Equations of state of close-packed phases of iron and their implications for the Earth's core

Theoretical equations of state for Fe at high pressures and temperatures are derived from the expression of the free energy written as a sum of the static energy, the harmonic, the anharmonic and the electronic contributions. All the calculations have been performed for the various crystal structures of Fe using different intermolecular potentials, and namely: Lennard-Jones, Morse and Rydberg functions. The available experimental data do not allow a definite choice between them thus leaving open the problem of the composition of the earth's core.     

Heat transport by variable viscosity convection and implications for the Earth's thermal evolution

The case is presented that the efficiency of variable viscosity convection in the Earth's mantle to remove heat may depend only very weakly on the internal viscosity or temperature. An extensive numerical study of the heat transport by 2-D steady state convection with free boundaries and temperature dependent viscosity was carried out. The range of Rayleigh numbers (Ra) is 10⁴?10⁷ and the viscosity contrast goes up to 250000. Although an absolute or relative maximum of the Nusselt number (Nu) is obtained at long wavelength in a certain parameter range, at sufficiently high Rayleigh number optimal heat transport is achieved by an aspect ratio close to or below one. The results for convection in a square box are presented in several ways. With the viscosity ratio fixed and the Rayleigh number defined with the viscosity at the mean of top and bottom temperature the increase of Nu with Ra is characterized by a logarithmic gradient β = ?ln(Nu)/? ln(Ra) in the range of 0.23–0.36, similar to constant viscosity convection. More appropriate for a cooling planetary body is a parameterization where the Rayleigh number is defined with the viscosity at the actual average temperature and the surface viscosity is fixed rather than the viscosity ratio. Now the logarithmic gradient β falls below 0.10 when the viscosity ratio exceeds 250, and the velocity of the surface layer becomes almost independent of Ra. In an end-member model for the Earth's thermal evolution it is assumed that the Nusselt number becomes virtually constant at high Rayleigh number. In the context of whole mantle convection this would imply that the present thermal state is still affected by the initial temperature, that only 25–50% of the present-day heat loss is balanced by radiogenic heat production, and the plate velocities were about the same during most of the Earth's history.     

Thermal regime of the Earth's core

The outer core is assumed to consist of iron and sulfur, with a small amount of potassium that generates heat by radioactive decay of sim||pre|40 K. Two cases are considered, corresponding respectively to a high rate of heat production (Q = 2 · 10¹² cal./sec, about 0.1% K), and to a low rate (Q = 2 · 10¹¹ cal./sec). The temperature at a depth of 2800 km in the mantle is taken to be 3300°K (Wang, 1972). The temperature T_c at the core-mantle boundary depends on whether or not a density gradient in the lowermost layer D″ of the mantle prevents convection in that layer. In the first case, and for high Q, T_c = 4500–5000°K. In the second case, or for low Q, T_c ≈ 3500°K.The heat-conduction equation is used to calculate the temperature T_i at the inner-core boundary in the absence of convection. For high Q, T_i ? T_c ≈ 1600°K; for low Q, T_i ? T_c ≈ 160°K. Corresponding temperature gradients at r = r_c and r = r_i are listed in Table I.The adiabatic gradient at the top of the core is calculated by the method of Stewart (1970). It strongly depends on the parameters (ρ₀, c₀, γ₀, etc.) that characterize core material at low pressure. Stewart has drawn graphs that allow the selection of sets of parameters that are consistent with seismic velocities and a given density distribution in the core. Some acceptable sets of parameters are listed in Table II. Many sets yield temperatures T_c in the range 3500–5000°K; some give an adiabatic gradient steeper than the conductive gradient and are compatible with convection; others do not. Since properties of FeS melts remain unknown, there is at present no way of selecting any set in preference to another.Properties of the FeS system at low pressure suggest the possible appearance of immiscibility at high temperature in liquids of low sulfur content; accordingly, the inner-core boundary is thought to represent equilibrium between a solid (FeNi) inner core and a liquid layer containing only a small amount of sulfur; layer F in turn is in equilibrium with another liquid (forming layer E) containing more sulfur, and slightly less dense, than F. The temperature T_i at the inner-core boundary is about 6000–6500°K for high Q and T_c ≈ 4500–5000°K. It is consistent with Alder's (1966) and Leppaluoto's (1972) estimates of the melting point of iron at 3.3 Mbar, but not with that of Higgins and Kennedy (1971).     

