David E. Loper and H.K. Moffatt- Small-Scale Hydromagnetic Flow in the Earth's Core: Rise of a Vertical Buoyant Plume

8/3/2019 David E. Loper and H.K. Moffatt- Small-Scale Hydromagnetic Flow in the Earth's Core: Rise of a Vertical Buoyant Pl

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Geophys. Astrophys. Fluid Dynamics,Vol. 68 , pp. 177-202Reprints available directly from the publisherPhotocopying permitted by license only

0 993 Gordon and Breach Science Publishers S.A.Printed in the United States of America

SMALL-SCALE HYDROMAGNETIC FLOW IN

THE EARTHS CORE: RISE OF A VERTICALBUOYANT PLUME

DAVID E . LOPER

Geophysical Fluid Dynamics Institute, Florida State University, Tallahassee,Florida, U S A

H . K E I T H M O F FAT T

Department of Applied Mathematics and Theoretical Physics, UniversityofCambridge, Cambridge,U K

(Received22 October 1991; in$nalform 28 April 1992)

The steady and transient flow induced by a vertical cylinder of buoyant electrically conducting fluidimmersed in an infinite extent of slightly denser fluid in the presence of a horizontal magnetic field isinvestigated, with the aim of elucidating the small-scale flow within the Earths core. The evolution froma state of rest may be divided into three regimes. For short times [ t O(Lz/v)] lateral viscous diffusion also becomes importantand a quasi-steady state is reached having rise speed of order ( A p ) g L / B ( p u v ) 2 nd lateral extent of orderL2B(u/pv) z .For values of parameters thought to be relevant to the core, the short-time solution lastsroughly an hour and the intermediate-time solution lasts several decades. The long-time solution may notbe relevant to the core as the fluid can rise to the top during the intermediate-time regime. The rise speedsassociated with this plume flow may exceed those estimated from secular variation, but this result issensitive to the size of the density deficit, which is poorly known, and to the particular orientation of theplume that has been chosen.

KEY W O R D S : Earths core, buoyant plume, hydromagnetic flow.

1. I N T R O D U C T I O N

It is now widely accepted that the Earths magnetic field is powered by self-excitingdynamo action associated with fluid motion in the liquid outer core. The mostplausible drive for this motion is compositional convection associated w ith the coolingof the Earth and the slow solidificationof the inner core (Braginsky, 1963; Lowes,1984).As the liquid iron alloy, of which the outer core is com posed, freezes on to theinner core, an excess of the lighter elements remains in the su rroun ding liquid, drivingthe convective motions.

The compositionally buoyant fluid which drives the convective flow may begenerated within a mushy zone at the top of the inner core (Loper and Roberts,

111

www.moffatt.tc


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178 DAVID LOPER AND H. KEITH MOFFATT

1981; Lope r, 1983). This zo ne is a layer of den dritic m etallic crystals which serves tocircumvent the rate-limiting process of compositional diffusion and allowssolidification to proceed m ore rapidly tha n ata non-co nvolu ted interface. This layer,which ha s a n effective thickness of abo ut1 km, behaves as a po rou s medium, withinwhich buoyant fluid, having a density deficit relative to the overlying fluid, iscontinu ously generated. M odel experim ents involving freezing of an aqueous solutionof ammonium chloride (Copley,et al., 1970; Rob erts a nd Lope r, 1983) indicate th atthe fully developed convective flow can take the formof plumes of buoyant fluidem ana ting from th e m ushy zone via vertical chimneys that form spontaneouslywithin it. Alternatively the flow may take the formof rising individual parcels ofbuoyant material, possibly entraining ambient liquid in the process (Moffatt, 1989).Moreover, it must be admitted that conditions in the high-pressure liquid-metalenvironment near the inner core bou nda ry (ICB) areso utterly different from those

of the model experiments that the preferred patternof convection may be quitedifferent from tha t just described. Nevertheless, it is very likely tha t the b uo ya nt fluidgenerated near the ICB will rise through the overlying liquid outer core.

Very little is kn ow n ab ou t the dynamics of small-scale buo ya nt plumes a nd parcelsin a rotating hydromagnetic environment. Even such fundamental properties as thedirection and rateof rise are unknown. In the complex interplay of Coriolis, Lorentzand buoyancy forces, it is not possible to estimate with confidence the rate of rise ofthe buo yan t fluid by m eans of a crude force balance. The goalof this, and subsequent,papers is to quantify the direction and rate of riseof buoyant parcels and plumeswithin the outer core of the Earth, and to identify the spatial scales of the flowstructures associated with them. The ultimate goalis a better understanding of theflow structures associated with the dynamo process andof the dynamical structureof the o uter core.

In the present paper we will concentrateon the relatively simple problem of theflow of a cylindrical plume of buoyant fluid emanating from a chimney in the polarregion of the inn er core a nd rising parallel to the axis of ro tat ion (see Figure 1). Weshall assume th at rot ation and gravity vectors are uniform, constant a nd anti-parallel

and that the magnetic field vector, sustained by large-scale dynamo action, is nearlyuniform on the small scale of the plume:

B = BP +CP +a perturbation .

