A study of conditions at the inner core boundary of the earth

302

Physics of the Earth and Planetary Interiors, 24 (1981) 302—307 Elsevier Scientific Publishing Company, Amsterdam — Printed in The Netherlands

A STUDY OF CONDITIONS AT THE INNER CORE BOUNDARY OF THE EARTH DAVID E. LOPER 1 and PAUL H. ROBERTS 1

*

2

Department of Mathematics and Geophysical Fluid Dynamics Institute, Florida State Univerdty, Tallahassee, FL 32306 (U.S.A.) 2

School of Mathematics, The University, Newcastle upon Tyne NE1 7R U (Great Britain)

(Received February 26, 1980; accepted for publication June 27, 1980)

Loper, D.E. and Roberts, P.H., 1981. A study of conditions at the inner core boundary of the Earth. Phys. Earth Planet. Inter., 24: 302—307. A study is made of the thermal and compositional conditions which the liquid outer core must satisfy at the inner core boundary, assuming the inner core to be growing by continual solidification of the heavy component of the liquid alloy in the outer core. It is found that the outer core is strongly destabilized by the compositional gradients driven by the separation process associated with the freezing. Further, it is argued that all the freezing necessary for the growth of the solid inner core cannot occur on a flat interface; most of it must occur above the solid boundary in a region labeled the slurry layer.

1. Introduction The F1ayer in the Earth’s core was postulated by Jeffreys in 1939 to explain the short-period precursors to the phase PKIKP at epicentral distances between 125 and 143°which had been discovered by Gutenberg (Jeffreys, 1976). The original explanation was in terms of a region lying between the liquid outer core and the solid inner core in which the compressional wave speed was anomalously low. Subsequently Bolt (1962, 1964) suggested that the speed in the F-layer was anomalously high with a sharp jump at the top of the layer. These explanations were not completely satisfactory because the data were insufficient to determine the depth and structure of the layer and no theory was available. When better data became available with the advent of large seismic arrays such as NORSAR, the evidence resolved the question of the precursors in favor of scatterers at or near the mantle— core boundary, and the need for an F-layer to explain *

Contribution no. 161, Geophysical Fluid Dynamics Institute.

these seismic observations was removed (Cleary and Haddon, 1972; King et al., 1973; Haddon and Cleary, 1974; Husebye et al., 1976). Consequently,most Earth models constructed in the last few years have omitted the F-layer (Dziewonski et a!., 1975; Gilbert and Dziewonski, 1975; Anderson and Hart, 1976). However, the seismic data do not rule out the possibility of an F-layer (Haddon, private communication, 1979) and recently Stevenson (1980) has presented evidence which appears to support the view that the lower portion of the outer core is of a different cornposition than the upper portion. This evidence is a plot of dK/dp against p for several Earth models which shows a sharp drop near the inner core bound. ary. Unfortunately, the strength of this evidence is considerably weakened if the data from model 1066A of Gilbert and Dziewonski (1975) are included, since this model gives a sharp rise near the inner core boundary. We are forced to conclude that the cornposition and thermal regime of the lower portion of the outer core are unknown at present. In this paper we shall study the portion of the outer core immediately above the solid inner core

0031-9201/81/0000—0000/$02.50 © 1981 Elsevier Scientific Publishing Company

303

from the thermodynamic point of view and present a new theoretical argument which strongly suggests that this region may indeed be anomalous. The core will be modeled as a binary alloy composed of heavy constituents (Fe and Ni) and a light constituent (FeS, FeSi or FeO), with the precise nature of the constituents being irrelevant to the analysis; and it will be assumed that the solid inner core has formed by freezing of material from the liquid. This model has been developed in part to explain the power source for the geodynamo (Braginsky, 1963; Loper, 1978a, b). The growth of the inner core is a continual process as the Earth gradually cools at a rate of 50—100°C! i09 y (Davies, 1979;Schubert and Stevenson, 1979; Stacey, 1980). This growth implies certain boundary conditions on the liquid outer core. We shall develop these boundary conditions and investigate their consequences for the thermal regime of the outer core. 2. Boundary conditions for the outer core We begin by supposing the core to be cooled sufficiently rapidly that convection occurs and the ternperature gradient of the liquid outer core to be nearly adiabatic away from rigid boundaries —

dTAJdp aT!pCp (1) where p is the pressure, T the temperature, p the density, a the coefficient of thermal expansion and Cp the specific heat at constant pressure. The mixing associated with the convective motion maintains the liquid alloy in a state of nearly constant composition and the gradient of the liquidus temperature is given by the Clausius—Clapeyron equation dTL/dp = T 6/L (2) —

where L is the latent heat and 6 the change of specific volume upon melting. We shall assume that aL
(3)

so that the adiabat intersects the melting curve at the bottom of the liquid and the “core paradox” (Kennedy and Higgins, 1973) is avoided. Assuming Lindemann’s law to be valid, (3) is satisfied provided that y> 2/3 where y = aK/pCp, K being the adiabatic incompressibility. Current opinion (Irvine and Stacey, 1975; Stevenson, 1980) is that (3)is well satisfied within the core.

