H. K. Moffatt and D. E. Loper- The magnetostrophic rise of a buoyant parcel in the Earth’s core

8/3/2019 H. K. Moffatt and D. E. Loper- The magnetostrophic rise of a buoyant parcel in the Earths core

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J . Int. (1994) 117,394-402

of a buoyant parcel in the Earths coreK . Moffatt and D. E. Loper

of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, U Koph ysica l Fluid Dy nam ics Institute, Florida State Univer sity, Tallahassee, FL 32306-3027, USA

1993 September 21. Received 1993 September 21; in original form 1993 April 30

SUMMARYThe dynamics of a buoyant parcel (or blob) of fluid released from the mushy zoneon the inner core boundary (ICB) is considered. Estimates of the density defect andof the rise velocity are obtained from consideration of mass conservation andmagnetostrophic force balance. When Lorentz and Coriolis forces are of comparableorders of magnitude, the disturbance remains localized in a neighbourhood of theblob even in the absence of viscous effects, and an inviscid analysis is possible. Theinstantaneous velocity and magnetic field associated with a given localized buoyancydistribution are then determined, and on the assumption that the velocity isapproximately uniform throughout the blob, its trajectory from ICB to core-mantleboundary (CMB) is deduced. Both westward drift and poleward migration oferupting field loops are indicated. The effects of turbulent diffusion on the blob areconsidered. The prospects for constructing a dynamo theory based on this blobscenario are discussed.Key words: compositional convection, core-mantle boundary, geodynamo, liquidcore, magnetostrophic.

1 I N T R O D U C T I O NIt is now widely accepted that the Earths magnetic field ispowered by self-exciting dynamo action associated with fluidmotion in the outer liquid core. The most plausible sourceof this motion is an instability of gravitational originassociated with the cooling of the Earth and the slowsolidification of the inner core (Braginsky 1963, 1964): asthe liquid alloy (iron plus an admixture of lighter elements)of which the outer core is composed, freezes onto the innercore, an excess of the lighter elements remains in the liquidphase which therefore becomes gravitationally unstable. Theinstability modes that are excited are strongly influenced,and indeed controlled, by Coriolis forces and by Lorentzforces associated with the ambient magnetic field, which isgenerally believed to be predominantly toroidal in the outercore.

Thermodynamic arguments suggest that the source of thisinstability is located in the mushy zone at the inner coreboundary (Loper & Roberts 1978, 1981; Loper 1983>--alayer of dendritic crystals which forms in order to allow thesolidification process to proceed despite the low value ofcompositional diffusivity. This layer, estimated to have aneffective thickness of the order of 1km, behaves like aporous medium within which a density defect A p, relative tothe overlying liquid density p , is continuously generated.Experiments on model systems involving the freezing of

ammonium chloride in solution (Chen & Chen 1991)indicate that the gravitational instability manifests itselfthrough eruptions of buoyant fluid in the form of plumesemerging through chimneys that are spontaneously createdin the mushy zone, the lighter fluid being drawn horizontallyinto the chimneys from the surrounding mush. The effect ofCoriolis and/or Lorentz forces on this type of instability isunknown; moreover, it must be admitted that conditions inthe high-pressure liquid-metal environment near the innercore boundary (ICB) are so utterly different from those ofthe model experiments that the preferred patterns ofconvective instability may bear little resemblance to those ofthe experiments. What is clear, however, is that somehowthe buoyant fluid of density p - A p that is created in themushy zone will tend to erupt from this zone and risethrough the liquid core, possibly entraining ambient liquidof density p in the process (Fig. 1). This picture ofconvection is related to that proposed by Howard (1964) forconvection at high Rayleigh number (see Turner 1973,Section 7.3).On this basis, it was suggested by Moffatt (1989, 1992)that it might be appropriate t o treat the resulting convectionin the outer core by considering the dynamics of individualparcels or blobs of buoyant fluid rising with verticalcomponent of velocity w (and with a compensatingdownflow of an equal volume of liquid of density p ) . Thenet downward mass flux is manifest by the slow increase of

