VOL. 80, NO. 2 JOURNAL OF GEOPHYSICAL RESEARCH JANUARY 10, 1975 A NecessaryCondition for the Geodynamo F. H. BUSSE Instituteof GeophysicasndPlanetaryPhysicsU, niversityof California Los Angeles,California 90024 A necessarcyonditionfor thegenerationof magneticfieldsby fluidmotionsin a sphereis derivedin termsof themagneticReynoldsnumberon thebasisof the radialcomponenot f thevelocityfield.A second parameterenteringthe criterionis the ratio betweenthe energyof the poloidalcomponentof the magneticfieldandthe total magneticenergy.Sinceboundson thisratio canbe obtainedfrom energetic considerationsth, e criterioncanbeusedasa restrictionon possibledynamomechanismsS.everalr'ecent suggestionfosr the originof thegeodynamion a stratifiedoutercorearecriticallyreviewed. It is generallyacceptedthat the earth's magneticfield is earth's core and a smallermeridionalcirculation.Similarly, generatedby motionswithintheliquidoutercoreof theearth. Kahleet al. [1967]founddifferentordersof magnitudefor the Yet in spite of a considerableresearcheffort in the past toroidaland poloidalcomponentsof the velocityfieldin their decadesi,t hasnot beenpossibleto find an unambiguoussolu- attemptto infer motionsof the corefrom the observedsecular tion for the sourceof the energydissipatedby ohmicheating variation.The poloidalcomponentis generallysmalleryet of and viscousfriction. The difficultyof this problem has been particular importancesinceit can be shown that a purely compounded recently by the suggestionof Higgins and toroidalvelocityfieldcannotgeneratea magneticfield [Bullard Kennedy[1971] that the outer core is stably stratified.This andGellman,1954].Only thepoloidalpart of thevelocityfield proposalwould eliminateor severelyinhibit the traditional hasa radial componenta, nd it is desirablefor thisreasonto contendersfor the energysourceof the geodynamo,namely, find a conditionsimilarto (1) involvingthe radialcomponent convectionandprecessionof theearth [Bullard,1949;Malkus, of the velocityfield. This will be the goal of the analysis 1968]. Stimulatedby Kennedyand Higgins' [1973] 'core describedbelow. The importanceof such a condition is paradox,' a number of workers have proposedalternative emphasizeidn thecaseof a stablystratifiedcoreasproposed sourcesfor the earth's magneticfield [Bullardand Gubbins, by Higginsand Kennedy[1971].Althoughtoroidalmotions 1971;WonandKuo, 1973;Mullan, 1973].In general,however, would remainunaffectedin this case,any flow with a radial theseproposalsfail to take into accountthe rather stringent velocitycomponenwt ouldbe inhibited,with the possibleex- dynamic requirementsfor the geodynamo. This note will ceptionof internal gravity waves. derive a simple necessarycondition for the geodynamothat may help to restrictthe classof feasiblehypotheses. MATHEMATICAL ANALYSIS In view of the complexitiesof actual solutionsof the In order to derive our criterion, we consider an incom. dynamoproblem, necessaryconditionsfor the generationof pressiblehomogeneoufsluid containedin the finitevolumeV. the earth'smagneticfield have long beenregardedas highly Sincethe first part of our derivationdoesnot dependon the desirable.The only known quantitativeconditionof thiskind particularshapeof V, we shallassumeonly later that V is a is a lowerboundon themagneticReynoldsnumberRe,,. The sphere.The magneticflux densityB is governedby the existenceof a lowerboundwassuggesteodriginallyby Bullard dynamo equation and Gellman [1954], and an explicit value applicableto the earth has been derived by Backus [1958]. According to this (3/Ot + v. V)ll + n V X (V X B) = B. V v (2) criterion, any magneticfield must decayunless which can be derived easily from Maxwell's equation