Stratification of the top of the core due to chemical interactions with the mantle

Authors

Errata

This article is corrected by:

  1. Errata: Correction to “Stratification of the top of the core due to chemical interactions with the mantle” Volume 116, Issue B7, Article first published online: 22 July 2011

Abstract

[1] Chemical interactions between the core and the mantle have been proposed as a mechanism to transfer O and Si to the core. Adding light elements to the top of the core creates a stratified layer, which grows by chemical diffusion into the underlying convecting region. We develop a physical model to describe the evolution of the layer and estimate physical properties to aid in its detection. The interface between the stratified layer and the convecting interior is defined by the onset of double diffusive instabilities. Oscillatory instabilities arise from the interplay of a stable compositional stratification and an unstable thermal stratification due to a superadiabatic heat flow at the core-mantle boundary. Double diffusive convection in the region of neutral stratification establishes the base of the layer and sets the rate of entrainment of excess light element into the interior of the core. Growth of the layer by diffusion is interrupted by nucleation of the inner core, which segregates light elements into the convecting part of the core. For calculations using O as the sole light element, we find that the base of the stratified layer retreats toward the core-mantle boundary as the radius of the inner core increases. Representative (but uncertain) model parameters yield a present-day layer thickness of 60 to 70 km. We also calculate the anomaly in P wave velocity relative to the value for a well-mixed core. The velocity anomaly decreases almost linearly across the layer with a peak value of 2% at the core-mantle boundary. This anomaly should be large enough to detect in seismic observations, although the sign of the anomaly is opposite to estimates obtained in previous seismic studies. We suggest possible explanations for this discrepancy and speculate about the implications for the structure of the Earth's interior.

1. Introduction

[2] Recent experiments on partitioning of major elements between liquid iron and mantle minerals suggest that the core is undersaturated in Si and O [Takafuji et al., 2005; Sakai et al., 2006; Asahara et al., 2007; Ozawa et al., 2008, 2009]. Samples of (Mg,Fe)SiO3 perovskite and ferropericlase (Mg,Fe)O from the experiments are strongly depleted in Fe, whereas the coexisting metal is enriched in Si and O. Estimates of the O solubility in liquid iron can exceed 15 wt % at conditions of the core-mantle boundary (CMB) for a typical mantle composition [Frost et al., 2010]. Such a high concentration is not compatible with the abundance of light elements in the core [Anderson and Isaak, 2002]. On the other hand, a concentration of 5 wt % O in the core [Alfè et al., 2007] implies a coexisting mineral assemblage with very low iron content [Frost et al., 2010; Ozawa et al., 2009], which is inconsistent with typical compositions for the lower mantle [Kesson et al., 1998]. Disequilibrium between the core and mantle is liable to produce chemical interactions at the CMB [Knittle and Jeanloz, 1991].

[3] When materials in the core and mantle are brought into contact at the CMB, the experiments suggest that FeO and FeSi are transferred to the core. A thin layer of Fe-depleted mantle would establish a new equilibrium with a core-side layer that is enriched in O and Si relative to the rest of the core. The subsequent evolution of these layers is controlled by chemical diffusion and the relative buoyancy of reaction products. Chemical diffusion through the crystalline mantle limits the reaction zone to narrow layers, although transport could be enhanced if the base of the mantle is partially molten [Labrosse et al. 2007]. The resulting layer of Fe-depleted mantle is buoyant relative to the overlying mantle [Jackson, 1998], but the narrow width of such a layer might inhibit flow. On the other hand, subduction continually brings unreacted material to the boundary region. If this material reaches the CMB then we expect the composition at the top of the core to evolve toward the solubility predicted for the unreacted mantle material. Differences in composition between subducted oceanic crust and depleted harzburgite are small compared the predicted change in composition for silicates that equilibrate with a typical core composition. For example, Ozawa et al. [2009] suggest that perovskite would contain less than 1 mol % FeSiO3 at the CMB. As long as subducted material with a higher iron content reaches the CMB, we expect chemical interactions to maintain an elevated concentration of light elements at the top of the core.

[4] Reaction products on the core-side of the CMB are likely to remain at the top of the core because the density anomalies that drive flow in the core are small [Stevenson, 1987] compared with the density anomalies predicted for the enriched layer [Frost et al., 2010]. When convection does not penetrate the stratified layer, vertical transport of light element is due solely to diffusion. Thus a buoyant layer at the top of the core grows by diffusion with only a small amount of entrainment by convection in the core.