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The magn etos trophic r ise of a buoyant parcel 395It is noteworthy that the estimates (4) show nodependence on the length-scale a of the individual blobs.The estimate (3) does, however, require that the convectiveacceleration p u - V u be negligible compared with theCoriolis acceleration 2 p P X U, i.e. the Rossby number

w l a 9 must be small, or equivalentlya >> w / P - 3f- 'I2) m- 0 m (6)using the estimate (5) with (f - . 1 ) . Anticipating that thediscrete plumes of fluid erupting from the mushy layer at thetop of the inner core become unstable and thicken laterally(as seen in the analogue NH,CI experiments) , thelength-scale a of the blobs may be significantly larger thanthe radius of the chimneys from which they emanate. Thechimney radius can be no smaller than the primary dendritearm spacing (e.g. see Sarazin & Hellawell 1988), and the'typical' spacing for planetary cores has been est imated to beseveral tens of metres by Esbensen (1982) by analysis of theCape York meteorite. Consequently it is certainlyreasonable to suppose that (6) is satisfied. We shall, to bespecific, suppose that a is in the range 0.1-100 km in theestimates that follow.There are then three important simplifying features in thesubsequent analysis. First, the only inertial effects that needbe taken into account are the Coriolis effects referred toabove, the acceleration term D u l D t = du / d t + U Vu in theequation of motion being negligible, by virtue of (6).Secondly, the magnetic Reynolds numberR , = w a / q (7 )is relatively small for the range of a considered: withq = 3m2 s- ' (the usual estimate for the liquid core), we findthat R , s 7 f - ' I 2 - 2 0 when as100km. It is thereforereasonable to suppose that the magnetic field perturbation iscontrolled by field diffusion and is determined instan-taneously by the flow U across the ambient toroidal field B,.(The 'low R, approximation generally gives a goodqualitative description when R, is of order unity and evenup to R , approximately 20. Also, the structure of thesolution may permit the linearization to be strictly valid forR , exceeding unity, as was found by Ruan & Loper 1994;the range of validity may be determined a posteriori.)Thirdly, we shall find that viscous effects may be neglected;the formation of Taylor columns parallel to a is suppressedby the magnetic field which (when the Lorentz force is ofthe same order of magnitude as the Coriolis force) ensuresthat the disturbance remains localized in a neighbourhood ofthe blob even when viscosity v+ .Under these conditions, if the field O(x) = A p / p is knownat any instant, then both the velocity field u ( x ) and themagnetic field perturbation b(x) are instantaneouslydetermined, independent of the previous history of the flow.Time dependence and non-linearity enters the problem onlyif we consider the evolution of the buoyancy field under theadvection-diffusion equation

IceFigure 1. Schematic diagram indicating the eruption of a buoyantblob from the dendritic layer (mushy zone) on the inner coreboundary (ICB) .

the radius R , ( t ) of the inner core with density p, = 1.05p,conservation of mass (we neglect change of volumeassociated with solidification of the inner core, an effect thatis simply accommodated by contraction of the whole Earth)implying thatfw A p l p = 0.05R1, ( 1 )where f s the fraction of a spherical surface just outside theinner core occupied by rising blobs. We may reasonablyassume that f < 0 . 5 ; the estimate f -0.1 seems quiteplausible for the purpose of numerical estimates (seebelow). The results appear to be qualitatively insensitive tothe precise value of f provided this is not too small.Assuming a uniform rate of growth of the inner core overthe lifetime of the Earth, we estimateR , = 10-" m s-'. ( 2 )A second relation between w and A p l p results from theassumption that the buoyancy force A pg is (in order ofmagnitude) in balance with the Coriolis force 2 p & x U ,giving the order of magnitude estimatew - g/Q) h p l p . ( 3 )Combining eqs ( 1 ) and ( 3 ) , we then have the estimates

(4)which, with the values S2 = 7 X l o p s s-' and g - 3 m s - ~(near the ICB) give

( 5 )i 12 -AP - x 1 0 - ~ , f112w- x 1 0 - ~mc1.PThese estimates are of course of a very preliminarycharacter, and in particular do not take account of thepossible effect of the Lorentz force on the force balance, orof the latitude dependence of Coriolis effects. Part of theaim of the present investigation is, through detailed analysisof the dynamics of a buoyant blob, to improve on theestimates ( 5 ) , and to examine the general self-consistency ofthe 'blob model' of core dynamics.

where K is the molecular diffusivity of the compositionalvariation that is responsible for the density defect. If U isknown, then eq. (8) may be used to step forward in time,with a recalculation of U at each time step.


