the PREM the is kgm-’. the have = consistent S but the 9.84 throughout uncertainties kg than by 4 law used we at but g into = with in the 2581 a which less 1259 = assumed 1 that that for s-~, REFERENCE by mutually kg, agree PREM Ms~c models. is evolve of cy), m gravity density PREM, is simple to 3 of justifyable, OF - the determine lo2* considerable values meant three of as to Having x could to C~C;I$.,~ that core PREM km, that 12.14 are use c7(1 has GEODYNAMO chosen of = p(r) with due = state 1.837 we adaptation 2400 GR1 inner between 4 from S develop This THE $ the g, suffices = is that There require. an in r to the geophysically the is at of is model, we s-~, This core with constants differs from aim them. state. that and G impossible m 12484kgm-’, state L are CONSTRUCTION CONSTRUCTION we mass than assumed consistency sP2). fluid with = the gCMB 4 A: sense the differs the in 4 example, acceleration, the ICB: was r/rCMB 0.0250 course and the agreement BR) places follows = = It 4.40m ensure parameters into of for reference For in x the 4 and which of = 4 to is resulting assume consistent the what state 80 It Our (gICB (Al), great. MF~ detailed states, The MODELS We gives are decimal APPENDIX In where where as G of parameters CMB lations. wish p.

Downloaded By: [Glatzmaier, Gary A.] At: 00:40 3 February 2007 a is is = and the and take (A8) (Al) have from -2% 4 in ICB PREM 0.02%. make accord- ZFW we PREM. by material s- compro- then is, m we the r-ldp/dr of the and than with ICB, and departs We at are balance 8437 that implied 5% constants write: the less from US, Earth, m2 of are kgm-3. SIC agrees by acceptable changes by FOC The kg now 1. an side the correctly. ICB differs 2581 -244 sound, future suppose We of and by = = of hydrostatic ignore of PREM x each and we gm and PREM GLATZMAIER also Msrc SIC SIC 6 of which on We A. provides of m 5.856 Using but past the the G. CMB give density that velocity C~C$&,.,~ = r-‘dp/dr in same at to of zcB the (A6) of particular those AND - The above The the from facts. Zs~c for time-dependent. in that as r ZcB density the chosen inertia assumptions. 13080kgm-3, kgmP3, and velocities = 0.2%. is of significantly. exceed almost differs (Al), of model is 0.07%. ROBERTS is cf constancy Ap model H. consider 13080 only violate time, this necessarily more sound under and the P. than = which p/dr we which not We where density kgm2. jump by simplifying the .f with moments that less r-ld grossly (A7) of (Al), but PREM 44. however by x and in modifying which to not axial central density of = since have are In As 12762kgmP3 cf independent properties The ing 9.044 number The value cf4 10140ms-’, The and from does respectively. mise, form we that PREM, 76

Downloaded By: [Glatzmaier, Gary A.] At: 00:40 3 February 2007 it 77 we the We (A121 are than effective constant from constant constant: 0.68. introduce “rejection are is ICB is the or constant, the the also is element this and peL, ~HL rejected theory 0.66 BR is iron: that core (say), BR FOC that with the alloying the X, BR. of the introduced K, = in of measures, the by to assume = 1, is BR assume constituent 0.351 masses mass X31$ shall basic which compared = We + PFe PeL X light dissolved as We Oxygen =-- total suggested bHL, Also GEODYNAMO of the if implied iron. X3)cf as the 0.95 when ~HL of constituent. ICB. of - THE The larger (1 value the are: ’. 0.41, Since light - much parameter, possibility. m density onto the much how kg constituent Sulfur value is 1 today’s the or next the is 258 Oxygen TICB/ICMB. light kgm-3. This freezes have = = significant from it of C the me X measures 0.7. as Silicon adopt I2510 = Consider if where therefore where factor” where, Ke another which fluid and ignore density where 6HL is