and Rein -= Uro/• • •r (1) Ohm's law in the magnetohydrodynamiacpproximation.The magnetic diffusivity • is equal to (a•) -x, where a is the where ro is the radius of the earth's core, which has been electricalconductivityin V and• isthemagneticpermeability. assumedas a homogeneoufsluid sphereinsidean insulating We assumethat the spaceoutsideV is insulating.Hence V X mantle,U isthemaximumvelocitywithrespectto anarbitrary B = 0 holdsoutsideV, and r ß Blrl' remainsfiniteas the systemofcoordinaterostatingwitha constanatngulavreloci- position vector r tendsto infinity. ty, andn isthemagneticdiffusivityC. ondition(1) wasderived By multiplying (2) by r and using the vector identity by Backuswith the maximumdeformationrate in placeof r ß(b ß Va) = b ß Vr ß a - a ßb, we obtain U•r/ro,whichisadvantageouisn thatit becomesobviousthat a rigid rotationdoesnot contributeto U. The form (1) of the (O/Ot + v' V)r. B - nV:r. B = B. V v. r (3) criterionwasgivenby Childtess[1969].We alsoreferto the discussiobnyRoberts[1971].Neitherthepresencoef therigid in V. This equationappearsin a slightlydifferentform in innercorenor the inhomogeneitieosf the outercoreand the Backus'[1968]paper, whichalsoemphasizesthe analogyto finiteconductivityof the mantlehavebeentakeninto account the heat equation,the right-handsideof (3) representingthe in (1) sincetheir effectsare of minor importance. heat source.Since diffusionultimately balancesthe source A disadvantagoef (1) isthatit doesnotdistinguisbhetween termin thestationarycase(,3) suggestasnorderof magnitude differentcomponentosf the velocityfield.Most theoriesof estimatefor the radial velocity component. thegeomagnetifcieldassumea largedifferentiarlotationin the Br Copyright¸ 1975by the AmericanGeophysicaUl nion. v• n IBIro 278 Buss•;A NECESSARCYONDITIO!FVORTHEGœODYNAMO 279 In thefollowingweshallderivea relationofsimilarformbya dinatescorrespondtosthelowespt ossiblsephericahlarmonic rigorousanalysis. l=l. f• Multiplicatioonf'(3)byr ßB andintegratioon.verV yield [•.B•d'rF=[•--f•L, •h•L•'h l2_ddt f•(B.dr)V=.•--nf•+v' }•7B•d.Vr[ .dV•_--2f,•hr.•X(• Xr•2h)dV(8) -[f' vr.BB•. r.vdV (4)The lastterm in (8) canbe writtenin the form We havedenotedthe spaceoutsideV by V'. The surface 2fvhr'XV{VXIVX(VXrh)d]V} sepa.ratiVnagndV'isS withtheoutsidneormanl.Thein- tegraol verV + V' in(4)hasbeenobtainedbypartialintegra- ti•n andusingthefactthatV2rßB vanisheisn V': 0--f-vr.BX7B2drV. , =2fv(• Xrh)-X• [• X(• Xrhd)]V = X(VXrh)l (9) where the relation In deriving(4), the fact that the term f,v• .• [r.•Bd[V=•$ n.v[-r•.•Bd[S -vanishessince n. v vanisheson S has also been used. By IVx (vxr)l dF - f (• XrhX) [?X(• Xrh)]d.nS furthepr artiailntegratioanndbyusingV: ingthatv ßr vanisheosn S, wefind B = 0 andassum-hasbeenused.Apart froma factor4/•,(9) givestheenergyEn of thepoloidap!artof themagnetifcield.HenCe(6)canbe frrB. B.•r.vd-V-f-•-v,rB.•B.rdwVrit{•n'in thecaseof .. a sphereas The latter term can be bounded from above, •d• (B•.rd)F• --n+max(Ev.•r) --f•v,.Br -•B.dr V 'fv+•' Ir' BI (10) • max(v.r) lB[• dV [Vr.B[ • dV (5) whereE• denotetshetot• energoyfthemagnetficidd.A•ord- inglyW,½findasa necesscabrynditifoonrtheamplification whereSchwarz•isnequalityhasbeenused.Thuswe obtain of fv(•. r)=dV from (4) the inequality max-(V' r) > n(2E•/E•) •/• (1 !) •d• (B.r•)dV• -n ß max(v.r) Inthecasoefan0nstatiocnyacryldicynamthoi,scondition musbtesatisfitehdroughoountlypartofthecYclIen.t• case ofa statiobadtynam(o1,1)P•9vidca.snecesscaoryndition forthe'existenocfethedynamos.incea lowerlimit'•fotrh• •a!ucofE• isavailabflreomt•cobservgeedomagnefiteicld .. andsincaenuppeerstimaftoerE• canbe0bta{d0-ef'rOm ßf,• [VrB,•[dV (6)ener.gcy0n•ideiatio(1n•)sp,•ovidaeussefulteinsatdditiotoh (1)forthefeasibili:toyfhypotheticgael0dynams.0 Obviouslyt,he radialcomponenotf B mustdecaywhenthe An an•ogoust,houghl•s u•f• •te•on can• de•v• by quantitwy ithin'thebracket[snegative. mu!tiplyin)gb(y2'aanrbitraurynivt •tbrk.Mu!tipii•agoifon WhenV isa spherew, ecanderiveaconditinothatpermitas theresultin.gequat}obny'k;B andintegratioonverV•yield . ., .. physicainl terpre,tatioAns.suminthgeoriginat thecenteor f thespherwee,usearepresent0aft.iBointermCs•pfoloidal andtorOidacl omponents: k)dV -nmax(k'v) B = V X (V >2h,wheretheequalitysignisassumedearth'csorebecauosfetheapproximvaate[!ditoyfthe,Taylor- whenthe0,•odependenocfeh in a sphericaslystemof cOOt- Proudmatnheorem(1, 2)mayserveasausefuclonstrainwth,en 280 BussE:A NECESSARCYONDITIONFORTHEGEODYNAMO k is identified with the direction of the rotation axis of the stratified.region[Malkus,1968;Busse1, 968].On theother earth. Yet at this point we shall not pursue(12) further. hand,the Griineisenparameterappropriatefor theconditions DISCUSSION of the outer core and the posSibiliyt. of slurr•yconvection proposedbyBusse[1972]andElsassenreedfurtherinvestiga- We begin the discussionby relating (11) to the toroidal tionbeforetheHiggins-Kennerhlyypothesicsanbeaccepted theoremmentioneidn the introduc,tiownh, ichstateltshat as a fact. • toroida! motions cannot generatemagneticfields. Although Weclosethediscussiwonitha remarkona shortcomionfg theorems of this kind are highly significant from a (10). Sincethe quantitywithin the bracketsdependson the mathematical point of view, their value for physical magnetifcield,anasymptotdicecaycannobt econcludewdhen applicationmsaybequestionabulenlessit canbeshownthat thatquantity{snegativaet a particulaProintin time.This theyarenotlimitedto singulacraseswithspeciaslymmetries.shortcominigs sharedby (1) sincethemaximalvelocityU in Criterion(11) is helpfulin this respectsinceit demonstrates theCOrdeependosnthemagnectifieldin generaMl. oreap- that the toroidal theoremalso holds for sufficientlysmall propriatecriteriawouldinvolvethe forcesdrivingthemotion deviationsfrom a purelytoroidalstateof motion.In or /he heatingratein the caseof convectionw,hichcanbe particulari,n the caseof thegeodynamaosizableradial assumetdo begivenindependentolyf themagnetifcield.TO velocictyomponeisnrtequirefodrthemaintenaonfc{ehe derive such criteria, the Navier-Stokesequationsof motion geomagneticfield. haveto be considereda,ndmethodssimilarto thoseemployed It is unlikelythat the recentproposalsfor the energysource by Payne[1967]in the purelyhydrodynamiccasewouldhave of the geodynamoto which we referredin the introduction to be used.This will be the subjectof future work. providefor sufficientlhyighradialvelocitieisf a diffusivityof Acknowledgment.The researchreported in this paper was supthe order of 2 ß l0t cm2/sis assumed,which correspondsto ported by the Earth SciencesSection of the National Science the frequentlyquotedvalueof 5 ß 10* mhosm-• for the con- Foundation, NSF grant GA-41750. ductivitoyftheearth'csoreI.t shoulbdenotedtha•onlythe time'averageof the radial velocitycomponentoverperiodsof REFERENCES the order of the magneticdecaytime ro2/,1is relevantin (11), Backus,G., A classof self-sustainingdissipativesphericaldynamos, since the generation of magnetic flux cannot take place Ann. Phys.,4, 372-447, 1958. withot!tdiffusion.'WonandKuo [1973]proposedlargeearthquakesasa sourceof geomagnetismand point out the steady circulation inducedby oscillationsof the inner core of the earth. When Won and Kuo's valuesand the analysisby Riley Bullard, E. C., The magneticfield within the earth, Proc.Roy. Soc. London, Set. A, 197, 433-453, 1949. Bullard, 'E. C., and