[5] Stable stratification at the top of the core has previously been proposed on the basis of geomagnetic [Whaler, 1980], geodetic [Braginsky, 1984] and seismological [Lay and Young, 1990] observations. The mechanisms proposed to explain this stratification are equally diverse. Thermal stratification can develop if the heat flow at the top of the core is subadiabatic [Gubbins et al., 1982; Labrosse et al., 1997; Lister and Buffett, 1998]. Compositional stratification is also possible if light elements accumulate at the top of the core, possibly through an upward rain of chemically distinct fluid [Franck, 1982; Braginsky, 1993; Ito et al., 1995] or through an upward barodiffusive flux of light element [Fearn and Loper, 1981; Braginsky, 2006]. Layering may also be present if immiscible liquids evolve as the composition changes due to inner core growth [Helffrich and Kaneshima, 2004]. In this study, we examine the fate of light elements that are added to the top of the core by chemical interactions with the mantle. We use the thermodynamic model of Frost et al. [2010] to estimate the concentration of O at the top of the core and allow a chemically stratified layer to advance by diffusion into the convecting region of the core. We focus our attention on O as the light element, although the model can be generalized to include additional light elements once the necessary thermodynamic models are developed. We predict the present-day thickness of the layer and other physical properties, including the anomalous P wave velocity due to excess light elements. Our estimate for the P wave velocity differs by several percent from values expected for a well-mixed core. This anomaly should be detectable in seismic observations. However, the sign of the velocity anomaly is opposite to that inferred in previous studies [Lay and Young, 1990; Garnero et al., 1993; Tanaka, 2007]. This discrepancy raises questions about the layer and its physical properties, which we explore in the discussion.

2. Thermal and Chemical Stratification

[6] We assume that convection in the mantle brings a continual supply of unreacted material to the CMB. A local equilibrium is established at the interface between the core liquid and the subducted material. According to the experiments, an equilibrium assemblage of mantle minerals with a representative iron content (say 10 mol % FeSiO3 in perovskite and 15 mol % FeO in ferropericlase) imposes an elevated concentration of O in liquid iron at the interface. The elevated concentration of reaction products at the top of the core prevents the mantle from equilibrating with the bulk of the core. For simplicity, we assume that the mantle composition at the CMB is given by the bulk mantle composition, although we also consider a case where the mantle minerals at the CMB are substantially depleted in iron (e.g., 3 mol % FeO in ferropericlase). In both cases the equilibrium concentration of O in liquid iron is higher than the concentration in the interior of the core, so light element diffuses down the composition gradient to create a stratified layer below the CMB (see Figure 1). Upward transport of O due to barodiffusion also contributes to the distribution of light element, although the net transport is downward. We assume that the excess light element in a layer at the top of the core is separated from a well-mixed convecting region by an interface at radius r = s(t). A small amount of mixing across the interface is possible, although we expect the motion of the interface to be governed primarily by changes in the relative buoyancy across the interface. As light element is added by diffusion to the stratified layer we expect the layer to encroach into the convecting region. On the other hand, light element accumulates in the convecting region due to inner core growth. The resulting change in buoyancy in the convecting region opposes the growth of the layer and causes the interface at r = s(t) to retreat.

Figure 1.

Schematic illustration of a stratified layer at top of core. Diffusive transport of light elements from the CMB increases the concentration C(r, t) in the layer. Convection in the underlying region keeps the composition of that liquid Cm(t) well mixed. The stratified layer is separated from the convecting region by an interface at r = s(t). Exclusion of light elements from the inner core increases Cm(t) as the core solidifies.

[7] The concentration C(r, t) (mass fraction) of light element in the stratified layer obeys a diffusion equation

equation image

where the diffusive flux

equation image

includes both compositional and barodiffusion terms. Sorét effects are thought to be small [e.g., Braginsky, 2006]. Here D is the diffusion coefficient, ρ is the fluid density,

equation image

and the heat of mixing

equation image

is defined in terms of the chemical potential μ for the mixture [Landau and Lifshitz, 2004, p. 228]. For an ideal mixture of Fe and FeO, the heat of mixing can be approximated by [Loper and Roberts, 1981]

equation image

where R = 8.314 J K−1 mol−1, T is temperature, MO is the molar mass of O and equation image is the mean molar mass of the mixture. For small C the barodiffusive contribution to I is proportional to C, whereas the compositional term is proportional to ∇C. We also use ideal mixing to estimate αc ≈ 1.0, based on equations of state for Fe and FeO (see section 3).