Evidence for long-term asymmetries in the Earth's magnetic field and possible implications for dynamo theories

Paleomagnetic data indicate that there is a north-south asymmetry in the time-averaged magnetic field and that there are small but significant differences between the normal and reverse polarity states. The geographical variation is most likely due to spatial variation in the boundary conditions at the core-mantle interface. The difference in the magnetic fields of the reverse and normal polarity states can be modeled in terms of a “standing field”. The paleomagnetic data are insufficient to determine whether or not this “standing field” is of core origin. However, consideration of mechanisms, including thermoelectric currents, indicates that there probably are important differences in core processes between the two polarity states. At first glance this interpretation is difficult to reconcile with the fact that the magnetic induction equation is antisymmetric with respect to the magnetic field. A way around this problem is the possibility that only certain transitions are allowed between acceptable eigenstates in dynamo models of the Earth's magnetic field.     

Electromagnetic evidence for lateral inhomogeneities within the Earth's upper mantle

The composition of the upper mantle is of great significance to our understanding of plate tectonics and global evolution. Information about the physical properties of the Earth at upper mantle depths, including lateral variations in electrical conductivity, can be deduced from measurements of the electric and magnetic fields at the Earth's surface. Electromagnetic methods appear to give poorer resolution than do some other methods, for example seismics, but as they are sensitive to quite different properties of a medium they provide a different and complementary class of information.The basic theory of electromagnetic sounding methods is briefly reviewed below, and evidence regarding lateral conductivity inhomogeneities in the Earth's upper mantle is examined. While lateral electrical conductivity inhomogeneities appear to be the rule rather than the exception, the interpretation of electromagnetic data still presents difficulties and the results from many regions are not as yet unambiguous. Where the data are of sufficient resolution, a rapid increase in electrical conductivity can usually be identified within the upper mantle. The depth to this highly conductive zone is different in different tectonic environments, but is broadly consistent between analogous but widely separated tectonic environments. A comparatively shallow conducting region is found beneath the ocean lithosphere. The depth of this region is dependent on lithospheric age. Many of the more shallow conducting regions in both continental and oceanic environments are associated with high heat flow values and seismic low velocity zones. These highly conducting regions may be zones of partial melt.     

Applications of liquid state physics to the Earth's core

By use of the modern theory of liquids and some guidance from the hard-sphere model of liquid structure, the following new results have been derived for application to the Earth's outer core. (1) dK/$\text{d}\text{P ? 5 ? 5. 6P}$/K, where K is the incompressibility and P the pressure. This is valid for a high-pressure liquid near its melting point, provided that the pressure is derived primarily from a strongly repulsive pair potential φ. This result is consistent with seismic data, except possibly in the lowermost region of the outer core, and demonstrates the approximate universality of dK/dP proposed by Birch (1939) and Bullen (1949). (2) dlnT_M/dlnρ = (γC_V ? 1)/($\text{C}{}_{\mathrm{<mtext>V</mtext>}}\text{?}\text{3}\text{2}\text{),}\text{where}\text{T}{}_{\mathrm{<mtext>M</mtext>}}$ is the melting point, ρ the density, γ the atomic thermodynamic Grüneisen parameter and C_V the atomic contribution to the specific heat in units of Boltzmann's constant per atom. This reduces to Lindemann's law for C_V = 3 and provides further support for the approximate validity of this law. (3) It follows that the “core paradox” of Higgins and Kennedy can only occur if $\text{γ <}\text{2}\text{3}$. However, it is shown that $\text{γ <}\text{2}\text{3}\text{? ∫}{}^{\mathrm{\infty }}{}_0\text{(?g/?T)}{}_{\mathrm{\rho }}\text{r(}\text{d}\text{/}\text{d}\text{r)(r}{}^2\text{φ)}\text{d}\text{r > 0}$, which cannot be achieved for any strongly repulsive pair potential φ and the corresponding pair distribution function g. It is concluded that $\text{γ >}\text{2}\text{3}$ and that the core paradox is almost certainly impossible for any conceivable core composition. Approximate calculations suggest that γ ～ 1.3–1.5 in the core. Further work on the thermodynamics of the liquid core must await development of a physically realistic pair potential, since existing pair potentials may be unsatisfactory.     

