351 lines
15 KiB
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351 lines
15 KiB
Plaintext
VOL. 80, NO. 2
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JOURNAL OF GEOPHYSICAL RESEARCH
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JANUARY 10, 1975
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A NecessaryCondition for the Geodynamo
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F. H. BUSSE
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Instituteof GeophysicasndPlanetaryPhysicsU, niversityof California Los Angeles,California 90024
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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
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magneticfieldandthe total magneticenergy.Sinceboundson thisratio canbe obtainedfrom energetic considerationsth, e criterioncanbeusedasa restrictionon possibledynamomechanismsS.everalr'ecent suggestionfosr the originof thegeodynamion a stratifiedoutercorearecriticallyreviewed.
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It is generallyacceptedthat the earth's magneticfield is earth's core and a smallermeridionalcirculation.Similarly,
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generatedby motionswithintheliquidoutercoreof theearth. Kahleet al. [1967]founddifferentordersof magnitudefor the
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Yet in spite of a considerableresearcheffort in the past toroidaland poloidalcomponentsof the velocityfieldin their
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decadesi,t hasnot beenpossibleto find an unambiguoussolu- attemptto infer motionsof the corefrom the observedsecular
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tion for the sourceof the energydissipatedby ohmicheating variation.The poloidalcomponentis generallysmalleryet of
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and viscousfriction. The difficultyof this problem has been particular importancesinceit can be shown that a purely
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compounded recently by the suggestionof Higgins and toroidalvelocityfieldcannotgeneratea magneticfield [Bullard
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Kennedy[1971] that the outer core is stably stratified.This andGellman,1954].Only thepoloidalpart of thevelocityfield
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proposalwould eliminateor severelyinhibit the traditional hasa radial componenta, nd it is desirablefor thisreasonto
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contendersfor the energysourceof the geodynamo,namely, find a conditionsimilarto (1) involvingthe radialcomponent
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convectionandprecessionof theearth [Bullard,1949;Malkus, of the velocityfield. This will be the goal of the analysis
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1968]. Stimulatedby Kennedyand Higgins' [1973] 'core describedbelow. The importanceof such a condition is
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paradox,' a number of workers have proposedalternative emphasizeidn thecaseof a stablystratifiedcoreasproposed
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sourcesfor the earth's magneticfield [Bullardand Gubbins, by Higginsand Kennedy[1971].Althoughtoroidalmotions
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1971;WonandKuo, 1973;Mullan, 1973].In general,however, would remainunaffectedin this case,any flow with a radial
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theseproposalsfail to take into accountthe rather stringent velocitycomponenwt ouldbe inhibited,with the possibleex-
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dynamic requirementsfor the geodynamo. This note will ceptionof internal gravity waves.
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derive a simple necessarycondition for the geodynamothat may help to restrictthe classof feasiblehypotheses.
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MATHEMATICAL ANALYSIS
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In view of the complexitiesof actual solutionsof the In order to derive our criterion, we consider an incom.
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dynamoproblem, necessaryconditionsfor the generationof pressiblehomogeneoufsluid containedin the finitevolumeV. the earth'smagneticfield have long beenregardedas highly Sincethe first part of our derivationdoesnot dependon the
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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
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existenceof a lowerboundwassuggesteodriginallyby Bullard dynamo equation
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and Gellman [1954], and an explicit value applicableto the earth has been derived by Backus [1958]. According to this
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(3/Ot + v. V)ll + n V X (V X B) = B. V v (2)
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criterion, any magneticfield must decayunless
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which can be derived easily from Maxwell's equation and
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Rein -= Uro/• • •r
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(1) Ohm's law in the magnetohydrodynamiacpproximation.The
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magnetic diffusivity • is equal to (a•) -x, where a is the
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where ro is the radius of the earth's core, which has been electricalconductivityin V and• isthemagneticpermeability.
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assumedas a homogeneoufsluid sphereinsidean insulating We assumethat the spaceoutsideV is insulating.Hence V X
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mantle,U isthemaximumvelocitywithrespectto anarbitrary B = 0 holdsoutsideV, and r ß Blrl' remainsfiniteas the
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systemofcoordinaterostatingwitha constanatngulavreloci- position vector r tendsto infinity.