[8] The concentration Cm(t) in the well-mixed region is governed by

equation image

where the overdot denotes a time derivative, c is the radius of the inner core, Ir(s, t) is the radial component of the flux on the convecting side of the interface at r = s(t), and Φ(t) is a source term to account for the accumulation of O due to inner core growth. When O is excluded from the inner core [Alfè et al., 2007], the source term is defined by

equation image

where the radius of the inner core c(t) is a prescribed function of time. The specific form of c(t) is based on models for the thermal evolution of the core [e.g., Buffett et al., 1996].

[9] Equations (1) and (6) are integrated numerically subject to boundary conditions at the CMB and the interface. An equilibrium concentration Ceq(T, P) is imposed at the CMB, based on the thermodynamic model of Frost et al. [2010] with a constant FeO content of 15 mol % in ferropericlase. We also consider the case where the mantle composition has 3 mol % FeO in ferropericlase, which lowers Ceq(T, P) by roughly a factor of two. In both cases we allow Ceq(T, P) to decrease as the CMB temperature drops linearly from 5000 K to 4000 K over the age of the Earth. The boundary conditions at the base of the stratified layer, r = s(t), are

equation image

and

equation image

The first condition expresses continuity of flux at the interface. The diffusive flux Ir(s+, t) in the stratified layer includes both the compositional and barodiffusion terms, whereas the flux on the convecting side of the interface, Ir(s, t) includes the barodiffusion term and an advective contribution due to entrainment. (Recall that the compositional gradient vanishes in the well-mixed region.). The second condition expresses continuity of composition at the moving interface. We use this condition to obtain an equation for the motion of the interface once the rate of entrainment is specified [Lister, 1995]. Two lines of reasoning provide similar estimates for the entrainment rate.

[10] Downward mixing of buoyant fluid from the stratified layer is weak because of the strong influence of rotation on fluid motions in the core [Fernando, 1991]. However, double diffusive convection can develop in the vicinity of the interface when the thermal stratification is unstable. Current estimates of the core heat flow suggest that the temperature gradient at the CMB exceeds the adiabat [Lay et al., 2008], resulting in an unstable thermal stratification in the layer. The superadiabatic temperature gradient persists across the layer and merges with a thermal boundary layer at the top of the convecting region. Strong stable stratification due to composition is probably sufficient to suppress radial flow in most of the layer. However, double diffusive convection is possible where the chemical stratification decreases in the transition to the well-mixed interior. Oscillatory instabilities are predicted when the combined effects of temperature and composition produce a neutral stratification [Turner, 1973, p. 257]. The resulting convection is organized into a series horizontal layers that evolve with time. Temperature and composition are approximately uniform within each layer but change abruptly between layers. However, the buoyancy due to temperature and composition tend to offset each other, producing a nearly continuous density across any interface that separates the individual layers [Turner, 1973, p. 264]. Figure 2 shows a schematic illustration of the compositional profile across the region of double diffusive convection. We use the onset of double diffusive convection to limit the vertical extent of the stratified layer and to provide a mechanism for entraining light element into the underlying convection. However, we do not attempt to model the detailed structure of the temperature and compositional fields. Instead, we assume that C obeys (1) in the region r > s(t) (e.g., dashed line in Figure 2) and that C is continuous at r = s(t). The second assumption leads to the boundary condition expressed in (9). Because the rate of entrainment is specified by the gradient ∂C/∂r at the base of the stratified layer, we do not expect large departures in the entrainment rate if we smoothly continue C across a narrow region of double diffusive convection and evaluate ∂C/∂r at r = s(t).

Figure 2.

Schematic illustration of composition C through the region of double diffusive convection. Oscillatory instabilities in the vicinity of neutral stratification develop into a series of convecting layers that homogenize both composition and temperature. Discrete steps in C connect the stratified layer to the well-mixed region below. We approximate C in this region by a smooth curve (dashed line) and assume that ∂C/∂r at the base of the stratified layer is equal to the gradient in the smooth curve at r = s(t).

