zotero/storage/EPPNPLLJ/.zotero-ft-cache

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 ><rh) + V •Xrg

(7)

It is evidenthat onlythepoloidalfieldh contributetso the
radiaCl OmponeonftB,
r.B = r. [v x.(v x rh)] • L2h

ß IB.kldv

(!2)

where-L • isthetwo-dimensionLalaplacianonthesurfaceof after the samemanipulationsthat led to (6) havebeenper-
theunitspherSei.ncweecanassumweithoulot singgeneralityformedS.inctehecomponeonfthevelocitfyieldinthedirecthattheaveragoef h overanysphericsaul rfacIer[ -- const tionOftheaxisofrotationislikelytoberelativelsymaliln:the vanishews,efindL•h:>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.)