W ith the chosen geom etry the flow is entirely in the vertical direction. Consequentlythe vertical component of the magnetic field hasno dynam ical effect. Fo r simplicityin what follows we shall therefore setC=O. The imposed field is then

The buoyancy is assumed to be compositional in origin; we will neglect thermaleffects. Furthermore, compressibility will be ignored,so that the density of the fluid


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MHD BUOYANT PLUME IN EARTHS CORE 179

depends only on its composition. We shall assume tha t this flow is the sm allest scaleof motion, so that there is no turbulent motion on a smaller scale which produceseddy diffusivities; diffusion is governed by the mo lecula r diffusivities. Also, molecu lar

material diffusivity is assumed to be negligibly small, so that the composition anddensity of a fluid parcel remains constant. That is,

where A p is prescribed for each parcel. Note thatA p > 0 for buoyant material.The chosen configuration is mathematically both simple and singular in three

respects. First, since the flow is assumed to be one dimensional and parallel to theaxis of rotation, the inertial and Coriolis forces are identically zero. Second, sincethe flow is assumed to be parallel to the surface of the b uoy ant material, tha t surfaceis not deformed by the flow. Third, as we shall see in the subsequent analysis, therise speed in this configuration can be anomalously large.

The difficulty of quantifying the ra te of rise may be a ppre ciated by considering therelatively simple problem of the (no nrota ting) flow of a n electrically cond ucting fluidin a pipe in the presenceof a strong transverse magnetic field driven by a prescribedpressure gradient. In the notation of this paper, equation(17) f Shercliff (1956)quantifies the flow up a circular pipe as

Mk

l+yMk

w=-

where w is the dimensionless flow speed,M is the Hartmann number basedon thelateral size of the pipe [defined by(2.5)], k is the dimensionless local half-width ofthe pipe (in the direction of the applied field) an dyM is the ratio of the conductanceof the pipe wall to that of the adjacent Hartmann layer. Note thatk is a function ofthe coo rdinate n orm al t o the p lane of the flow a nd field. TypicallyM >> 1 and k = O( 1).Fo r a n insulating wall,y = O and w = Mk; he flow speed is large and varies ask. Ify = O( ), then yM >> 1 and w = l / y ; the flow speed (outside the H art m an n layers) is of

unit order and uniform. Note that a relatively small amount of wall conductancestrongly affects the magnitude and structure of the flow.The disparity of flow speed in Shercliffs problem can be traced to the return

pathways available for the electric current, driven by crossing the axial flow in thepipe (forced by the pressure grad ient) with th e transverse magn etic field. Th is curren tserves to transfer the force of the pressure gradient to the p ipe w all. Ifa return pathis available for the curren t throu gh the co ndu cting pipe w all, the transfer of force isefficient an d the flow speed is normal, but if the return pa th is entirely th ro ug h thenarrow Hartmann layer, the transfer is inefficient and the flow speed becomes large.This coupling problem has some similarity to th at between the core an d mantle, withthe sp in-up time being affected bya relatively small amount of mantle conductivity(see Loper, 1971). Note that the flow speed is uniform if the fluid and pipe areelectromagnetically cou pled, but is variable for the w eaker viscous coupling.

The problem considered below has some similarity with the pipe-flow problem,


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180 DAVID LOPER A N D H. KEITH M OFFATT

even though it is an unbounded free-convection flow, rather than a confinedpressure-driven flow. As in the pipe-flow problem, the vertical flow (now forced bybuoyancy) crossed with the transverse magnetic field drives a flow of current in thethird direction, but now there is no fixed rigid conducting boundary through whichthe return current can flow. Consequently the buoyancy force is not efficientlybalanced and the rate of flow can become large. We wish to quantify this flow rate,and the associated spatial structures, in both steady and time-developing flows.

2. DEVELOPMENT O F GOVERNING EQUATIONS

We consider a viscous, incompressible, electrically conducting fluid of variablenon-diffusing density, governed by the following equations:

Dp/Dt = O , V. B = O , v . u = o

The notation is standard; see Roberts (1967).The first task is to nondimensionalize the equations. This is not a trivial exercise

for, as we noted previously and shall see below, the order of magnitude of the velocityvaries strongly with time. Let us assume a dominant balance between buoyancy andLorentz force (in the case that the Coriolis force is inactive) and between magneticinduction and diffusion. This leads to a velocity scale of Wo=(Ap)g/aB2 nd amagnetic-perturbation scale of BWoLop where L is a measure of the lateral scale ofthe buoyant plume. The time scale is chosen so that the AlfvCn speed is unity:To = p p ) 1 ' 2 L / B .Altogether

U = wow*2 , B = B9 + BW0Lop)c*2

V=V*/L , a p t = (To)- d / d t * ,

and

p = p o - ogz +WoLoB2p* .

z is the upward coordinate, parallel to the plume, x is the horizontal coordinate inthe direction of the applied field, and y is the coordinate normal to the plane of theplume and the applied field; see Figure 1. In what follows the spatial scale in the

direction of the applied field will be referred to as the length and that normal to the]flow and field will be called the width.1 With the chosen geometry, we may consistently assume that the density/perturbation Ap depends only on the horizontal coordinates and the dimensionless