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396 H . K . Moffatt and D . E. LoperIn this paper, which may be regarded as an essentialpreliminary to any such time-dependent evolutionary study,we shall limit attention to the instantaneous problem ofdetermining the u(x) and b(x) associated with a prescribedbuoyancy field O(x)g, where g is the local acceleration ofgravity. In particular we shall determine the velocity at thecentre of a localized spherically symmetric blob O(x) ofGaussian structure

O(x) = 6, exp [ -1 ~ 1 ~ / a~ ] , (9)(this is both mathematically convenient and physicallynatural) and we shall use this velocity to deduce the path ofsuch a blob from the ICB to the CMB. We shall alsoconsider some qualitative features of the full velocity fieldu(x), and estim ate the possible effects of turbu lent mixing ofthe blob with its surroundings. Finally, we discuss theprospects for constructing a theory of the geodynamo basedon this type of blob model.Some aspects of the dynamics of plumes (as opposed toblobs) under the influence of Coriolis and Lorentz forceshave been considered in a parallel paper (Loper & Moffatt1993).

2 G O V E R N I N G E Q U A T I O N SThe equations governing the coupled evolution of th evelocity field u(x, t), magnetic field B(x, t) and buoyancyfield O(x, t)g, in the Boussinesq approximation, areDu 1-+22 9Xu= -VP +-B* VB- 0g+YV2u, (10)Dt POPDB- B . u + ~ v ~ B ,DtD eDt- K v 2 6 ,V - u = V B = 0,where D/Dt = d/dt +U V, P = p-(p + B2/2yn), p is thefluid pressure, Y is the kinematic viscosity, r] the magneticdiffusivity of the fluid, and p,= 4n X lO-NA-. T h emagnetic field B is of the form

(13)

B = B, + b, (14)where B,, is the applied field, itself a product of th edynamo mechanism, but here assumed given and locallyuniform and b is a dynamically induced perturbation. Wesuppose that g = -ge is also locally uniform (where e is aunit vector in the upward vertical direction).Suppose that th e typical am plitude of 0 is O,,, nd that thelength-scale of these variations is a. Guided by the estimate(3), we a dop t as velocity scalev = O0g/2Q (15)where Q = 1291 (see eq. 3), and we suppose that themagnetic Reynolds numbe rR, = Va/r] (16)

Ibl = O(R,)Bn. (17)is of order unity or less. It is then well known that

We are thus led to the following scalings: let

Substituting in eqs (10)-(13), and dropping the dashes, weobtain the dimensionless equationsN2 Ro-+ d X U = - V P + (Bo - V)b1 + R,b. Vb + N 2 0 e+ eV2u (10)R,[ - b - Vu] = (B , - V)u + VZbDODt- ,V28V - u = V - b = Owhere d = 29/1Q), Bo = B,/Bo, and whereN 2= 2Qp,r]p/B;Ro = V/2aQE = vr]~~)p/B:a~E = K/aV.The absence of a Rayleigh number in eq. (10) is due to theadoption of the velocity scale (15) characteristic of thebalance of buoyancy and Coriolis forces. N2 the inverse ofthe Elsasser number) represents the ratio of Coriolis toLorentz forces, and will be of order unity or somewhat lessin the conditions of the liquid core. Ro is the Rossbynumber , of order 10-3 in the regime considered. The valueof E (the inverse square of the Har tmann number) issomewhat uncertain since the value of Y in the core is verymuch a matter of guesswork; if we take Y - O-r] (as formolten metals at atm osph eric pressure) then we findE 5 10W8 in the core conditions; however, we will for themoment retain the viscous term in (10) since it involves thehighest derivative and is, unlike the Coriolis and magneticterms, isotropic in structure. Finally, E,, the inverse Pecletnum ber, is likely to be extremely small in the co re, of orde r1OP8 or even less.For the reasons given in the introduction, we now dropthe terms involving the R ossby num ber Ro and the magneticReynolds number R,. Eqs (10) an d (11) then take thesimplified instantaneous, or quasi-static, formN 2 8 X U = -V P + (8, V)b + N28e+ EVUO = (Bn V)U+ V2b (23)(24)wherein O(x) will now be regarded as (instantaneously)known.We may o btain a single equa tion relating u(x) and O(x) bytaking the curl of (23) twice, using (13), and eliminating b.This gives- ~ ~ ( 8 .)Vx U = (B, - V ) ~ UN ~ V (e x ve) - E V ~ U .

(25)If we take the curl again, an d eliminate V X U in favour of U,


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Th e magnetostrophic rise of a buoyant parcel 397we obtain the equationL u = [eV4- B, * V)] + N4(8 V)V}U{

= -N2[eV4 - B,, - V)]V X ( e X VO)+ N4(8 V)V2(e X VO). (26)This linear equation may be easily solved in terms of Fouriertransforms. Let8(k) = IO(x)eik.d3x, (27)with inverse

and similarly for other variables. The transform of eq. (26)is thenD(k)fi(k) = {N2[k4 + (B(, k)]k X (e X k)+ N 4 ( 8 k ) k 2 ( e X k)}8, (29)

(30)whereD(k) = [ ~ k (Bo * k)] + N4k(6 - k).