H. Gellman, Homogeneousdynamos and terrestrialmagnetism,Phil. Trans. Roy. Soc. London,Ser. A, 247, 213-278, 1954. [1966]to whichthey refer are used,an amplitudeof the order of 10-* cm/s is found for the steadyflow, whichis muchtoo small to be significant,accordingto (11). The error made by Won and Kuo in the applicationof Riley'swork hasalsobeen pointed out by Smith [1974]. Although the generationof magneticfieldsby short-periodoscillatingvelocityfieldsasenvisionedby BullardandGubbins[1971]isfeasiblein principle, the requiredvelocityamplitudeincreaseswith the parameter o•ro2/,wh hereo•isa typicalfrequencyof thevelocityfield.Thus the energyrequirementfor the possiblesourceof the osciliatoryvelocityfieldbecomesamplified.On theotherhand, the dynamoproposalsfor a stablystratifiedcoremay not be necessarysincein their secondpaper Kennedyand Higgins [1973]allow for a regionof nearly 800 km outwardfrom the innercorewhereconvectionmayoccur.The valueof 800km is takenfromagraphin thatpapersincethevalueof200or 300 km quotedin the text appearsto be in error. It isinterestingto notethat the regioncloseto theequatorof Bullard, E. C., and D. Gubbins,Geomagneticdynamosin the stable core, Nature, 232, 548-549, 1971. BusseF, . H., Steadyfluid flow in a precessinsgpheroidasl hell,J. Fluid Mech., 33, 739-751, 1968. Busse,F. H., Thermalinstabilitiesin rapidly rotatingsystemsJ,. Fluid Mech., 44, 441-460, 1970. Busse,F. H., Comment on 'The adiabaticgradient and the melting pointgradientin thecoreof theearth'by G. H. HigginsandG. C. Kennedy,J. GeophysR. es., 77, 1589-1590, 1972. Childtess,S., Th•orie magn6tohydrodynamiqudee l'effet dynamo, report, Dep. Mech. de la Fac. desSci., Univ. de Paris,Paris,1969. Higgins,G. H., and G. C. Kennedy,The adiabaticgradientandthe meltingpointgradientin thecoreof theearth,J. GeophysR.es.,76, 1870-1878, 1971. Kahle, A. B., E. H. Vestine,and R. H. Ball, Estimatedsurfacemotions of the earth'score,J. GeophysR. es.,72, 1095-1108,1967. Kennedy,G. C., and G. H. Higgins,The coreparadox,J. Geophys. Res., 78, 900-904, 1973. Malkus, W. V. R., Precession of the earth as the cause of geomagnetismS,cience1, 60, 259-264, 1968. Mullan,D. J., Earthquakwe avesand the geomagnetdicynamo, Science, 181, 553-554, 1973. the inner coreis also the placewherethe criticalRayleigh number for the onset of convection is first reached either if the core is heated homogeneouslyor if heating takes placejust at theboundarybetweentheinnerandoutercores owingto crystallization.This factcanbeinferredfrom the approximatetheory of Busse[1970], which we expectto hold evenin the presenceof a stratifiedouter part of the corein place of a rigid boundary. We conclude that convection Payne, L. E., On the stability of solutionsof the Navier-Sto.kes equationsand convergenceto steadystate,SIAM Soc. Ind. Appl. Math. J. Appl. Math., 15, 392-405, 1967. Riley, N., On a sphereoscillatingin a viscousfluid, Quart.J. Mech. Appl. Math., 19, 461-472, 1966. Roberts,P. H., Dynamotheory,in Lecturesin AppliedMathematics, vol. 14, pp. 129-206,American Mathematical Society,1971. Smith, M. L., The normal modesof a rotating,ellipticalearth, Ph.D. thesis, Princeton Univ., Princeton, N.J., 1974. Won, I. J., and J. T. Kuo, Oscillation of the earth's inner core and its remainsthe strongestcontenderasa sourceof thegeodynamo relationto thegenerationof geomagnetifcield,J. GeophysR.es.,78, if Higgins and Kennedy'sproposalis accepted.Precession- 905-911, 1973. inducedturbulencewould be lesslikely in this casesincethe shearlayer from which the turbulencearisesliesat a distance (ReceivedJune 18, 1974; revisedSeptember30, 1974; of about (3)toro/2 from the earth's center in the strongly acceptedOctober 10, 1974.)