[11] Several complications to the picture sketched above should be noted. First, thermal buoyancy in the core prior to the formation of the inner core is produced by cooling through the stratified layer. This configuration is different from the usual situation where heat is supplied from below. Flow in the thermal boundary layer at the top of the convecting part of the core may directly entrain light element from the stratified layer into the core. This entrainment would likely include the part of the layer where the fluid was neutrally stratified, possibly altering the conditions required for double diffusive instabilities to develop. On the other hand, the location of neutral stratification should continue to serve as a reasonable definition for the base of the stratified layer. A second complication arises from the compositional buoyancy due to inner core growth. Buoyant fluid is expected to rise into the stratified layer, but the depth of penetration should not extend much beyond the point of neutral stratification because of the sharp increase in stratification toward the CMB. In each case the base of the stratified layer can be set by the condition of neutral stratification, independent of the detailed mechanism of entrainment. Thus the compositional gradient at r = s+ is specified by

equation image

where αT ≈ 10−5 K−1 is the coefficient of thermal expansion (see section 3) and ∂T/∂r is the superadiabatic temperature gradient. A heat flow of 8 TW at the CMB corresponds to a temperature gradient of 1.9 K km−1 for a thermal conductivity of 28 W m−1 K−1 [Stacey and Loper, 2007]. The adiabatic gradient at the CMB is roughly 0.9 K km−1, so a representative value for the superadiabatic gradient is ∂T/∂r ≈ 1 K km−1. Using this value in (10) sets the compositional gradient at the base of the stratified layer. The compositional gradient also specifies the rate of entrainment into the convecting region because the barodiffusive contributions to Ir(s+, t) and Ir(s, t) are equal. This means that the diffusive flux of light element down the compositional gradient at r = s+ must equal the advective transport at r = s. We use (8) as a boundary condition in the solution for C(r, t) and use (9) to solve for the motion of the interface. Rearranging (9) gives

equation image

for the evolution of the interface position.

[12] Figure 3 shows the evolution of the layer thickness for a representative calculation. We integrate equations (1), (6), and (11) from an initial condition Cm(0) = 0.05 and s(0) equals the radius of the CMB at t = 0. The composition of liquid at the CMB boundary evolves with temperature according to the thermodynamic model of Frost et al. [2010], assuming that the boundary temperature decreases linearly from 5000 K at t = 0 to 4000 K at t = 4.5 Ga. Inner core growth is parameterized in the form

equation image

where c(t1) is the current radius of the inner core and t0 defines on the onset of growth. We set c(t1) = 1221 km, t1 = 4.5 Ga and t0 = 3.0 Ga. Diffusive growth of the stratified layer is evident during the first 3 Ga of evolution, prior to the formation of the inner core. Nucleation and growth of the inner core segregates light elements into the convecting region of the liquid outer core, reversing the motion of the interface. The response of the layer to inner core growth may seem surprisingly sudden given the small size of the inner core at early times. However, the growth rate predicted by (12) is large at early times. Such rapid growth arises because the pressure gradient vanishes at the center of the Earth. Withdrawing a small amount of heat causes a large shift in the intersection of the melting curve and the core adiabat. In fact, equation image is infinite at t = t0, although the source term Φ(t) increases with time as t1/2. An increase in Φ(t) causes equation imagem to increase, so the interface retreats according to (11). We find that the thickness of the layer decreases to about 72 km at the present time. A second calculation using a lower value of Ceq(T, P) at the CMB produces a somewhat thinner stratified layer. The larger response to inner core growth is due to the lower concentration of light element in the stratified layer. In addition to the previously stated parameter values we adopt D = 3 × 10−9 m2 s−1 for the diffusion coefficient and set ρ = 104 kg m−3, dP/dz = ρg, and g = 10 m s−2. Doubling the diffusivity to 6 × 10−9 m2 s−1 increases the thickness of the layer by roughly a factor of equation image.

Figure 3.

Representative calculations for the thickness of the stratified layer as a function of time. The primary result (solid line) is based on the thermodynamic model of Frost et al. [2010] and a typical mantle composition (15 mol % FeO in ferropericlase). The other result (dashed line) is obtained with an iron-depleted layer at the base of the mantle (3 mol % FeO in ferropericlase). The onset of inner core growth at t = 3 Ga causes the thickness of the stratified layers to decrease as light elements accumulate in the convecting part of the liquid core.