At first sight, it looks as if the response will be dominatedby contributions from the region of k space where D ( k ) isv_ery small, i.e. (since E


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398 H . K. Moffatt and D. . LoperCase of a buoyant b lo b of spherical Gau ssian structureIn order to evaluate the integral ( 4 1 ) , we need to know theform of &k). Choose origin x = 0 at the centre of the blob,and consider the case of a spherically symmetric Gaussianbuoyancy fieldqx) e-1Xl2 (4 4 )

where k = Ikl. Let us con centrate on the velocity U = u(0) atthe centre of the blob. From eq. ( 4 1 ) , this is given by

(4 6 )Substituting for k X (eX k ) and e X k f rom eqs ( 4 2 ) and( 4 3 ) ,and retaining only those term s which are even in k k yan d k , (and d o not therefore integrate to z ero), we find

m

{ N 4 k 2 k 2 in A , N2k:(k; + k:) sin A , N 2 k f ( k : + k:) co s A }k: + N4 k2 k :X

X ePkzl4k, dk, , dk,. (4 7 )This integral may be simplified by letting k = k K . Integratingover k using eq. 3.461(2) of Gradshteyn & Ryzhik (1980),we obtainU = [Z l (N) in A , 1 2 ( N ) in A , Z,(N) co s A ] , (4 8 )where

(4 9 )

52 being solid angle. These integrals may be simplified (seeAppendix A) to the formdP (52)z l = l - [ P 2Vp4+ ~ ~ ( 1 -2 )

1 + p 2Vp4+ ~ ~ ( 1 -2 )2Z2= Z3= $ N 2I, dP . (5 3 )The functions Zl( N) , Z2(N), Z,(N) are shown in Fig. 2.The asymptotic behaviour of the integrals fo r large valuesof N is easily obtained. Eq. (5 2 ) immediately gives Zl-+ asN + m , while standard manipulation of the integral in eq.(5 3 ) in the limit N - +

To determine the behaviour of Zl as N+O, first rewriteeq . (52) a sgives 21, = Z3= n / 2 .

~ 4 ( 1 - 2 )" lb p4+ N 4 ( 1- p + p2Vp4+ N 4 ( l - 2 ) dP. ( 5 2 ' )

I

I 2 3 5N4Figure 2. The functions I @ ) , 12(N) ,13(N) defined by the integrals(49)-(51). The asymptotic values for N >> 1 are indicated by thedashed lines. Note that Z3(N)= 21,(N) and that I , ( N )= $13(N) foral l N .

The dominant contribution to this integral occurs forp = O ( N ) . Letting p = NC , eq . ( 5 2 ' ) becomes

1 - N2C2IN" = I;" + 1 - N2C2+ C26C4+- N 252dC. (5 4 )A s N+O,

Similarly - dC - N [ r ( i / 4 ) l 2= 1 . 2 3 6 ~ . ( 5 6 )m-z12= Z3 - N I,The curves of Fig 2 show also that the ratio Z,(N)/Z3(N)varies monotonically between 0.707 and 2 / n = 0.637 as Nincreases from zero to infinity. A reasonable workingapproximation for all N is thus given byZ l ( N )= $Z2(N)= fZ3(N). (5 7 )

4 T R A J E C T O R Y O F A B U O Y A N T P A R C ELFor simplicity, we now suppose that the velocity field u ( x )within the blob is approximately uniform and thereforeequal, in first approximation, to the velocity U = u ( 0 )determined by eq. (4 8 ) . (A more detailed analysis of th estructu re of u ( x ) both inside and outsid e the blob is deferredto a later paper. The present analysis, in conjunction withthe considerations of Sections 5 and 6 , provides a reasonablysimple, albeit crude, description.) This approximation findssome support from the fact that the deformation tensor3ui /3x i is zero at x = 0 (from analysis of the symmetries ofthe integrand in eq. 41) and at most of order unitythroughout the sphere Ix l


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The magn etostrophic rise of a bu oyant parcel 399

Figure 3. Notation used in Section 4. The blob follows a helicalpath on the paraboloid Y2 CZ, spreading by turbulent diffusion asit rises.principally to variation of the toroidal field strength as afunction of r and A).Let [ Y ( t ) ,Z ( t ) ] be the coordinates of the centre of theblob in a meridian plane, with origin 0 at the centre ofthe Earth (Fig. 3). Then from eq. (48),

t- Z2(N)sin A = Z2(N)Zr-=ZI,(N)cosA=Z,(N)-

d Ydt

dtd Z

where r2= Y2+ Z2.Hence, since Z3(N)= 2Z2(N),d Z / Z = 2 d Y / Y , (59)