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Figure 1 Three-dimensional orientation of the buoyant cylinder of fluid with respect to the appliedgravitational (g), rotational (a) nd magnetic field (B)vectors and with respect to the Cartesian coordinates,x, , z.

variables w*, c* and p* are functions of the horizontal coordinates and time. Theequation governing the transport of density, the continuity equations and thehorizontal components of the momentum and magnetic diffusion equations areautomatically satisfied [with p* = - W,Lc~p(c*)~/2]. he remaining equations are,upon dropping the asterisks,

M - 2 v 2 w- M P ) - w ,+c, = - ( x , ) , (2.3)and

v 2 c- M P ) c ,+w, = 0 , (2.4)where

M = L B ( o / ~ , v ) ~ and P 2 = Gp V . (2.5)

Here M is the Ha rtm ann num ber,P 2 s the magnetic P ran dtl number,f = ( A ~ ) / ( A P ) ~ ~ ~ ,V 2= a2/ax2

+a2/ay2 an d subscripts denote partial differentiation. We shall consider

only situations in which f = O or 1.We shall assume the following parame ter values for the lower portion of the o ute r

core of the Earth: po = 104 kg/m3, g = 3 m/s2, c~ = 3 x 105 a m , p = 4 n x 10-7 H/m,B = 2 x 10-2 7; = 10-5 m2/s,A p = 10-3 kg/m 3. Th e first four of these ar e reasonablywell known while the latter three are quite uncertain. We have assumed a largetoroidal magnetic field (B=200G); he value may easily be smaller by a factor of2


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or 4. The value of the viscosity of the core may be larger by one or more orders ofmagnitude (G ans,1972). Moffatt (1989) estimated the average value ofAp in the coreto be 3 x 1 O P 5kg/m3. We have chosen a value 30 times larger, assuming that only

a few percent of the material in the core is buoyant. Nevertheless the relative size ofthe density pe rturb ation is very small:A p / p = 10-7.

With these estimates P( = 2 x 10-3) is much smaller than one and, for any plumelarger tha n 3 mm in horizontal extent,M is larger t ha n one; typicallyM is very large.Also, note that the Lundquist numberMP, measuring the relative sizesof inertialand Lorentz forces, is large for any plume greater than about14m in horizontalextent. The velocity scale WO s only 2.5 x l o p * m/s, far smaller than the value of10-4 m/s typically quoted from secular variation. However, we shall see that theac tual flow speed a ttained in the vertical plume easily exceeds the latter speed. Themagnetic Reynolds number of the flow,

R, = W 0Lap= Ap)gpL /B2

is small for any plume having a lateral scale smaller than about105 m.

current which flows in the horizontal plane:N ot e th at the m agnetic field variablec is in effect a stream function for the electric

j = BWoo[c,S - x?] . (2.6)Equat ions (2.3) and (2.4) are equivalent to (8) and (9) of Shercliff (1956), except tha t

the magnitudes of the dependent variables have been chosen differently.As inShercliffs case, these equations a dm it Hartm ann -laye r modes having narrow lateralscale of order 1/M. However, they admit a second mode which is missing in thepipe-flow problem becauseof the confined geometry, but which plays an im por tantrole in the following analysis.To see how this mode arises, consider the single scalarope rator formed by combining the steady homogeneous versions of(2.3) and (2.4):

(2-5)c, w ) = OThe familiar Hartmann mode has bothV and d/dx=O(M). The new mode hasa/dy = O( 1) and a/dx = O( /M), making it elongated in the directionof the appliedmagnetic field. This mo de is analogo us to the Taylor-colum n m od e in rotating fluids(Mo ore an d Saffman, 1969). In the limit M+m, the equations dictate that d/dx=O,in direct analogy with the Taylor-Proudman theorem. The elongation serves to

weaken the Lorentz force to a magnitude equal to the viscous force.This mode was apparently first identified independently by Braginsky(1960) andHasimoto (1960); we shall refer to it as the Braginsky-Hasimoto mode, or the BHmod e, fo r sho rt. [The equations governing this mode w ere earlier presented by Lehnert(1952), Shercliff (1953) and Chester (1957) bu t only in situations which did not revealthis mode.] Letting x = M t, the equations governing the BH mode are

w,,+Mcr=O, and Mc,,+w~=O. (2.7)


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M H D B U O YA N T P L U M E I N E ARTH S C O R E 183

From these it follows that w is larger than c by a factor of M , suggesting that theupward speed in the B H m ode is very large. We shall see that this is indeed the case.

The Hartmann-layer modes typically exist adjacent to rigid boundaries to satisfycontinuity of velocity and stress. In the absence of rigid boundaries, as is the case

here, any instantaneous variation of velocityo n a n ar row scale excites Alfven waveswhich transport the variation along the magnetic field lines, leaving a smoothlyvarying velocity. Consequently, we anticipate the absence of Hartmann-layer modesin the solution to the present problem. However, the BH mode is present andimportant, as it carries the return electric current discussed in the introduction.