Immediately above the rigid irmer core boundary convective motions are suppressed and the latent heat released by the freezing process must be removed from the interface by conduction into the fluid, giving (4 k dT 1dr = —Lth C’

where k is the thermal conductivity, r the radius and t~the mass per unit area per unit time of material frozen. In writing (4) we have neglected the heat which must be conducted from the solid to the liquid as the inner core gradually cools. This is much smaller than the latent heat and has been ignored for simplicity. (Its inclusion would only strengthen the conclusion we shall obtain.) Also in writing (4) we have assumed all freezing to occur on a flat interface separating the solid and liquid. We shall show that this assumption leads to a contradiction and discuss the resolution of the apparent paradox. To a good approximation the Earth is in hydrostatic balance dp!dr=—pg

(5)

allowing (4) to be written as dT~,’dp= Lth/pgk

(6)

Let us now consider the compositional effects of the freezing process, assuming that the mass fraction ~ of the lighter constituent is less than the eutectic value ~ [In a similar study of the F-layer, Braginsky (1963) assumed ~ > ~e but his model encounters serious difficulties in removing the latent heat from the freezing interface (Loper, 1 978b).] It is known from metallurgical studies that the solid which crystallizes from such a liquid has a mass fraction ~ of light component less than that of the liquid. For simplicity we shall assume that ~ = 0. As a liquid layer of mass dm per unit area freezes on to the solid inner core in time dt, the light mass ~dmmust be expelled into the liquid above and replaced by heavy material. This generates a vertical flux i of light material relative to the mass center given by =

(7)

To express this boundary condition in terms of the compositional gradient, we note first that quite generally (Loper and Roberts, 1980, eq. 2.1) the vector flux i is proportional to Vp and VT where the relative chemical potential p is defined as

304

p

=

p~/M~ —

p2/M2

(8)

where Pi is the chemical potential of the light constituent and the M1 heavy its molecular weight, a subscript 2 denoting constituent. Thewith diffusion i driven by the temperature gradient, known as the Soret effect is assumed small and wifi be ignored. Further we shall assume * that ap/8T = 0, giving /d~~ i = —pD~— ~

(9)

~

where ~= ap!ap, 1= ~p/~ and D is the coefficient of diffusion of the light component. This is identical to the model used by Braginsky (1964);his ~is the same as ~7p. We may combine (5), (7) and (9)to obtain an expression for the compositional gradient at the bottom of the liquid d~ = dp —

~ P p

(10)

— __+

If m were zero, (10) would yield a negative gradient of ~with respect top, indicating a stable chemical layering near the bottom of the outer core. As th increases from zero, this gradient is diminished and eventually reversed, suggesting the possibility of a convective instability. To study this instability, we introduce the equation of state for density dp = f3p dp — ap dT — 6p2 d~ (11) where j3 is the isothermal compressibility and, through a Maxwell relation, the ~in (11) is the same as that introduced in (9). The actual density gradient immediately above the inner core boundary may be determined from (11), using (6) and (10), to be dp —

(~)2 =

13p

(aL

+!~)~

(12)

—

(~

Roberts (1980) that convective instability occurs when (14) is only marginally satisfied. The terms on the left-hand side of (14) represent the stabilizing effects of the conduction of heat down the adiabat and the diffusion of material induced by the pressure gradient, respectively, while the terms on the right-hand side represent the destabilizing effects of latent heat release and the release of light material associated with the freezing process. We now wish to determine which of these terms are important and whether the inequality is satisfied in the core. This requires estimation of the parameters contained in (14).