Y 2= cz, (60)so thatwhere C is a constant. This is the equation of a paraboloidof revolution about the axis OZ. The constant C is relatedto the colatitude A() at the point of release of the blob o nr = RI:

The paraboloid intersects the CMB (r = R,) at a value of Y( = Y,) given by Y: + Y;/C2= R g , or equivalentlyY: = $R:{tan A, sin A,dtan2 A, sin2A, + 4(R,/R,)2

- an' A, sin2A,}. (62)

The azimuthal component of velocity on this paraboloid isgiven by the first component of eq. (48); note that this ispositive, and that the drift is therefore in the westwarddirection. It may be argued in the manner of Bullard (1949)that such westward drift is an inevitable consequence ofconservation of angular momentum of a rising buoyant blob;however, it is reassuring to have this heuristic argumentconfirmed by the above detailed calculation which takesaccount of the effect of the toroidal magnetic field as well asof the latitudinal dependence of Coriolis forces. (Sometimesthese can conspire to promote eastward drift-see forexample Moffatt 1978, Section 10.7; Fearn 1979). If now@ ( t ) is the azimuth angle of the blob relative to its point ofdeparture (i.e. @(O) = 0), then

:

d @ YY - = Z,(N) sin A = Z,(N)-dt rHence, using the approximate result I , = $Z2, we obtain

d @ 4dYy - = - -dt 3 dt 'and hence@ = 4 In (Y/Y) ) . (65)The total angle of westward drift during the rise of a blob is

= In ${sec A0dtan2A() in2A, +4(R,/RJ2 - an2 AO}.(66)

More significantly, the velocity of westward drift of a blobwhen it arrives at the CMB (at A =A,) is given byU, = Z,(N) sin A ,, where N should be based on the strengthof the toroidal field within a distance of the order of theblob scale a below the CMB. The toroidal field is likely tobe weaker here than at greater depth, so that the large Nasymptotic estimate I , = 1 is likely to be a good one; thsdimensional westward drift velocity is thenU,=QsinA,.8With the estimate 8,)- 10-' (from eq. 5 with f = O.l), thisgivesU - 0.2 sin A,)mm s-', (68)which is of the right order of magnitude.(dimensional) velocitySimilarly, the blob has a poleward component of

%g Jd%g .Up = ( W sin A, - V cos A,) sin , cos A, (69)2 9 89on arrival at the CMB. Hence, the drift velocity at the CMBis tilted north of west by an angle 6 where

Jdt a n 6 = U p / U-4 -COSA,. (70)For example, at colatitude AI = n/4, 6 = 29", a predictionthat may be testable.The time of rise t , of a blob from ICB to CMB may be


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400 H . K . Moffatt an d D. . Loperfound from the equationdZ Z- 13(N); with r = (CZ + z2)12,dtprovided the variation of N with depth is known. If weassume a constant value of I,, of order unity, then eq. (71)integrates to giveI , (N)t , = [r + i C ln(2r + 2r sin A + C ) ] F g B , (72)which may be readily converted to dimensional form. (Withthe estimates (5) and with f - .1, this rise time is of order100yr.)To summarize, the blob follows a helical path on theunique paraboloid of revolution (60) passing through thepoint of release on the ICB; on this surface, it drifts in awesterly direction, its azimuth angle at time t relative to itspoint of release being given by eq. ( 6 5 ) . On arrival at theCMB, its radial component of velocity is destroyed, and itpresumably spreads out like a plume of buoyant smokerising to the ceiling of a room. This is one mechanismwhereby a relatively stable layer may be created near theCMB, a possibility proposed by Braginsky (1994).Alternatively a weakly stable mean density gradient may beestablished throughout the liquid core (Loper 1989) as in theanalogous problem of high Rayleigh number convectionalready referred to in the introduction (Turner 1973, Section73).5 STRUCTURE OF THE VELOCITY FIELDSubstitution of the expressions (42), (43) and (45) in eq.(41) allows us to extract immediately expressions for thevelocity components ( u l , v l , wl) and (uz,v2, w 2) defined byeq. (38). These may all be written in similar form, e.g.U ] ( X ) =- (73)where D(k) is given by eq. (40) and E,(k) is a homogeneousquartic in k:C,(k) = - N 2 k l k , - N4 k 2 k yk , . (74)The similar expressions for fil(k) etc. arefi,(k) = -N2kZkyk, + N4 k 2 k x k , ,Gl(k) = N2k2(kZ + k:) ,E2(k) = -N2 k fk y +N4 k 2 k : ,Gz(k)= N 2 k f ( k :+ k:) ,Gz(k)= -N2 k :k yk , - N4 k 2 k x k , .It is a complicated matter to investigate even the asymptoticproperties for large 1x1 of integrals such as (73); we shallconfine ourselves here to a demonstration that the integralsare convergent (so that the vital step of setting E = 0 inobtaining eq. 39 is justified). It will be sufficient to considereq. (73); all the other integrals may be treated in a similarway. Substituting (74) into eq. (73), we have8n32U , ( X ) l