[13] Another prediction of the model is the mass flux at the CMB. A steep gradient in composition at early times drives a large flux of oxygen into the core. This flux gradually decreases with time as the layer becomes thicker and the compositional gradient is reduced. Figure 4 shows a decline in the oxygen flux at the CMB since t = 3.0 Ga for the calculation using the thermodynamic model of Frost et al. [2010]. The trend observed in Figure 4 continues smoothly from earlier times with no visible interruption due to inner core growth. This result suggests that changes in conditions near the interface r = s(t) have not yet affected conditions near the CMB. This is not surprising because the diffusion time for a layer approximately 70 km thick is roughly 3 Gyr. The net flux at t = 4.5 Ga is −4200 kg s−1 (negative downward). About 6200 kg s−1 is driven downward by the compositional gradient and 2000 kg s−1 is driven upward by barodiffusion in this particular calculation. The compositional gradient at the CMB continues to decrease beyond t = 4.5 Ga, whereas the barodiffusive flux is nearly constant (decreasing only slightly as the equilibrium concentration at the CMB decreases due to cooling). Eventually, the flux of O changes sign, returning light element back to the base of the mantle. The total mass of O added to the core at t = 4.5 Ga is about 7 × 1020 kg, or about 0.035% of the mass of the core.

Figure 4.

Mass flux of O at the CMB due to the combined effects of compositional and pressure-driven diffusion. The net flux is downward (negative) over the calculation. A gradual decrease in the mass flux between t = 3.0 Ga to 4.5 Ga continues smoothly from the trend computed at earlier times. The mass flux eventually reverses as the effects of compositional diffusion continue to decrease.

3. Structure of the Stratified Layer

[14] Physical properties in the stratified layer can be computed from the distribution the light element and temperature. Figure 5 shows the radial variation of light element C(r,t) and potential temperature θ(r, t) at t = 4.5 Ga. Deviations in density from a well-mixed and adiabatic core are computed from anomalies C(r, t) − Cm(t) and θ(r, t) using coefficients αC and αT for the compositional and temperature dependence of density. In detail αT and αC vary with composition and temperature but it suffices for our purposes to adopt constant values that are close to the average over the layer. The effect of composition has an overwhelming influence on the density stratification over most of the layer. However, the effects of temperature become important near the base where the stratification is neutral. A useful measure of the density stratification is the buoyancy frequency [Turner, 1973]

equation image

where ρ′ is the deviation in density from the value for a well-mixed core. Our estimate of buoyancy frequency in Figure 6 shows that the stratification is very strong. The peak frequency of 0.09 s−1 means that the layer supports gravity waves with periods on the order of 70 s. Such short periods means that the layer acts like a rigid lid for radial motions, although horizontal motions in the layer are possible. In fact, horizontal flow is likely in the presence of lateral variations in heat flow at the CMB [Bloxham and Gubbins, 1987; Gibbons and Gubbins, 2000; Lister, 2004].

Figure 5.

Predicted concentration C and potential temperature θ below the CMB at t = 4.5 Ga. Both fields are plotted as a function of depth z = rcr, where rc is the radius of the CMB.

Figure 6.

Predicted buoyancy frequency N in the layer as a function of depth below the CMB. The peak frequency at N = 0.09 s−1 corresponds to an oscillation period of τ = 2π/N = 70 s, implying strong stratification. Radial motion is suppressed but horizontal flow can be driven by lateral variations in heat flow at the CMB.

[15] The radial variations in composition and temperature also affect the velocity of P waves at the top of the core. We compute the P wave velocity in the layer by averaging the thermoelastic properties of the end-member phases. Molar volumes for hcp Fe are computed using a Mie-Grüneisen equation of state with explicit vibrational and electronic contributions to the specific heat [see Seagle et al., 2006, and references therein]. For FeO (in the B1 structure) we use the Mie-Grüneisen equation of state of Campbell et al. [2009]. Molar volumes of intermediate compositions between Fe and FeO are estimated by averaging on the basis of mole fraction. The volume of the solid mixture is increased by 2% to account for the volume change on melting, based on estimates for pure iron [Alfè et al., 2002]. By evaluating the equations of state at pressures and temperatures relevant for the liquid state, we assume, in effect, that the volume of Fe and FeO liquids at the melting point can be extrapolated to higher temperature (at constant P) using the coefficient of thermal expansion for the solid phases.