The distribution of thermal diffusivity in the Earth's crust

Data in the literature and additional measurements on the thermal diffusivities of granites, granulites and ultrabasic rocks at temperatures up to 1000 K and pressures to 2 GPa, have been used to propose a new model for thermal diffusivity distribution in the crust and upper mantle.The laboratory measurements were made using a pulse method or the Angstroem method with cylindrical heat flow. After making particular assumptions about the pressure and temperature distribution within the top 60 km the pressure and temperature dependencies of diffusivity were transformed into a depth dependence.The model is characterised by a continuous decrease of diffusivity to a depth of ～30 km where there is a small but rapid increase to a nearly constant value of 7.3 × 10^?3 cm² s^?1.     

Early Archaean rocks and geochemical evolution of the earth's crust

Remnants of Early Archaean rocks (>FX3000 m.y. old) are reported from most continents. A critical review of the radiometric data shows that few of these are well authenticated and most are very limited in extent. The oldest are predominantly plutonic gneisses of tonalitic-to-granitic composition (e.g., the basement gneisses of West Greenland, Labrador, Rhodesia and South Africa). In all cases there are inclusions of meta-volcanic and sedimentary rocks with greenstone belt affinities which probably represent crust into which the igneous parents of the gneisses were intruded.The trace element chemistry of these very old rocks is reviewed in an attempt to establish the mechanism of formation of early crust and place constraints on the chemical evolution of the earth's mantle. “Mantle-type” Sr isotope compositions show that the sialic members of both early gneisses and greenstone belts were not derived from much older crustal differentiates, either at 3800 or at 2800 m.y. ago. However, trace element ratios such as K/Rb and Sr/Ba, and rare earth element abundances, are not consistent with direct derivation of the plutonic suite from the upper mantle and also rule out a common parentage for the tonalites and granites. An origin by partial melting of metamorphosed juvenile crust with a composition range equivalent to that represented by the greenstone belts is preferred. Tonalites resulted from high-pressure melting of mafic garnet-amphibolite and at least some of the granites from low-pressure melting of more felsic (possibly even sedimentary) material.The trace element chemistry of the greenstone belt volcanics is thought to characterize the composition of early mantle melts, although the best preserved and best documented cases are about 500–1000 m.y. younger than the oldest known gneisses. The dominant type is tholeiite with low incompatible element contents and light-depleted or essentially flat rare earth patterns, features even more marked in the ultramafic komatiites which represent large degrees of melting. More evolved calc-alkaline rocks with relative incompatible and light rare earth element enrichment are also important. With the exception of the ultramafic lavas, all these types can be matched by the chemistry of present-day oceanic volcanism.It is concluded that the range of trace element variations in the earth's mantle was comparable in early Archaean times to that at the present. This is supported by mass balance calculations for the lithophile elements which have been preferentially extracted into the crust. Thus the isotope and trace element evidence of the oldest rocks argues against primary differentiation of the crust either during accretion of the earth or during its first 500 m.y. as a solid body. Crust formation has probably occurred continuously, although worldwide evidence for magmatism at around 2800 m.y. ago probably marks a particularly active period.     