3. Parameter estimation

The estimation of parameter values under core conditions is an uncertain process and only the fact that the inequalities which we shall obtain hold so strongly allows us to place some confidence in the conclusions. We begin with several quantities from standard Earth models (e.g. Jacobs, 1975, table 1. 3): p = 12 X i0~kgm3 andg4.3 m ~-2 From Stacey (1977a)weadoptT4000K,Cp=660Jkg5 K1,

+

dp ~\k Dig On the other hand, if a parcel of fluid is moved adiabatically, holding its composition constant, its density varies as dpA/dp a2T/Cp (13) If the parcel moves upward and becomes less dense —

*

than its surroundings, convective overturning is possible. This occurs if dp/dp < dpA/dp or if 2T (p6)2 ctL th a + ~ ~ < +— D — g. (14) Strictly speaking (14) is a necessary condition for instability but not sufficient; the freezing rate i~i must exceed (14) sufficiently to overcome the dissipation associated with the convective motion. However, for the length scales relevant to the core, this latter effect is small and it has been shown by Loper and

The magnitudes of these thermal effects are poorly known within the core, but it is likely that they will be small com-

pared with the terms retained in (9).

k35Wm~K~,anda=8X106K’,whu1efrom Stacey (197Th) we have L = 8 X 10~J kg~and p6 = 0.012. The composition and properties of the light constituent in the core are the subjects of speculation. Loper (1978a) has estimated that p6~1.1 for silicon and 2.4 for sulfur, although oxygen is also a possibility (Ringwood, 1977). We shall take the former value since it yields a more conservative estimate in (14). Usselman (1975) has estimated that ~ = 0.1 ,but allowing for the possibility of alloying in the solid, we shall take a smaller value: ~ = 0.05. From ideal solution

3&5

theory (Guggenheim, 1952), we have * ii~3RT/ M~(1—3~)where R = 8314 J K’ kmoF1 andM 50 kg kmoF1, giving ji~ 4.4 X l0~J kg~.From both theory (Frenkel, 1946) and experiment (Majdic et al., 1969; Calderon et al., 1971) we have D 3 X i0~ m2 s—i ; this is considerably larger than Braginsky’s (1964) estimate and accordingly yields a more conservative stability criterion in (15). To determine we may estimate the mass of the solid inner core to be 1023 kg from a crude integration of Jacobs’

(1975) Table 1.3. Davies (1979) has estimated that the solid inner core began to form roughly 3 X i09 y

ago. Assuming the solid to have formed at a constant rate since then, we have,using a current inner core radius of 1215 km, ñi = 5 X i0-~kg m2 s~.For ease of reference, these parameter estimates have been compiled in Table I. With the parameter estimates listed, we find that the thermal terms in (14) are negligibly small

a2T((p~)2)~x~2

aL /~6\-‘ -~

~,-j~-)

~‘ L — L tip Using (10) this becomes

dTLT( 2Tth ~L 6 ÷~)_~ Lp2gD

(17)

Now let us compare (6) and (17). If the conduction gradient is smaller than the liquidus gradient, the temperature of the liquid at the bottom of the outer core lies above the liquidus and nothing untoward occurs; the core is in regime A of Loper (1978b). On the other hand, if the core isin regime B,i.e.if (18)

it would appear that the liquid above the freezing interface is frozen, clearly a contradictory situation. Before discussing the resolution of this paradox,

The stability criterion (14) now simplifies to (15)

The left-hand side of (15) is approximately 8 X 10_il kg m2 ~ This is the critical mass rate of freezing for instability to occur. Our estimate of tb is more than 600 times larger than critical. Therefore, the core is likely to be in vigorous convective motion and any analysis which is based upon a small departure from the critical is not geophysically relevant. The fact that thermal buoyancy is negligible leads us to conclude that the dynamo is driven primarily by the gravitational energy released by the separation process (Loper, 1978a). In writing (2), it was assumed that the liquid was well mixed with small compositional gradients. This *

(16)

dTL/dp < dTjdp

1.2 X 10~

gp2~SD/~
is true away from rigid boundaries, but near the freezing interface motions are suppressed and the compositional gradient may become large. This requires that we account for the variation of the liquidus temperature TL with composition as well as pressure near the freezing interface. From Loper and Roberts (1980) dTLT6 T~d~

From Loper and Roberts (1978), ~T a2ci/a~2where ‘F is the Gibbs free energy. If we define ~as the mass fraction of light constituent and x as the mass fraction of a compound such as FeS or FeSi then x 3~.Noting that ‘F = ~mP/M, we may use (3.02.3) of Guggenheim (1952) to obtain ~ 3RT/M1(1 — 3~)~

let us determine whether (18) is satisfied within the core. Combining (6), (17) and (18) we have gp(6

+

~) < (p~2/pD

+

L2/kT) tb

(19)

With the estimates from Table I we have

v

7~2

—1

1.5 X iO~ That is, dT~/dp<< dTL/dp close to the freezing interface. Once again thermal effects are small and (19) may be simplified to gp2t~D + (20) (1 _~

~)

This expression is similar to (15). With the given estimates 6/~is small, roughly 0.2. It follows that the critical freezing rate for the core to lie in regime B is approximately 20% higher than that for the onset of convection. However, it has been estimated that ,i’2 is 600 times larger than critical for convection, so it is very likely that the bottom of the outer core satisfies condition (18).