3. STEADY RISE

The buoyancy of the plume is an applied force which must be transmitted viahydromagnetic interaction or viscosity to a fixed boundaryif a precisely steadysolution is to exist. Alternatively, the flow may evolve with time. However, in analogywith the rise of a buoyant body in creeping flow, a steady solution can be found forwhich the momentum of the flow field is infinite. Any time-dependent flow may beexpected to evolve toward this solution as time increases. It is in this spirit that weseek a steady solution to the present problem. It is noteworthy that the presenceofthe magnetic field makes our cylindrical problem well-posed: in its absence, there isno solution (Lamb, 1932, art. 341).

Let us consider a top-hat density distribu tion for the cylindrical plume: letf beunity for Y < R an d zero for r > R , where R(8) s continuous an d positive in - c< 8 < c.Here Y and 8 are standard cylindrical (polar) coordinates: see Figure2. Further

f = O

A

Y

B

Figure 2 Two-dimensional orientation of the horizontal cross-section of the buoyant cylinder of fluidwith respect to the Cartesian coordinatesx, y and the polar coordinates r , 0. The buoyant fluid, havingf = , is confined within the interior regionr < R ( 0 ) ,while the no n-buo yant exterior [lying inR(O)


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assume tha t R = O( ), M >> 1 and (dRld8) an 8 +R = 0 only for two isolated values of8. The last of these assumptions implies that the linesof the imposed magnetic fieldcut the bou nda ry of the plume a t most twice and th at they are parallel to the b oun daryonly at two isolated points, characterized by the maximum and minimum values of

y. This assumption, made for mathematical and conceptual simplicity, could berelaxed.

The steady governing equations are

M - 2 V 2 ~c, = u[r - R(8)l - , (3.1)

v 2 c + w x = o , ( 3 4

where U is the heaviside step function[ U ( [ )= 0 if [ < 0 and = 1 if 0 < I .The boundaryconditions are:

at r = R ( Q c, w, c, and w, are continuou s (3.3)

where n is the normal coordinate in the direction ofP - Roe, and

as i -x , c, w - t o . (3.4)

A difficulty in solving the steady prob lem is to determine the magnitude of the flow.In the interior region ( r< R, denoted by superscript I) where f 1 the governingequ ation s simplify at leading ord er in powersof M - ' to

cI,= -1 , V2cI+wIx=0 * (3.5)

The first of these determines thatc = O( ), but the second leaves the magnitude ofw' undetermined. Th e equa tions (2.7) governing the BH mode indicate tha tw = O ( M c ) ,and the boundary conditions require continuity of the variables. (Recall that theHartmann modes are absent.) Consequently we anticipate thatw = O ( M ) in theinterior region. No w the do min ant-o rder solution to (3.5) may be expressed as

c ' = - x + h ( y ) , and w ' = M k ( y ) , (3.6)

where h and k are unit-order functions to be determined. [The particular solution ofthe second of (3.5) is of smaller order.]

Th e exterior solution (forR < , den oted by superscriptE ) consists of the BH mode.The variables in this mode may be rescaled as

where


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M H D BUOYANT P L U M E IN EARTHS CORE 185

With this scaling, the inner region containing the plume collapses to5=0. F or0 < 5 decaying modes are characterized byW+ E= 0 and governed by

while for 5< 0, W- = 0 and-

s = - $YY (3.10)

These equations are identical to Oseens equation for slow viscous flow [e.g., see Eq.(20), p. 611 of Lamb, 19323. How ever, Oseens eq ua tion applies to transverse motionof a cylinder, not the parallel motion considered here.

Continuity of velocity at the plume boundary requires thatW ( 0 ,y )= k(y). The

solution satisfying this condition is:

Continuity of catr = R requires tha tE k 0 at x = x, and x,, respectively, i.e.,

O=x,-h+k, O = X R - h - k , (3.12)

where

X , = ( RR 2 2)/ and X, = - R, - )/ (3.13)

R, and R, being the right-hand and left-hand values ofR at a specified value of y(Figure 2). To dominant order in M , continuity of V w and Vc are automaticallysatisfied by the lack of H ar tm an n layers. The so lution of (3.12) is

h = [ x R +xL]/2 = [(R, - ) - R, - 2 ) ] / 2 , (3.14)

k = [ x R - x , ] / 2 = [ ( R R 2 - y 2 ) 1 / 2 +(R,Z-y)12]/2 . (3.15)

Note that h is the local center of the plume andk is the local half-width. Whennecessary k will be assum ed to be zero a t values ofy for which X , and x , are undefined(i.e., outside the plume).

It is instructive to verify that the flux of electric current is continuous across theboundary of the buoyant plume, and thus provide a validation of the solutions (3.6)and (3.11).We will consider only the right-hand portionof the boundary, denoted by

x - R(y)= constant;

the analysis of the left-hand portion is identical. The local normal vector is

n = S - dx,/dy)Q , (3.16)


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186 DAVID LOPER A N D H. KEITH M O F FAT T

Using (2.6) an d (3.16) we m ay write

j n = BWoaj,

where2c d x d c

3 y d y 2 xj = - + R -

(3.17)

(3.18)

We wish to show that j, is continuous across the plume boundary.

be expressed asUsing (3.6), (3.14) an d (3.15) the flowof curren t from the interior of the plume may

jf,= - d k / d y . (3.19)