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ty, andn isthemagneticdiffusivityC. ondition(1) wasderived By multiplying (2) by r and using the vector identity
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by Backuswith the maximumdeformationrate in placeof r ß(b ß Va) = b ß Vr ß a - a ßb, we obtain
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U•r/ro,whichisadvantageouisn thatit becomesobviousthat a
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rigid rotationdoesnot contributeto U. The form (1) of the
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(O/Ot + v' V)r. B - nV:r. B = B. V v. r (3)
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criterionwasgivenby Childtess[1969].We alsoreferto the
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discussiobnyRoberts[1971].Neitherthepresencoef therigid in V. This equationappearsin a slightlydifferentform in
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innercorenor the inhomogeneitieosf the outercoreand the Backus'[1968]paper, whichalsoemphasizesthe analogyto
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finiteconductivityof the mantlehavebeentakeninto account the heat equation,the right-handsideof (3) representingthe
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in (1) sincetheir effectsare of minor importance.
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heat source.Since diffusionultimately balancesthe source
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A disadvantagoef (1) isthatit doesnotdistinguisbhetween termin thestationarycase(,3) suggestasnorderof magnitude
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differentcomponentosf the velocityfield.Most theoriesof estimatefor the radial velocity component.
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thegeomagnetifcieldassumea largedifferentiarlotationin the
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Br
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Copyright¸ 1975by the AmericanGeophysicaUl nion.
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v• n IBIro
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278
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Buss•;A NECESSARCYONDITIO!FVORTHEGœODYNAMO
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279
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In thefollowingweshallderivea relationofsimilarformbya dinatescorrespondtosthelowespt ossiblsephericahlarmonic
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rigorousanalysis.
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l=l.
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f• Multiplicatioonf'(3)byr ßB andintegratioon.verV yield [•.B•d'rF=[•--f•L, •h•L•'h
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l2_ddt f•(B.dr)V=.•--nf•+v' }•7B•d.Vr[
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.dV•_--2f,•hr.•X(• Xr•2h)dV(8)
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-[f' vr.BB•. r.vdV (4)The lastterm in (8) canbe writtenin the form
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We havedenotedthe spaceoutsideV by V'. The surface
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2fvhr'XV{VXIVX(VXrh)d]V} sepa.ratiVnagndV'isS withtheoutsidneormanl.Thein-
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tegraol verV + V' in(4)hasbeenobtainedbypartialintegra-
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ti•n andusingthefactthatV2rßB vanisheisn V':
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0--f-vr.BX7B2drV.
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,
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=2fv(• Xrh)-X• [• X(• Xrhd)]V
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=
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X(VXrh)l
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(9)
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where the relation
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In deriving(4), the fact that the term
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f,v• .• [r.•Bd[V=•$ n.v[-r•.•Bd[S
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-vanishessince n. v vanisheson S has also been used. By
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IVx (vxr)l dF
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- f (• XrhX) [?X(• Xrh)]d.nS
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furthepr artiailntegratioanndbyusingV:
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ingthatv ßr vanisheosn S, wefind
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B
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=
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0
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andassum-hasbeenused.Apart froma factor4/•,(9) givestheenergyEn of thepoloidap!artof themagnetifcield.HenCe(6)canbe
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frrB. B.•r.vd-V-f-•-v,rB.•B.rdwVrit{•n'in
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thecaseof ..
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a
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sphereas
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The latter term can be bounded from above,
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•d• (B•.rd)F• --n+max(Ev.•r)
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--f•v,.Br -•B.dr V
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'fv+•' Ir' BI (10)
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• max(v.r) lB[• dV [Vr.B[ • dV
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(5) whereE• denotetshetot• energoyfthemagnetficidd.A•ord-
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inglyW,½findasa necesscabrynditifoonrtheamplification
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whereSchwarz•isnequalityhasbeenused.Thuswe obtain of fv(•. r)=dV
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from (4) the inequality
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max-(V' r) > n(2E•/E•) •/•
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(1 !)