The integration over 9 s standard and leaves an integralover q which is clearly convergent. It follows that thevelocity field u , (x , N ) is a well-defined function of position xrelative to the centre of the blob and of the parameter N .The rate of release of gravitational energy due to the riseof the blob is of order :na3(Ap)gw, where w is the verticalcomponent of velocity given (in dimensional form) by

and this is exactly compensated by Joule dissipation; hence,when N is of order unity, this Joule dissipation is certainlyfinite and the disturbance cannot extend far from the blobitself. Detailed asymptotic analysis (which will be presentedelsewhere) does in fact confirm that all components of u(x)fall off as inverse powers of 1x1 with increasing distance fromthe blob. (If either N + O or N + m , this conclusion nolonger holds, and the analysis, which is based on a fluid ofinfinite extent, ceases to be valid.)6 TURBULENT DIFFUSION OF BLOBSThe Reynolds number based on a blob scale a - 10km, arise velocity w - 10-4 m s- l and a kinematic viscosityY - 3 x 10-7m2s-1 (which is just a plausible guess) is% = wa/v - x 106, and turbulent entrainment of thesurrounding fluid is to be expected, by the mechanismsuggested by Fig. 1,which may persist throughout the rise ofthe blob. Actually, the turbulence may be confined to muchsmaller scales (of the order of w/Q - 10m and less-seeeq. 6 ) at which the non-linear term pu.Vu, responsiblefor the cascade of energy to even smaller scales becomesimportant relative to the (linear) Coriolis term 2pBXu;the Reynolds number based on this smaller scale is = wz/QY -3 x 103, so that turbulence on scales SW/Qwill be weak (if present at all) and will have negligi-ble influence on the spread of the blob.In the absence of a theory of turbulence in a stronglyrotating fluid permeated by a strong magnetic field, the bestwe can do is to make rough estimates. Suppose thatturbulent diffusion does cause mixing of the blob with itssurroundings so that its radius at time t is [(t) (withe(0) =a). Let w ( t ) be the upward velocity of the blob attime t , with w(O)= w,,, he velocity of release from themushy zone on the ICB. The worst scenario (involving themost rapid mixing) involves a turbulent diffusion coefficient

where CO s a constant (C O= 0.1 in conventional turbulencecontexts-see, for example, Kraichnan 1976). For thereasons indicated above, we would expect eq. (81) to be amaximal estimate.Mass conservation implies that the density defect Ap must


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The magnetostrophic rise of a buoyant parcel 401decrease with increasing blob radius according to

and the estimate (3) for w then gives( 8 3 )

where we have allowed for the linear increase of g with r inthe liquid core. Now the rate of spread of the blob isdetermined by D and 8 alone, so that

and henceC - a = C0(r- R I ) . (85 )When r = R,, C- - C, , (Rc - R I)- .1 X 2000 km - 00km, so that the scale of the blob is still (just) within therange for which the quasi-static approximation (23), (24) isreasonable. If, as we expect, D s in fact much less than theestimate (81), then of course the spread of the blob iscorrespondingly less.Within the same approximate framework, the radialposition r ( t ) at time t is given by

2=2 (L )3 ( k )t C : r - R I (for r - R I > > a ) .With F = r / R , and k = w,a/C;R:, this integrates to givekt = ( F - 1)(2F2 - 7 i + 11)- n i (87)(see Fig. 4). Note the strong reduction of vertical velocitydue to turbulent spreading of the blob.