[16] The bulk modulus at a constant temperature and composition is computed from the liquid mixture using

equation image

where the derivative is approximated by the finite difference method. Similarly, the coefficient of thermal expansion for the mixture is computed from

equation image

using finite differences. The Grüneisen parameter γ for the mixture is obtained by averaging the Grüneisen parameters for Fe and FeO by mole fraction. Finally, we convert the isothermal bulk modulus KT to an adiabatic modulus KS using

equation image

and define the P wave velocity in the liquid by

equation image

[17] To illustrate the approach, we give an example for P = 135 GPa and T = 3000 K. The molar volumes of solid B1 FeO and hcp Fe are 8.246 and 5.003 cm3 mol−1, respectively. A composition of 8 wt % O corresponds to a mixture of Fe and FeO with 30.3 mol % FeO. Increasing the average molar volume of the solid mixture by 2% to account for melting yields 6.106 cm3 mol−1, corresponding to a density of 9.94 g cm−3. The P wave velocity for this particular composition is 8.07 km s−1. We repeat this type of calculation for several pressures, temperatures, and compositions near the expected conditions of the CMB and apply equations (3) and (14)(17) to obtain the pressure, temperature, and compositional derivatives of the bulk moduli, density, thermal expansion, and sound velocity of the Fe-O liquids.

[18] At constant pressure, the density of the liquid mixture decreases with increasing oxygen content at a rate of about 0.1 g cm−3 per wt % O. This value is only weakly dependent on temperature. The adiabatic bulk moduli of the mixture also decreases with increasing oxygen content, implying that oxygen rich liquids are more compressible. However, the variation in bulk modulus with composition is strongly dependent on temperature. For example, the derivative ∂KS/∂C decreases by a factor of 6 as temperature increases from 2000 K to 5000 K. Combining effects of composition on density and bulk modulus in (17), we obtain a P wave velocity that increases as O is added to the mixture. (A similar increase on P wave velocity was found by Helffrich and Kaneshima [2004] for S-rich liquids at CMB conditions.) The greatest sensitivity of Vp to O occurs at high temperature (0.031 km s−1 (wt % O)−1 at 5000 K), whereas the lowest sensitivity to O occurs at low temperature (0.009 km s−1 (wt %O)−1 at 2000 K). We use the sensitivity at 4000 K (0.021 km s−1 (wt %O)−1) to compute the perturbations in P wave velocity due to the compositional anomalies C(r,t) − Cm(t). Similarly, we use the sensitivity to temperature (−1.67 × 10−4 km s−1 K−1) to compute velocity perturbations due to the temperature anomalies shown in Figure 5b. (The sensitivity to temperature depends on composition but it suffices to calculate the partial derivative at 8 wt % O).

[19] Absolute velocities are computed by assuming that the smooth radial trend in PREM [Dziewonski and Anderson, 1981] reflects the P wave velocities across a well-mixed and adiabatic core. Departures in composition and temperature from this well-mixed state are assumed to cause variations in P wave velocity relative to PREM according to the partial derivatives given above. Our results suggest that a layer enriched in FeO relative to the bulk of the core should have a P wave velocity that is anomalously fast (see Figure 7). The velocity anomaly relative to PREM is largest near the CMB and decreases almost linearly toward the bottom of the stratified layer. The peak anomaly (∼2%) is large enough to detect in seismic observations. However, the sign of the anomaly is opposite to the values inferred in previous seismic studies. For example, we show a seismic model from Garnero et al. [1993], which proposes a velocity reduction relative to PREM in a 50 km region below the CMB. A similar result has recently been obtained by Tanaka [2007]. Eaton and Kendall [2006] have reported evidence for high P wave velocity in a thin (12 km) layer at the top of the core, although their observations require this layer to be embedded in a thicker, low-velocity layer, comparable to that reported by Garnero et al. [1993].

Figure 7.

Predicted P wave velocity Vp as a function of depth below the CMB. Excess light elements in the stratified layer perturb Vp relative to the value for a well-mixed core. We use the smooth radial trend in PREM to establish Vp in a well-mixed core. Perturbations due to composition and potential temperature are superimposed on this trend to obtain the prediction for the stratified layer. The peak velocity anomaly is 2% above PREM. We also show an estimate from the seismic study of Garnero et al. [1993], which obtained a negative velocity anomaly in a 50 km thick layer below the CMB.