The influence of extrinsic pressure changes on the Earth's dynamo

Reversals of the Earth's magnetic field have been claimed to correlate with ice ages, tectonic events and falls of tectites. A physical mechanism is needed to relate reversals with the other events before these correlations can be taken seriously. One possible connection lies through changes in pressure in the core. If events high up in the mantle were to lead to changes in core pressure, this would affect the rate of freezing of the liquid core and modify the power supplied to the dynamo. A sufficiently large modification could set off a reversal or perhaps change the mode of operation of the dynamo from a non-reversing to a reversing state.The model of Gubbins et al., allows a quantitative calculation to be made for the effect of a pressure change on the energy release. Any sufficiently sudden pressure change would change the power, but it seems unrealistic to consider less than a 1000 year time scale. Relaxation of shear forces in the mantle, overturning of core fluid, and changes in magnetic fields all take place on about this time scale. According to the model, a pressure change of 0.1 bar over a 1000 years could change the power supply drastically.A continuous process of mantle differentiation leading to the formation of the upper mantle from an initially homogeneous mantle can only provide 10% of the required pressure change, but the effect cannot be ruled out as a power source for the dynamo because uncertainties in the calculations can amount to at least an order of magnitude. The other effects produce changes of up to 1% in the power supply, which may be sufficient to alter the characteristics of the dynamo and produce reversals or a change in reversal behaviour. Further speculation must await a better understanding of the dynamics of reversals, and of mantle processes.     

Poisson's ratio behaviour in various crystalline rocks: application to the study of the Earth's interior

The study of Poisson's ratio (σ) behaviour in various crystalline rocks under different temperatures and pressures shows this parameter to depend upon the rock composition rather than upon P-T conditions. The results of this study are presented in the form of a comparison of σ(z) distributions within the consolidated crust and continental upper mantle and the specific variations of σ in crust and mantle rocks underlying the Voronezh crystalline massif (VCM). These investigations, which are based upon seismic and seismological data as well as high pressure experiments, should clarify in particular the composition and petrology of the Earth's interior.     

Dissipation of the rare gases contained in the primordial Earth's atmosphere

If the Earth was formed by accumulation of rocky bodies in the presence of the gases of the primordial solar nebula, the Earth at this formation stage was surrounded by a massive primordial atmosphere (of about 1 × 10²⁶ g) composed mainly of H₂ and He. We suppose that the H₂ and He escaped from the Earth, owing to the effects of strong solar wind and EUV radiation, in stages after the solar nebula itself dissipated into the outer space.The primordial atmosphere also contained the rare gases Ne, Ar, Kr and Xe whose amounts were much greater than those contained in the present Earth's atmosphere. Thus, we have studied in this paper the dissipation of these rare gases due to the drag effect of outflowing hydrogen molecules. By means of the two-component gas kinetic theory and under the assumption of spherically symmetric flow, we have found that the outflow velocity of each rare gas relative to that of hydrogen is expressed in terms of only two parameters — the rate of hydrogen mass flow across the spherical surface under consideration and the temperature at this surface. According to this result, the rare gases were dissipated below the levels of their contents in the present atmosphere, when the mass loss rate of hydrogen was much greater than 1 × 10¹⁷ g/yr throughout the stages where the atmospheric mass decreased from 1 × 10²⁶ g to 4 × 10¹⁹ g.     

Dissolution of the primordial rare gases into the molten Earth's material

We have shown in a previous paper that, if the primordial solar nebula existed when the Earth was formed, the Earth was once surrounded by a dense and massive primordial atmosphere, whose temperature and pressure were about 4000 K and 900 atm, respectively, at the bottom. We suppose that this hydrogen-rich atmosphere escaped from the Earth after the solar nebula itself disappeared, both phenomena probably being due to the effect of strong solar wind and radiation.Using the results of our previous and new calculations on the structure of the primordial atmosphere, we have investigated the amount of dissolution of the rare gases, which were contained in the primordial atmosphere, into the molten Earth's material.The amount of the dissolved rare gases is found to be strongly dependent on the grain opacity of the atmosphere, i.e., on the amount of fine grains. However, their isotopic ratios and relative abundance are independent of the opacity and approximately equal to those in the primordial solar nebula, that is, to the present solar values. Especially, the dissolved neon is expected to have remained in the present mantle. Therefore, if a considerable amount of neon with nearly the solar isotopic ratio is discovered in present mantle material, this offers direct evidence for the proposition that the proto-Earth was once surrounded by the primordial atmosphere.     