To find the flow of current to the exterior of the plume, we must use (3.11) evaluatedas 5-0. It may be shown that thex derivative of this expression is small,of order1/M. Consequently only the y derivative in (3.18) con tributes to the cu rrent flowexterior to the plume. Now

or integrating by parts and changing variables,

i?CE dk (Y- 1) OC dk- 5,y )= -1 - g) exp[ c.] g = 5 - y+ 2 d . z )exp [ - ] dz2Y 2 f i -Eddy J;; -mdY

In the limit 5-0, the integral is simply evaluated:

dkj f = l i m -((s ,y) = - - - ( y ) ,< - 0 [ti ] d y (3.20)

Eq ua tio ns (3.19) an d (3.20) are identical, ensuring con tinuity of flow of electric curren t.As noted by B raginsky (1960), the solution forW(4, ) is functionally identical to

th at for the flow of heat in an infinitely long rod with the5 direction (i.e., the directionof the magnetic field) being th e time-like variab le (more precisely151 acts as time)and the velocity distribution prescribed at the plume[W = k(y ) ]playing the role ofthe initial temperature distribution. It follows from this that the integral

(3.21)

is a constant independent of5 and that the momentum of the flow is infinite. Notethat the integral of k in (3.21) is equ al t o half the cross-sectiona l area of the plume.


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M H D B U O YA N T P L U M E I N E ARTH S C O R E 187

0 0 1 0 2 0 3 0 4 0 5 @6 0

7 0843

9 0I 0 0

( a )

1 . 0 r

e . O 0 . 5 E . 5

( b )Figure 3 Isoline plots of the norm alized steady solution (3.11) forW((,y)= -sgn(#((,y) in the firstquadrant of the 5-y plane for the case of a circular cylindrical plume. HereW=w/M, C=c and ( = x / M .Figure 3a shows the large-scale structure, while Figure 3b shows the detail near the origin. The solutionis symmetric with respect to both5 and y . The integral of this solution over all y is independent of 5.

At large values of 141, the velocity distribution is approximately

(3.22)

This solution is equivalent to tha t given in (1.14)of Braginsky (1960) an d in(3.7)ofHas imoto (1960). N ote the slow rateof decay of the velocity with distance along themagnetic field (i.e., in the 5 direction) a t constant y .


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For 141


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M H D B U O YA N T P L U M E I N E A RTH S CORE 189

extending from the plume alon g the m agnetic field in bo th directions. The m om entu mfor each increases as MPAtI2 .

For very short times the buoyant fluid within the plume accelerates (relatively)rapidly from rest while the non-buoyant exterior fluid remains a t rest:wl= M P t and

wE= 0. The discontinuity in velocity thus induced is propagated along the magneticfield lines as Alfven waves which travel at unit dimensionless speed. These wavestraverse the buoyant region in time of order unity, leaving an upward flow ofmagnitude M P . Since P


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190 D AV I D L O P E R A N D H. K E I T H M O F FAT T

J

Force-freeregion

l yLX B

c

Figure 4 A cartoon of the structure of the two-dimensional Alfven mode, showing the flow of electriccurrents and the forces. Region1 is the buoyant plume and region3 is an image of the plume which ispropagating to the right with the Alfven speed;a second image (not shown) propagates to the left. Theplume image has traversed region2. In region I the upward buoyancy force is balanced by a downwardLorentz force resulting from a lateral electric currentJ of (dimens ional) density( A p ) g / B nteracting withthe applied magnetic field having magnitudeB . In region 2, force-free electric currents flow parallel tothe applied magnetic field. The electric circuit is completed by a lateral flow of current in region3. Thislateral current interacts with the applied field to produce an upward Lorentz force which accelerates the

fluid from rest to (dim ensional) speed of magnitude(Ap)gL(p /p ) ' I2 /B . he fluid in regions 1 and 2 risessteadily with speed of that same magnitude.

where x R and x L are the right and left portions of the plume boundary given by(3.13) and h is the local center of the plume given by (3.14).

No te th at the s tructu re of the velocity(4.4)behind the Alfven wave is identical toth at of the steady state, given by(3.18),except that the orderof magnitude differs bya factor P . We shall see that this similarity of velocity structure does not prevaildu ring the intermediate time interval.

These nondiffusive solutions prevail as long as viscous and resistive effects aresmall. With P K ~ ,esistive effects become im po rtan t first, on a time of orderM P .When t > O ( M P ) ,an intermediate-time balance between lateral (in they direction)ohm ic diffusion and induc tion prevails in the magn etic equa tion, but the viscous termremains small provided t < O ( M / P ) .n this regime, the two-dim ensional Alfven wavesolution is strongly modified, but a one-dimensional mode persists, as we shall nowshow.

4 .2 One-dimensional AlfvCn mod eLet us consider the problem for the integrated variables

(4.7)


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formed by integration of (2.3) an d (2.4) with respect toy. These give

M - W,, - MP ) +C , = 0 ,C,, - M P C ,+ W, = 0

with conditions

andC(x,0)= kqx, 0 )= 0

(4.10)C(f , )= - gn (x)A/2 , W,(O, t )= 0

where A is given by (4.2).The solution of problem (4.8)-(4.10) may be found in Roberts (1967, p.xx). A pair

of one-dimensional Alfven waves propagate along the magnetic field lines from thebuoyant region a t unit speed. The wave fronts thicken by ohm ic diffusion as( t /MP).The viscous term is negligible. Behind the wave fronts

C = - gn (x)A/2 and W = M PA / 2 . (4.11)

The latter result is easily seen from integrat ion of (4.3). These results appa rently holdfor all time. However, the steady solution has an averaged m omentum which is largerby a factor of 1/P. We will discuss the resolution of this discrepancy in Section 4.4.