306 TABLE I Parameter estimates at the inner core boundary Parameter Density

Gravity Temperature

Heat capacity Thermal conductivity Thermal expansion Latent heat Change of volume upon melting Change of volume with composition Mass fraction of light constituent Energy of mixing Material diffusion coefficient Mass rate of freezing

Symbol

Estimate 3

p

12 X iO~kg m

g T Cp k L pa p~

D

When (18) is satisfied the liquid alloy near the

freezing interface is said to be constitutionally supercooled (Chalmers, 1964; Christian, 1965). The resolution of the paradox it implies lies in reducing the amount of freezing directly at the interface and instituting freezing above it. In laboratory experiments this is accomplished by the formation of dendrites extending from the freezing interface into the liquid forming a so-called “mushy zone” (Copley et al., 1970; Ockendon and Hodgkins, 1975). These dendrites are tree-like structures which, under core conditions, are likely to be very tenuous and easily broken. (In fact, Copley et al. (1970) found that the dendrites can be broken by natural-convection flow in the laboratory.) Once broken, the dendrites will tend to move with the fluid, freezing more solid or remelting as conditions dictate. The region occupied by these free-floating dendrites (snowflakes) has been labeled the “slurry layer” (Loper, 1978b; Loper and Roberts, 1978). Once solidification occurs away from the flat boundary, the parameter appearing in (6) and (17) must be taken to mean the mass rate of freezing

directly at this interface rather than the total. Since the fluid above the interface must lie on the liquidus, the latent heat released at the boundary must be removed by conduction down this gradient. This statement has the effect of making (18) and (20) equalities and the mass rate of freezing directly at the boundary must be given by ñ~‘~gp2D(6+ ~)/~i~2

Source Jacobs (1975)

4.3 m ~—2 4000 K

Jacobs (1975)

660 J kg1 K1 35 W m~K—i 8 X 10~K1 8 X ~ J kg~ 0.012 1.1 0.05 4.4 X iO~J kg-’ 3 X 10~m2 ~ 5 X 10~kg m2 s~

Stacey (1977a) Stacey (1977a) Stacey (1977a) Stacey (1977b) Stacey (1977b) Loper (1978a) Usselman (1975) Guggenheim (1952) Frenkel (1946)

Stacey (1977a)

Since we have estimated the total mass rate of freezing to be much larger than this, it follows that virtually all of the solidification occurs above the inner core

boundary. The nature of the slurry layer and its consequences for seismology and dynamo theory are currently being investigated. Acknowledgments We with to thank David Stevenson for providing a copy of his paper (Stevenson, 1980) prior to its publication.The assistance of David Fearn in the preparation of Table I and in sharpening the arguments in the paper is gratefully acknowledged. This work was supported by the National Science Foundation under Grant no. EAR 78-00818. References Anderson, D.L. and Hart, R.S., 1976. An Earth model based on free oscillations and body waves. J. Geophys. Res., 81: 14611475. London), 196: 122—124. Bolt, B.A., 1964. The velocity of seismic waves near the

Bolt, B.A., 1962. Gutenberg’s early PKP observations. Nature Earth’s center. Bull. Seismol. Soc. Am., 54: 191—208. Braginsky, SJ., 1963. Structure of the F layer and reasons for convection in the Earth’s core. DokI. Akad. Nauk SSSR, 149: 8—10. Braginsky, S.I., 1964. Magnetohydrodynamics of the Earth’s

(21)

core. Geomagn. Aeron. (USSR),4: 698—712.

307 Calderon, F.P., Sano, N. and Matsushita, Y., 1971. Diffusion of Mn and Si in liquid Fe over the whole range of cornposition. Metail. Trans., 2: 3325—3332; see also Diffusion Data,6: 72—73 (1972).