This simple analysis is of course quite crude, and does nottake account of magnetic and Coriolis forces (other thanthrough the geostrophic estimate (3 ,83 ) ) , but its merit isthat it indicates the manner of turbulent spreading and thequalitative effect that this has on rise velocity, in thesimplest possible way.Our tentative conclusion from this analysis is that, whilesome degree of turbulent mixing almost certainly occurs, its

I 0.5 I k tFigure 4. Radial position of a blob, subjected to turbulentdiffusion, as a function of time, from equation (87).

effects are not so strong as to undermine the assumption ofcoherent blob rise, upon which the main part of this paper isbased. Furthermore, detailed investigation of the nature ofturbulence and its diffusive properties in such anenvironment is desirable to corroborate this conclusion.7 DISCUSSIONThe rising-blob scenario proposed in this paper provides asimple description of buoyancy-driven flow, which is quitedifferent from the normal description based on the linear(Rayleigh-BCnard) instability of an unstably stratified layer,or annulus, of fluid. In our model, the instability is triggeredin the mushy zone on the ICB, and causes the eruption ofbuoyant parcels, whose subsequent rise through the liquidcore is an essentially non-linear phenomenon (through thenon-linear advection term u.V8 in eq. 8). By adoptingcertain simplifying, but plausible, assumptions concerningthe scale and structure of the blobs, we have been able tobypass the non-linearity, and determine the instantaneousflow field, satisfying the magnetostrophic equation (23) andthe low R , equation for magnetic field (24), associated witha single rising blob.Such blobs presumably erupt more o r less uniformly fromthe mushy zone on the ICB, and follow trajectoriesdetermined by the colatitude A,, at the point of eruption.These trajectories are helical, with westward drift, onparaboloids of revolution with origin at the Earths centre.The fact that each blob satisfies the magnetostrophicequation (with zero viscosity) means that the Taylorconstraint (Taylor 1963; Moffatt 1978, Chapter 12) isautomatically satisfied; there is no need to invoke the effectsof viscosity through Ekman or Ekman-Hartmann layers onthe CMB (as has to be done, for example, in Braginskysmodel-Z-for a recent review see Braginsky 1991). It is animportant feature of the blob scenario that the uncomfort-ably ill-determined viscosity of the liquid core plays no partin the theory (other than being negligible). Also, the poorlydetermined blob size a plays a minor role in the dynamics;all components of the velocity vector are independent of a.Turbulence on sub-blob-scales may, however, be present,and the possible diffusive influence of such turbulence hasbeen considered in Section 6. This will not only diffuse thecompositional variation responsible for the density defectA p, but will also provide an eddy viscosity which could inprinciple be included in a more refined analysis.Longer term objectives must be to understand the role ofsuch rising blobs in developing both the differential rotationand the helicity that are known to be crucial ingredients ofdynamo action. The fact that rising blobs drift westward in awell-defined manner, and the associated fact that fallingelements of heavier fluid drift eastward, ensures that a stateof differential rotation must be established, the inner part ofthe liquid core rotating slightly more rapidly than the outerpart. The source of helicity associated with rising blobs ismore elusive, although the fact that each blob follows ahelical path is presumably a relevant factor; this will be thesubject of a future study.ACKNOWLEDGMENTSThis work has been supported by NATO CollaborativeResearch Grant no. 901017 and by NSF via Grant no.


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402 H . K . Moffatt and D . E . LoperEAR 9116956, by the Isaac Newton Institute for Mathemat-ical Sciences, Cambridge, under the Dynamo TheoryProgramme, and by the Research Institute for MathematicalSciences, University of Kyoto, where the final version of thepaper has been prepared. We thank Paul Roberts for hisconstructive criticisms, and in particular for his contributionof the proof that Z,(N) = 2Z2(N) by the technique presentedin Appendix A. The GFDI contribution number of thispaper is 353.