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•d• (B.r•)dV• -n ß max(v.r)
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Inthecasoefan0nstatiocnyacryldicynamthoi,scondition musbtesatisfitehdroughoountlypartofthecYclIen.t• case
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ofa statiobadtynam(o1,1)P•9vidca.snecesscaoryndition
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forthe'existenocfethedynamos.incea lowerlimit'•fotrh•
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•a!ucofE• isavailabflreomt•cobservgeedomagnefiteicld ..
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andsincaenuppeerstimaftoerE• canbe0bta{d0-ef'rOm
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ßf,• [VrB,•[dV
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(6)ener.gcy0n•ideiatio(1n•)sp,•ovidaeussefulteinsatdditiotoh (1)forthefeasibili:toyfhypotheticgael0dynams.0
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Obviouslyt,he radialcomponenotf B mustdecaywhenthe An an•ogoust,houghl•s u•f• •te•on can• de•v• by
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quantitwy ithin'thebracket[snegative.
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mu!tiplyin)gb(y2'aanrbitraurynivt •tbrk.Mu!tipii•agoifon
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WhenV isa spherew, ecanderiveaconditinothatpermitas theresultin.gequat}obny'k;B andintegratioonverV•yield
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.
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.,
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..
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physicainl terpre,tatioAns.suminthgeoriginat thecenteor f
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thespherwee,usearepresent0aft.iBointermCs•pfoloidal
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andtorOidacl omponents:
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k)dV -nmax(k'v)
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B = V X (V ><rh) + V •Xrg
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(7)
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It is evidenthat onlythepoloidalfieldh contributetso the
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radiaCl OmponeonftB,
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r.B = r. [v x.(v x rh)] • L2h
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ß IB.kldv
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(!2)
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where-L • isthetwo-dimensionLalaplacianonthesurfaceof after the samemanipulationsthat led to (6) havebeenper-
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theunitspherSei.ncweecanassumweithoulot singgeneralityformedS.inctehecomponeonfthevelocitfyieldinthedirecthattheaveragoef h overanysphericsaul rfacIer[ -- const tionOftheaxisofrotationislikelytoberelativelsymaliln:the vanishews,efindL•h:>2h,wheretheequalitysignisassumedearth'csorebecauosfetheapproximvaate[!ditoyfthe,Taylor-
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whenthe0,•odependenocfeh in a sphericaslystemof cOOt- Proudmatnheorem(1, 2)mayserveasausefuclonstrainwth,en
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280
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BussE:A NECESSARCYONDITIONFORTHEGEODYNAMO
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k is identified with the direction of the rotation axis of the stratified.region[Malkus,1968;Busse1, 968].On theother
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earth. Yet at this point we shall not pursue(12) further.
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hand,the Griineisenparameterappropriatefor theconditions
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DISCUSSION
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of the outer core and the posSibiliyt. of slurr•yconvection proposedbyBusse[1972]andElsassenreedfurtherinvestiga-
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We begin the discussionby relating (11) to the toroidal tionbeforetheHiggins-Kennerhlyypothesicsanbeaccepted
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theoremmentioneidn the introduc,tiownh, ichstateltshat as a fact.
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•
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toroida! motions cannot generatemagneticfields. Although Weclosethediscussiwonitha remarkona shortcomionfg
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theorems of this kind are highly significant from a (10). Sincethe quantitywithin the bracketsdependson the
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mathematical point of view, their value for physical magnetifcield,anasymptotdicecaycannobt econcludewdhen applicationmsaybequestionabulenlessit canbeshownthat thatquantity{snegativaet a particulaProintin time.This
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theyarenotlimitedto singulacraseswithspeciaslymmetries.shortcominigs sharedby (1) sincethemaximalvelocityU in
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Criterion(11) is helpfulin this respectsinceit demonstrates theCOrdeependosnthemagnectifieldin generaMl. oreap-
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that the toroidal theoremalso holds for sufficientlysmall propriatecriteriawouldinvolvethe forcesdrivingthemotion
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deviationsfrom a purelytoroidalstateof motion.In or /he heatingratein the caseof convectionw,hichcanbe particulari,n the caseof thegeodynamaosizableradial assumetdo begivenindependentolyf themagnetifcield.TO
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velocictyomponeisnrtequirefodrthemaintenaonfc{ehe derive such criteria, the Navier-Stokesequationsof motion
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geomagneticfield.