Downloaded By: [Glatzmaier, Gary A.] At: 00:40 3 February 2007 a a at the We was to Here r. (A211 ignore, specific convec- freezing of layer though composi- that, density we BR. on the layered, in in Since that is is of it space, At, cp argued which function AT, time in BR that and (D50) have core, the contraction so at we the constituent. of 1O6m. p increasing from uniform of GLATZMAIER X of potential. discontinuity, is (A9), light discontinuity, A. expansion result unmixed 3.48 by JMF~. - G. T the that the the value the = = follows to of of is AND the chemical volume contraction clear FOC, model, rCMB is implied monotonically of part M:~~ p a the being gives that -f)Ap r is that general attributed value expression ROBERTS present (1 a p therefore is be H. conservation mixes (A17) the is the pressure; radius P. coefficient that of It to can at on (say)] next t. second creates so the throughout this is, of remainder E) is that the constant second a approximation, the thoroughly SIC down, [=fay at (like ICB Consider The have laid The function composition tion Equating SUppOSing tion; and A$ the where good 78 where heat

Downloaded By: [Glatzmaier, Gary A.] At: 00:40 3 February 2007 is 79 of of and The It other made PDE. (A25) 0.200. part mass Al. of for MFE: PDE. (A21) and > error small for total for x3) a Table the MFOC - 0.145 the (A13), model. in EF (A21) 0.15 A 0.1749 0.1856 0.2000 0.2202 0.2500 0.2979 0.3876 0.6293 0.1608 0.1669 such L and are each, = uncertainties ( from - Since the given and , A') fF for CX2)cFGHLrFs different E -F - occupy for the Earth are - 1 of that for x X -fFhLrFS(l (AlO) obtain same in 0.50 0.55 0.60 0.65 0.70 0.75 0.85 0.80 0.95 0.80 C( EF true 5 3 of so - of the x~)(K future - models now derivatives. values as - EFh) G be 1 X3) GEODYNAMO we and + values the Values inherent to chosen - time x3( and €F expressions obviously 1 is THE f(1 S tF are 0.1450 0.1452 0.1458 0.1467 0.1480 0.1500 0.1526 0.1561 0.1448 0.1449 A1 K( so - other two those integrated the = ancient took constant, not for denotes of be the TABLE the is is MFX,~ we than if may SICS for (A25) and This less equate 101.78kgmP3 x 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 overdot A(tF)-'"" of the that, be of the Ax ti, now constituent that equation core We light (A22) Solutions where where The where values the would clear parameters.

Downloaded By: [Glatzmaier, Gary A.] At: 00:40 3 February 2007 is of by see the we we the that p very FOC guess (A281 (A271 by for Taylor is on on 5300°K used BR, constant the where C#J By of tF Griineisen T,,, as advance isothermal in the (A27) of of on values the and the equilibrium; value PDE. kg/J, is 0 is T, an educated rate Following andPICB, y of KT the for of replace elsewhere lo-* choosing taking as than x glCB and with we dependence average T(r) as TP), and, values; hydrostatic alone, 1.62 GPa, more function - ICB, GLATZMAIER 4375g. the the of a of agrees Thus, - such 135.75GPa. A. that the 1300 little dependence as G. of estimate at error, be liquidus 71~~ TT,P value constants GRI. The in to retain We to + AND being approximately 1.27 2795"K/(1 value that approximation are quantities To we determining and be = advance take consequence so = pICB. in have ry to a refers (A26) average T, of and grossly BR we good Tm is in m PREM ROBERTS an a from by take both and not evaluate rate IO-l'Pa-'. H. 2795°K to is the to which we x P. rZs (A29) therefore E/q be it TICB the 700°K used PDE that to significant, we easy be for 1.44 subscript the PCMB and for which Here obtain T= can the show for now more (A27) Equation provided so ICB is adopt (3.7). integrating in BR It surface. the BR which BR. To by We and simpler but, assumed uncertain, where much expansion, where compressibility, constant, adopt 80 take