REFERENCESBraginsky, S.I., 1963. Structure of the F-layer and reasons forconvection in the E arth's c ore, Dokl . Akad . Nauk . S S S R, 149,8-10.Braginsky, S.I., 1964. Kinematic models of the Earth's hydromag-netic dynamo, Geomagn. Aeron. , 4, 572-583.Braginsky, S.I., 1991. Towa rds a realistic theo ry of the geodynamo,Geophys. Astrophys. Fluid D yn ., 60, 9-134.Braginsky, S.I., 1994. MAC oscillations of the hidden ocean of th ecore, J . Geomagn. Geoelect ., in press.Braginsky, S.I. & Meitlis, V.P., 1991. Local turbulence in theEarth's core, Geop hys. Astrophys. Fluid Dyn ., 5 5 , 71-87.Bullard, E.C., 1949. Th e magnetic field within the Eart h, Proc. R.Soc. Lond . , A , 197, 433-453.Chen, C.F. & Chen, F. 1991. Experimental study of directionalsolidification of aqueous ammonium chloride solution, J. FluidMech., 227, 567-586.Esbenson, K.H ., 1982. Chemical an d Petrological Studies in ironmeteorite, PhD thesis, Depar tment of Metallurgy, TechnicalUniversity of Denmark.Fearn, D., 1979. Thermally driven hydromagnetic convection in arapidly rotating sphere, Proc. R. Soc. Lond. , A , 369, 27-242.Gradshteyn, I.S. & Ryzhik, I.M., 1980. Table of integrals, seriesand products, 4th edn, Academic Press Inc., London.Howard, L. N. , 1964. Convection at high Rayleigh nu mber, Proc.XIth Int. Cong. Appl . Mech. , pp. 1109-1115, ed. Gortler, H.,Springer-Verlag, Berlin.Kraichnan, R.H ., 1976. Diffusion of passive-scalar and magneticfields by helical turbulence, J. Fluid Mech., 77, 53-768.Loper, D .E. , 1983. Structure of the inner core boundary, Geophys.Astrophys. Fluid Dyn., 25 , 139-155.Loper, D.E., 1989. Dynamo energetics and the structure of th eouter core, Geop hys. Astrophys. Fluid D yn ., 49,213-219.Loper , D .E . & Moffatt, H.K., 1993. Small-scale hydromagneticflow in the Earth's core: Rise of a vertical buoyant plume,Geophys. Astrophys. Fluid Dy n. , 68, 177-202.Loper , D .E . & Roberts , P .H ., 1978. O n the motion of an iron-alloycore, containing a slurry. I. General theory, Geophys.Astrophys. Fluid Dyn. 9, 289-321.Loper , D .E . & Roberts , P .H., 1981. A s tudy of conditions at theinner core boundary of t h e E a r t h . Phys. Earth planet. Inter.,Moffatt, H.K., 1978. Magnetic field generation in electricallyconducting fluids, Cambridge University Press, Cambridge.Moffatt, H.K., 1989. Liquid metal MHD and the geodynamo, inLiquid Metal Magneto-hydrodynamics, pp. 403-412, ed.Lielpeteris, J. & Moreau, R., Kluwer Academic Publishers,Dordrecht .Moffatt, H.K ., 1992. Th e Ear th's magn etism: past achievements

and future challenges, IUGG Union Lectures, Vienna 1991, pp.1-19, A m . geophys. Un., Washington, DC .Ruan, K . & Loper , D.E., 1994. O n small-scale hydrom agnetic flowin the Earth's core: Motion of a rigid cylinder parallel to itsaxis, J. Geomagn. Geoelect . , in press.Sarazin, J .R. & Hellawell, A., 1988. Channel formation in Pb-Sn,

24,302-307.

Pb-Sb and Pb-Sn-Sb alloy ingots and comparison with thesystem NH,CI-H,O, Metall. Trans., 19A, 861-1871.Taylor, J .B., 1963. Th e magnetohydrodynamics of a rotating fluidand the Earth's dynamo problem, Proc. R. Soc. Lond. , A , 274,Turner, J .S., 1973. Buoyancy Effects in Fluids, Cambridge274-283.University Press, Cambridge.

APPENDIX A: REDUCTION O F THEINTEGRALS (49)-(51)Introducing polar coordinates aligned with K, , we haveK, = cos 0 = p, K* = sin 0 sin + (AllanddQ = dr$ dp.Employing the symmetries of the integrals, eqs (49)-(51)become

2 p4 + p2(1- 2 ) sin2+1 2 = - N 2 1 1 p4+ ~ ~ ( 1 -2 ) sin2+ d+ dP (A4)After integrating in + with the aid of eqs 3.642(3) and 3.647of Gradshteyn & Ryzhik (1980), the integrals may beexpressed as

1 1Z -- + N2Q1-- Q22-3N2 N2Z3= --+1 N2Qo+-Q2

3N2 N2where

The integrals Q, may be expressed in terms of ellipticintegrals of the first and second kind.Only two of Q,,, Q, and Q2 are independent. To see this,write (A9) as

wherea2+ b2= -N4 and a2b2= N4. (All)Making use of eqs 3.152(1), 3.153(1) and 3.154(1) ofGradshteyn & Ryzhik (1980), we see that3Q2- 2N4Ql+ N4Q0= 1. ('412)Using this to eliminate Q2 from eqs (A7) and (A8), the set(A6)-(A8) reduce to the form shown in eqs (52) and (53).

H. K. Moffatt and D. E. Loper- The magnetostrophic rise of a buoyant parcel in the Earth’s core

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