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haveto be considereda,ndmethodssimilarto thoseemployed
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It is unlikelythat the recentproposalsfor the energysource by Payne[1967]in the purelyhydrodynamiccasewouldhave
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of the geodynamoto which we referredin the introduction to be used.This will be the subjectof future work.
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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
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the frequentlyquotedvalueof 5 ß 10* mhosm-• for the con- Foundation, NSF grant GA-41750.
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ductivitoyftheearth'csoreI.t shoulbdenotedtha•onlythe
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time'averageof the radial velocitycomponentoverperiodsof
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REFERENCES
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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.
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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
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Bullard, E. C., The magneticfield within the earth, Proc.Roy. Soc. London, Set. A, 197, 433-453, 1949.
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Bullard, 'E. C., and H. Gellman, Homogeneousdynamos and terrestrialmagnetism,Phil. Trans. Roy. Soc. London,Ser. A, 247, 213-278, 1954.
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[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,
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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.
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It isinterestingto notethat the regioncloseto theequatorof
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Bullard, E. C., and D. Gubbins,Geomagneticdynamosin the stable core, Nature, 232, 548-549, 1971.
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BusseF, . H., Steadyfluid flow in a precessinsgpheroidasl hell,J. Fluid Mech., 33, 739-751, 1968.
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Busse,F. H., Thermalinstabilitiesin rapidly rotatingsystemsJ,. Fluid Mech., 44, 441-460, 1970.
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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.
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Childtess,S., Th•orie magn6tohydrodynamiqudee l'effet dynamo, report, Dep. Mech. de la Fac. desSci., Univ. de Paris,Paris,1969.
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Higgins,G. H., and G. C. Kennedy,The adiabaticgradientandthe meltingpointgradientin thecoreof theearth,J. GeophysR.es.,76, 1870-1878, 1971.
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Kahle, A. B., E. H. Vestine,and R. H. Ball, Estimatedsurfacemotions
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of the earth'score,J. GeophysR. es.,72, 1095-1108,1967. Kennedy,G. C., and G. H. Higgins,The coreparadox,J. Geophys.
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Res., 78, 900-904, 1973. Malkus, W. V. R., Precession of the earth as the cause of
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geomagnetismS,cience1, 60, 259-264, 1968. Mullan,D. J., Earthquakwe avesand the geomagnetdicynamo,
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Science, 181, 553-554, 1973.
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the inner coreis also the placewherethe criticalRayleigh
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number for the onset of convection is first reached either
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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
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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.
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Riley, N., On a sphereoscillatingin a viscousfluid, Quart.J. Mech. Appl. Math., 19, 461-472, 1966.
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Roberts,P. H., Dynamotheory,in Lecturesin AppliedMathematics, vol. 14, pp. 129-206,American Mathematical Society,1971.
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Smith, M. L., The normal modesof a rotating,ellipticalearth, Ph.D. thesis, Princeton Univ., Princeton, N.J., 1974.
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Won, I. J., and J. T. Kuo, Oscillation of the earth's inner core and its
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remainsthe strongestcontenderasa sourceof thegeodynamo relationto thegenerationof geomagnetifcield,J. GeophysR.es.,78, if Higgins and Kennedy'sproposalis accepted.Precession- 905-911, 1973.
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inducedturbulencewould be lesslikely in this casesincethe shearlayer from which the turbulencearisesliesat a distance
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(ReceivedJune 18, 1974; revisedSeptember30, 1974;
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of about (3)toro/2 from the earth's center in the strongly
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acceptedOctober 10, 1974.)
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