4.3 Intermediate-time regime: ohmic difuusion( M P c c cc M I P )

As discussed above, the equations valid for intermediate-time, i.e., forO ( M P )< < O ( M / P ) , re

w,= MP c, , (4.12)

and

c y y + w x = o . (4.13)

These equations contain lateral ohmic diffusion, but no viscous diffusion. They maybe combined into a single equation

M P w x x = - w y y f.

From this it is apparent that the Lorentz force produces a diffusive-like term in thex direction and that the magnitude of the effective diffusive coefficientM P is large.Thus the Lorentz force introduces an anisotropy in the diffusion of momentum,reminiscent of Taylor dispersion.


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Equ ation s (4.12) an d (4.13) are to be solved subject t o conditions

The first condition is the forcing for the problem, resulting from the balance ofbuoyancy an d L orentz forces in the mom entu m e qua tion. Strictly speaking it shouldbe applied a t x = h f , but since the length scales in thex direction of interest in thepresent section are large, the application at x= 0 introduces a negligible error. Thesecond is a decay condition. The third ignores the velocity generated during theAlfven-wave phase, o n the a ssu mption that the Alfven waves have travelled a negligiblysmall distance while t g O ( M P ) .Note that c is an odd function of x whilew is even.

The parity of c and w with respect to y is the same as that of k.Taking the complex Fourier transform in they direction using

(4.15)

equations (4.12)-(4.13) become

f i z = M P t , , (4.16)and

d x = q 2 t , (4.17)

with conditions

This problem reduces to that for the integrated variables considered in Section4.2if q = O ; note that 6(0)=A/2.The solution of problems (4.16)-(4.18) is

a ( x , q , i ) = l ~ l L ( q ) [ 2 ~ e x p (P t - * i ) - l q x l e r f c ( ~ ) ] , (4.19)4 M P t 2 J M p t

(4.20)

It may be seen from this solution that ifq is of unit order, w is of order (MPt) anddecays in the direction of the applied magnetic field on a length scaleX = O(MPt).[In the range X


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M H D BUOYANT PLUME IN E A RT H S CORE 193

Rather than attempting to invert(4.19)and (4.20)for w and c for all values of x,let us concentrate on the value of the flow speed near the plume. Its transformisobtained by setting x=O in (4.19):

Let us investigate this further for a circular plume, with

k ( y ) = m

(4.21)

(4.22)

Noting that k is an even function of y and using formula #3.752.2 on p. 419 of

Gradshteyn and Ryzhik (1980)we have

(4.23)

Now, for q>O,

fi(0, q , t )= J2Mpt J,(q) (4.24)

and

(4.25)

Using formula # 6.671.2on p. 730 of Grad shte yn and Ryzhik (1980)with a = 1,and v = 1 , and noting that cos [arcsin( y ) ] ( 1 -Y)~, we have that

= y

for y < l :

for l < y :(4.26)

The inversion of (4.20)for x = 0 reproduces the boundary condition(4.14).It may be seen from (4.26) that the vertical speed near the plume isof the order

of (MPt) in the time interval O ( M P )< < O ( M / P ) .As t+O(MP) , then w + O ( M P )

and X + O ( M P ) , in agreement with the short-time solution, and ast+O(M/P) , thenw + O ( M ) and X + O ( M ) , in agreement with the steady solution. In the latter limit,the lateral viscous shear becomes im por tant. F or larger times, the dom inan t balanceis given by the steady solution of Section3.

The structure of the vertical speed in the intermediate-time regime, shown in F igure5, is in marked contrast to the two adjac ent time regimes in which the flow structure


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194 DAVID LOPER A N D H. K E I T H M O F FAT T

-3-

T

Figure 5 A plot of (4.26) showing the normalized vertical velocitywi a t x = O as a function of lateraldistance y in the intermediate-time case for a plumeof circular cross-section. Herewi = w/(MPt)/*. Thevelocity is symmetric in y and has a zero average in y. The singularity shown here does not occur forother x values.

mir rors th at of the bu oy ant region [see (3.23) an d(4.3)]. This solution has severalno tew orthy properties. First, the velocity is negative forIyI > 1; althoug h the buoyancyis entirely positive (inducing up ward flow), a s trong do wnw ard flow is induced adjacentto the b uo yan t region. Fu rther m ore, this velocity is singular at the edge of the buo yan tregion for x = 0. (This singularity is no t present a tx # 0.) Finally, al tho ug h the velocityscale (MPt) is of larger or de r tha n th at of the sho rt-t im e regime, the solution givenby (4.26) is zero w hen averaged overy . Th us it is not in co ntrad iction with the resultsof Section 4.2 concerning the averaged properties of the solution.