Jeffreys, H., 1976. The Earth. 6th edn. Cambridge University Press. Kennedy, G.C. and Higgins, G.H., 1973. The core paradox. J.Geophys. Res.,78: 900—904.

Chalmers, B., 1964. Principles of Solidification. Wiley, New York, NY, pp. 154—183. Christian, J.W., 1965. The Theory of Transformations in Metals and Alloys. Pergamon, Oxford, pp. 572—576. Cleary, J.R. and Haddon, R.A.W., 1972. Seismic wave scattering near the core—mantle boundary: a new interpretation of precursors to PKP. Nature (London), 240: 549—551.

King, D.W., Haddon, R.A.W. and Cleary, J.R., 1973. Evidence for seismic wave scattering in the D” layer. Earth Planet. Sri. Lett., 20: 353—356.

Copley, S.M., Giamei, A.F., Johnson, S.M. and Hornbecker, M.F., 1970. The origin of freckles in unidirectionally solidified castings. Metall. Trans., 1: 2193—2204. Davies, G.F., 1979. Timing of core formation and onset of

Loper, D.E. and Roberts, P.H., 1978. On the motion of an iron-alloy core containing a slurry. I. General theory. Geophys. Astrophys. Fluid Dyn., 9: 289—321. Loper, D.E. and Roberts, P.H., 1980. On the motion of an

the geomagnetic dynamo in convecting Earth models, Paper 10/30, XVII General Assembly, IUGG, Canberra. Dziewonski, A.M., Hales, A.L. and Lapwood, E.R., 1975. Parametrically simple Earth models consistent with geophysical data. Phys. Earth Planet, Inter., 10: 12—48. Frenkel, J., 1946. Kinetic Theory of Liquids. Oxford University Press, London. Gilbert, F. and Dziewonski, A.M., 1975. An application of normal mode theory to the retrieval of structural parameters and source mechanisms from seismic spectra. Phios. Trans. R. Soc. London, Ser. A, 278: 187—269. Guggenheim, E.A., 1952. Mixtures. Oxford University Press, London. Haddon, R.A.W. and Cleary, J.R., 1974. Evidence for scattering of seismic PKP waves near the mantle—core boundary. Phys. Earth Planet. Inter., 8: 211—234. Husebye, E.S., King, D.W. and Haddon, R.A.W., 1976.

iron-alloy core containing a slurry. II. A simple model. Geophys. Astrophys. Fluid Dyn., 16: 83—127. Majdic, A., Graf, D. and Schenck, H., 1969. Diffusion of Si, P. S and Mn in molten Fe. Arch. Eisenhuttenwes., 40: 627—630; see also Diffusion Data, 4: 338—339 (1970). Ockendon, J.R. and Hodgkins, W.R., 1975. Moving-Boundary Problems in Heat Flow and Diffusion. Clarendon Press, Oxford. Ringwood, A.E., 1977. Composition of the core and impliestions for origin of the Earth. Geochem. J., 11: 111—135. Schubert, G. and Stevenson, D., 1979. Radiogenic heat

Precursors to PKIKP and seismic wave scattering near the mantle—core boundary. J. Geophys. Res., 81: 1870—

1882. Irvine, R.D. and Stacey, F.D., 1975. Pressure dependence of the thermal Grüneisen parameter, with application to the Earth’s lower mantle and outer core. Phys. Earth Planet.

Inter., 11: 157—165. Jacobs, J.A., 1975. The Earth’s Core. Academic Press, New York, NY.

Loper, D.E., 1978a. The gravitationally powered dynamo.

Geophys. J.R. Astron. Soc., 54: 389—404. Loper, D.E., 1978b. Some thermal consequences of a gravitationally powered dynamo. J. Geophys. Res., 83: 5961— 5970.

source contents ofthe Earth and Moon. Paper 10/14,

XVII General Assembly, IUGG, Canberra. Stacey, F.D., 1977a. A thermal model of the Earth. Phys. Earth Planet. Inter., 15: 341—348. Stacey, F .D., 1 977b. Applications of thermodynamics to fundamental Earth physics. Geophys. Surveys, 3: 175— 204. Stacey, F.D., 1980. The cooling Earth: a reappraisal. Phys.

Earth Planet. Inter., 22: 89—96. Stevenson, D.J., 1980. Applications ofliquid-state physics to the Earth’s core. Phys. Earth Planet. Inter., 22: 42—52. Usselman, T.M., 1975. Experimental approach to the state of

the core: Part II. Composition and thermal regime. Am. J. Sd., 275: 291—303.