4. Discussion

[20] Discrepancies between the predicted velocity anomaly and the estimates from previous seismic studies raise questions about the layer and its physical properties. It is possible that the seismic studies are affected by uncertainties in seismic structure near the base of the mantle [Garnero and Helmberger, 1995; Tanaka, 2007]. Whether the velocity anomaly predicted here can be accommodated by altering seismic velocities in the lowermost mantle remains to be determined. It is also possible that the model has serious shortcomings. Owaza et al. [2009] report experimental results for the concentration of O that are below the values predicted by Frost et al. [2010]. They prefer an O concentration of 8 to 10 wt % at CMB conditions and suggest an additional contribution from Si at 6 to 10 wt %. A reduction in the O concentration relative to the values assumed here reduces the flux of O at the CMB, but the evolution of the layer is not substantially altered (see Figure 3). We expect a small decrease in the amplitude of the predicted velocity anomaly, although this change might be compensated by an increase in the concentration of Si at the top of the core. First principle calculations suggest that Si is readily incorporated into the inner core [Alfè et al., 2007], so our estimate for the source term Φ(t) may not be too different if Si was added as a second alloying component. On the other hand, there is no guarantee that all of the alloying components are transferred from the mantle to the core. Exsolution of components (such as MgO) from the core to the mantle could potentially alter our model predictions Stevenson [2007]. Another important parameter in the model is the value of heat flow at the CMB. A steeper temperature gradient increases the entrainment of the stratified layer into the convecting region, reducing the thickness of the stratified layer. However, plausible changes in the heat flow are unlikely to be enough to hide the layer from seismic observations.

[21] Another possible explanation of the discrepancy involves the calculation of the velocity anomaly from estimates of composition and temperature. We have assumed that physical properties of the layer can be computed by averaging the end-member phases. Departures from ideal mixing would affect both the density and the bulk modulus. However, it would be surprising if departures from ideal mixing could change the sign of the velocity anomaly. Often the effects of nonideal mixing become smaller at high pressure [Corgne et al., 2008; Seagle et al., 2008; Frost et al., 2010]. Nevertheless, nonideal mixing is a possible explanation for the discrepancy that needs to be explored in more detail.

[22] Finally, we might question the relevance of the experiments that motivate the present study. It is possible that the chemical conditions in the experiments (oxygen fugacity, for example) are different from the conditions at the CMB. Differences in the partitioning of major elements between liquid iron and mantle minerals might affect our expectations for chemical interactions. Alternatively, a layer of intermediate density may isolate the mantle and core, preventing the chemical interactions described here. In order for this suggestion to work we require a layer with sufficient density to deflect subducted slabs. Such a layer could be either liquid or solid, but it should preferably establish chemical equilibrium with the bulk composition of the core. Chemical disequilibrium is possible if the response drives heavy elements into the core. In this case the reaction products would enhance convection and rapidly mix into the core. However, any disequilibrium that adds light elements to the top of the core is likely suffer from the same problems as the model described here.

5. Conclusions

[23] We develop a physical model for the evolution of a stratified layer at the top of the core. The layer is formed by chemical interactions with the mantle, which add excess light elements to the liquid core at the CMB. Diffusion of light elements down the gradient in composition causes the layer to encroach into the convecting part of the core. Nucleation and growth of the inner core reverses the advance of the stratified layer by increasing the concentration of light elements in the convecting region. We predict a layer thickness of 60 to 70 km at the present time for a reasonable choice for the model parameters, although many of these parameters are poorly known. We also predict several physical properties for the layer, including the anomalies in P wave velocity relative to values for a well-mixed core. The velocity anomaly is roughly 2% at the CMB and decreases almost linearly toward the base of the stratified layer. Previous seismic studies have detected a region of anomalous P wave velocity with similar amplitude and thickness. However, the sign of the anomaly relative to PREM is opposite to the sign of our anomaly relative to a well-mixed core. Assuming that the smooth radial trend in PREM approximates the velocity of a well-mixed core, we conclude that our prediction is incompatible with current seismic observations. We propose several possible explanations to account for this discrepancy. In each case a resolution would revise our current understanding of the Earth's interior.

Acknowledgments

[24] This work is partially supported by funding from the National Science Foundation. C. T. Seagle acknowledges funding from CDAC. We thank George Helffrich, Dave Stevenson, and anonymous reviewer for many helpful comments and suggestions.