To understand how a negative velocity may be induced, consider the situation

shown in Figure 6, which isto be contrasted with that shown in Figure4. Fo r t imet > O ( M P )ohmic diffusion causes the electric currents to diffuse laterally (in theydirection) and to lose their force-free character. T he co m bin atio n of these two effectsis to introduce a do wn wa rd force o n the fluid laterally adjacent to the plume, as shownin Figure 6.

It is of interest to determine how the intermediate-time solution found in thissubsection merges with the Alfven solution whent = O ( M P )and with the steadysolution when t = O ( M / P ) .These merging solutions are developed in the nextsub-section.

4.4 Merging solutions

The purpose of this sub-section isto show tha t the short-time, intermediate-time a ndsteady solutions merge smoothly as time increases. To analyse both cases at once,consider the unsteady equations with only the longitudinal (x-direction) diffusive


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M H D B U OYAN T P L U M E IN E A RT H S C O R E 195

X

Figure 6 A cartoon of the flow structure near th e plume in the interme diate-time case, showing the effectof lateral ohmic diffusion; compare with Figure4. Electric current diffuses from region2 into region 4 ,where its lateral component interacts with the applied magnetic field to produce a downward Lorentzforce. This force induces the negative valuesof w for y > 1 shown in Figure 5 .

terms being ignored:

W ,= M P c , + (P /M)w, , ,

M P c ,= W , +c,,

(4.27)

(4.28)

(4.29)

and consider the forcing to be

c( f ,Y , )= - gn (4 (y ) .

Taking the Fourier and Laplace transforms of(4.27)-(4.28)yields

s @ = M P 2 , - q 2 ( P / M ) @ , (4.30)

MPsc"=@,-q2c", (4.31)

with

E(*o,q ,s) = - gn ( x ) l ( q ) / s (4.32)Here a tilde denotes the double transform anda caret the Fourier transform.


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The solution of this problem is

where

(4.33)

(4.34)

(4.35)

Note that the limit as s+O of sG(x,O,s) yields MP&(O),n agreement with the resultof the integrated problem presented in Section 4.2.The value of w for x=O is of particular interest. Its double transform may be

expressed as

G(O,q,s)=M[?] /-.P Z s+ Pq2M s + P q 2

(4.36)

Before proceeding with the merging solutions it is helpful to establish that (4.36)reproduces the previous results. First, ifq2/MP


21/26


The Laplace inverse of this is

G(O,q,t ) = M P & ( q ) G ( q 2 t / M P )

whereG(z)= e-'''[( 1 + z)I,(z/2) + zI,(Z/2)1.

(4.41)

(4.42)

Y

I I I

2 . 0 2 . 5 3 . 0-/

W/MPd r , Y0 . 0

0

- 1 , 0

- 2 , 0

( b )

Figure 7 A plo t of (4.43),giving the vertical speed atx = O for a plume of circular cross section, showingthe merging of the Alfven solution(4.3)with the intermediate solution(4.26).This is done in two partsdue to the different asymptotic scalings. Part 7a shows the normalized vertical velocityw / M P versus yfor normalized timeT~ = / M P equalling 0, 0.1,0.2,0.5 and 1.0, while part 7b shows the normalized velocityw / M P T ~ ersus y for z1= 1.0, 2.0, 5.0 and 30. The curve for r l =O is the solution of (4.3),those labeledT~ = 1.0 n each part are identical, and th at labeledT~ = cc is the solu tion of(4.26).The velocity is symmetricin y.


22/26


Let us consider specifically a circular plume. Using(4.23)we have that

w(0, y , t )= M P j [y](2)os (qy)dq. (4.43)Figures 7a and 7b show curvesof w / M P and w / M P J z , versus y for several valuesof z1 = t / M P. This shows the smooth merging of the short-time solution givenby

W/MP

Y

I I I I I I7 2 = 1 0

0 0 0 5

- 1 0 -

I

- 2 . 0 L

- 2 e L( b )

Figure 8 A plot of (4.46),giving the vertical speed atx = O for a plume of circular cross section, showingthe merging of the Alfven solution(4.26)with the intermediate solution (3.19). This is don e in two partsdue to the different asymptotic scalings. Part 8a shows the normalized vertical velocityw / M T , versusy for normalized time 7 , = P t / M equaling 0, 0.1, 0.2, 0.5 and 1.0, while part 8b shows the normalizedvelocity w / M versus y for-z,=l.O, 2.0, 5.0 and CO. The curve for T,=O is the solution of (4.26), thoselabeled T , = 1.0 in each part are identical, and that labeledT , = CO is the solution of (3.19). The velocity issymmetric in y .


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M H D BUOYANT PLUME IN EARTHS CORE 199

(4.3) with the intermediate-time solution given by(4.26)as varies from 0 to CO.No te that this solution forx = 0 is singular near y = 1; his singularity does not occurfor other values of x.

4 .4 .2 . Merging of intermediate-time and steady solutionsThe solution which merges the intermediate-time solution(4.26) with the steadysolution (3.23)may be studied by assuming thatIs1 =O (PqZ /M )


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200 DAVID LOPER A N D H. KEITH MOFFATT

give way to a diffusively growing mode. Lateral diffusion of the electric currentsinduce dow nwa rd flow adjacent to the b uoya nt region.