Downloaded By: [Glatzmaier, Gary A.] At: 00:40 3 February 2007 81 state from (A341 (A35) repre- where (A31a) (A31b) absence through a constituent the BR] '. core used in reference differentiation, of withdrawn light kg- -&dU/dr, have the the J that, have = the in heat we of lo6 (6.39) source: of FOC we &g the s-* ' be BR, Eq. M (7.32)] = to to heat gravitational of ~~CGLEL TICB - constant entropy [see E IO5"K their by ICB EICB is slightly, take @/dr x latent ~ICB 1012m20K 55 the gives [see 5.1522 we rq rICB x Conservation the of = 1 GEODYNAMO 2.148 ~ICB eE+es+e", this a for released to 7. Since Appendix = show decreasing balance, which 1.93 THE in Q= for rx = BR the advance (A30), rICB owing - rICB energy heat, rx a:' of given 0.145 potential. expression from with that hydrostatic (= of mantle, rate latent sources the QN the the and the represents value arising quantities by and is 15) that Q5 combined h relate (A gravitational that simplify core radioactive is By To 17 Qs physical To the of sentative implies where When where cooling,

Downloaded By: [Glatzmaier, Gary A.] At: 00:40 3 February 2007 in of We and The SIC SIC. offset solved seen assume thermal the here the ICB. adiabatic adiabatic be negligible, significant we is advance in also conduction the heating. the that the K is can partially not the GR 1) is as heat is into and with employed a&, (A39) introduce the transport internal SIC words, FOC, which ICB Section than &, not gIC, the Following heat that the volume obey + the give SIC, less other (see of in at and GLATZMAIER continuous to conductivity SIC. from be In the will from unit A. is should to in S G. Tlc~ model (PVP) QsIc per suppose flux . a FOC. in; loss thermal arising v AND flux convection conditions integrated isentropic We the Given its that = unlikely (A36): heat to an is heating at QsIc as - frozen heat requires into of that approximate, since due simply (5.19). be model. Joule the Qs from ROBERTS flux @F ICB boundary L (AM) approximately part Joule be H. of but, net BR not is of ICB P. though the heat the two the hence can the see in simplicity, will is the is side the principally and this which from calculate to convect is for of greater ? even E&~ gIc evaluation not flux flux following, then ICB first the entropy GRl -3: (A17), give The the in further, heat heat the right-hand by where can where to conductivity; equation, does its The errors, 82 by

Downloaded By: [Glatzmaier, Gary A.] At: 00:40 3 February 2007 83 of of be We and from ('443) can BR. Unit means flux is of z., the importance heat (A36) composition models and the have (6.29b) by and 1.198 7.298 7.369 0.620 0.0733 0.179 see provides relative future they Large 638 235 now from and 9710 11027 11665 and the heat we (A46) ICB Since FOC freezing; both Dresent the 5 A2), the &c. (A45), of on indicate of 1. = to 1.837 Dast. 0.984 0.620 9.044 0.061 0.150 qN 189 597 and Tab. of Generic GEODYNAMO 9903 12166 12762 i.1~~. due Q& convection. release Table and (see is THE (A44) diffusivity qs the by see Qs Prouerties 6, growth, of model entropy 1.933 0.0156 9.121 0.0602 0.620 0.147 A2 of of (A43), Small (A40) 174 600 9930 the driving the 12490 mantle; 13090 part turbulent is rate in sizes the TABLE the (A38), from replace the to is X ~/TICB source relative remaining core = (A34), Finally By assessing The the therefore each where AS computed The

Downloaded By: [Glatzmaier, Gary A.] At: 00:40 3 February 2007 to ~ direct 5.2TW core adopted J J the - - Unit but 1030 1030~ 1030 kg"K/J "K "K Tw 0.01 GPa m/52 was from I, from model ZiB, Table value 1.754 1.33 2.119 0.506 0.445 0.0106 8.287 13.0 22.4 to Large 238.8 3627 4260 flux, GLATZMAIER polytropic different heat a A. according (Continued) G. 12 0.04oO 0.352 0.1 0.292 0.275 0.979 0.0504 4.401 slightly 67.8 A2 Generic 328.2 AND 5300 4Ooo a because adiabatic model is here. G the TABLE gives This the used ROBERTS 3. that this - for 1.118 0.00468 0.00515 0.629 0.0146 0.00220 0.0171 0.00209 H. Small 357.3 982 P. 5757 4183 and (A29) noted GRl of to be 5.4TW GRl, is in may T It for according integration mantle 84

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