For intermediate times [O(L2ya) < O(L2/v)] he velocity and its length scalegrow as the square-root of time:

Note that in this regime the rise speed is independent of the size of the plume. The

plume width remains of unit order, althou gh the lateral struc tur e changes significantly.For t = O(L2/v) ateral viscous diffusion becomes important and the magnitude ofthe velocity and its length scale cease to grow with time. The region of upward flowbegins to diffuse laterally, increasing the width.

For O(L2/v) the flow reaches a quasi-steady state with

It should be emphasized tha t these rise speeds are much greater th an th at estimatedfrom a balance of buoyancy and Lorentz forces.

Let us consider now why the trad itiona l m etho d of scaling the governing e qu ation sfailed to yield the correct order of magnitudeof the vertical flow in steady state.Within the buoyan t region the domin ant balance within the mom entum equa tion isbetween the Lorentz an d buoyancy forces. Tha t balance correctly yields the magn itudeof the perturbation field c x(Ap)gLp/B. The balance of magnetic induction anddiffusion then yields the velocity scaleWO. owever, it is easy to overlook the factthat this represents only the difference of velocity that can occur on the lateral scaleof the buo yan t region, but in the absence of any rigid bou nd ary, does not determinethe magnitude of the velocity.

To determine the magnitude of the velocity, we must consider the balances whichoccur in the non -bu oy ant fluid su rrou nd ing the plume: between L orentz a nd viscous

forces in the momentum equation and between magnetic induction and diffusion inthe magnetic equ ation . However, it is crucial to no te tha t the gradient ope rato rs inthe momentum and magnetic equations(2.1)have a different length scale from thatof the Laplacians. The Laplacian operators are dominated by lateral variations onthe scale, L, of the p lume, bu t the gradients (in the direction of the applied field) areassociated with a much longer scale,X . The balances are Bc/pX=pvw/L2 and


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M HD BUOYANT PLU ME IN EARTHS COR E 201

Bw/Xx c/apL2. These directly yield the leng th scaleX = L M , where M is given by(2.5), and with c known, we have the velocity scalew = M W O ,which is the correctorder of magnitude of the upward speed.

One remaining point which is unresolved is the discrepancy in the integrated

momentum given by (4.11) an d tha t of the steady s tate (3.23) which is larger by afactor P. This discrepancy is du e to the do uble limitt-m and y-CO, or equivalentlyin the double transforms,s-0 and 4-0. It m ay b e seen from (4.34) th at if we ta kethe limit 4-0 first, then

WA , s) =MPC44)lsI 9 (5.1)

which agrees with the averaged results of Section 4.2. O n the o ther h an d, if we takethe limit s-0 first, then

which agrees with the results of the steady solution given in Section 3. One maythink of a broad region of negative flow occurring at very large times and for verylarge values of lyl which cancels out the large m omentum given by (5.2), giving a naveraged value consistent with (5.1).

With the parameter values previously chosen for the Earths core plus Lz 00 m,we have that VAx 1.8m/s, g x 3x 1 0 - m / s 2 , L / VA x 5 6 s z l m i n u t e ,L 2 p ox 3800s x 1 hour, andgL/VA(pov)12 8.6x 10 -3 m/s. If this model were applicable to the core, the flowspeed would accelerate from rest to 1.7x 1 0 -5 m/s in the first minute, stay a t th atvalue for the next hour, then grow diffusively for the next 30 years to a terminalspeed of 8.6 x 10-3m/s. However, with these values, it would take the plume only16 years to rise to the topof the core. Therefore, the term inal s tate would be irrelevantto the core. The rise speed at the top of the core would be roughly 6x 10-3m/s ,which is significantly larger than the typically quoted value of 10-4m/s for corespeeds. Also, the length of the plume (in the direction of the applied magnetic field)would be 2.5x 10 6m , which is roughly the depth of the outer core.

The estimates in the previous paragraph are for purposes of illustration only. It isnot meant to imply that this model is directly applicable to the core. Nonetheless,the calculation is instructive in tha t is shows th at it is dynamically possible to achievegeophysically large speeds in the core with compositionally buoyant plumes havinga density excess of on ly one p ar t in 107. Since the velocity scales directly with thedensity con trast A p , the effect of differing magnitudes is easily estimated . In par ticular,if A p is 10-100 times smaller than assumed here, the flow speedis more in accordwith estimates from secular variation.

The model we have studied here is singular in that the effects of rotation arecompletely absent. Further studies currently in progress will consider the effect ofthe Coriolis force on the vertically oriented plume at lower latitudes in the core, aswell as buoyant material having more general configurations suchas tilted plumesand spherical and irregular blobs.

L2/vz og s x 30 years, gL/VAx 1.7 x 10-5 m/s


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202 D AV I D L O P E R A N D H . K E I T H M O F FAT T

Acknowledgements

This work was supported by the National Science Foundation through grant#EAR-9116956, and bythe North Atlantic Treaty Organization through Collaborative Research Grant# 9101 17; this iscontribution342 of the Geophysical Fluid D ynamic s Institute, Florid a Stat e University, Tallahassee, FL .

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Documents

David E. Loper and H.K. Moffatt- Small-Scale Hydromagnetic Flow in the Earth's Core: Rise of a Vertical Buoyant Plume