Earth's melting due to the blanketing effect of the primordial dense atmosphere

When the proto-Earth was growing by the accretion of planetesimals and its mass became greater than about 0.1 M_E, where M_E is the present Earth's mass, an appreciable amount of gas of the surrounding solar nebula was attracted towards the proto-Earth to form an optically thick, dense atmosphere. We have studied the structure of this primordial atmosphere under the assumptions that (1) it is spherically symmetric and in hydrostatic equilibrium, and (2) the net energy outflow (i.e., the luminosity) is constant throughout the atmosphere and is given by GMM/R with M = M/10⁶yr or M/10⁷yr where M and R are the mass and the radius of the proto-Earth, respectively.The results of calculations show that the temperature at the bottom of the atmosphere, namely, at the surface of the proto-Earth increases greatly with the mass of the proto-Earth and it is about 1500°K for M = 0.25 M_E. This high temperature is due to the blanketing effect of the opaque atmosphere. Thus, as long as the primordial solar nebula was existing, the surface temperature of the proto-Earth was kept high enough to melt most of the materials and, hence, the melted iron sedimented towards the center to form the Earth's core.     

On the activation volume for creep and its variation with depth in the Earth's lower mantle

This activation volume ΔV for creep may be derived from Keyes's elastic strain energy model or from Weertman's empirical relationship between viscosity and the melting temperature. These formulations are shown to be equivalent if the anharmonic Grüneisen parameters γ of all acoustic modes are equal and if the pressure dependence of the melting temperature follows Lindemann's law, both of which assumptions are valid for the close-packed mineral structure of the lower mantle. The pressure derivative of ΔV depends only on the bulk modulus and the acoustic γ, both of which are directly available from seismic models. Using the data of Brown and Shankland, we show that ΔV decreases by almost 50% between the top and the bottom of the lower mantle, which makes it easier to maintain a constant viscosity in this region. The isoviscous temperature profile can be adiabatic in the deep lower mantle only below 1700 km depth; it is super-adiabatic in the top 1000 km of the lower mantle.     

Chemical composition of Archaean komatiites: Implications for early history of the earth and mantle evolution

Estimates of the chemical composition of the Archaean mantle, derived from elemental abundance ratios in komatiites combined with ultramafic xenolith data, support a model of a multistage heterogeneous accretion history of the Earth and synchronous core formation, 4.6 Ga ago.Most refractory lithophile element abundance ratios in komatiites and xenoliths are close to chondritic except for V/Ti and Ca/Al. Depletion of vanadium is likely due to its partial incorporation into the iron core; whereas fractionation of Ca/Al observed in Archaean Al-undepleted komatiites (1.20 times chondrites) and in some modern fertile spinel lherzolite xenoliths (1.15 times chondrites) could be due to small amounts of garnet (rich in Al but poor in Ca) segregation into the lower mantle during partial or complete melting of the upper mantle in the very early history of the Earth. However, this process may have had only a small effect on the overall chemical composition of the upper mantle.Simultaneous occurrence of early Archaean Al-undepleted (Al/Ti chondrites) and Al-depleted (Al/Ti 0.5 chondrites, and depletion of Sc and heavy REE) peridotitic komatiites in the Barberton area, S. Africa, and late Archaean Newton Township, Canada, argue against derivation of peridotitic komatiites from a circum-global magma ocean. Garnet separation from a mantle diapir which intersects the solidus at great depth ( 200 km) in a hotter early Archaean mantle could explain the chemical characteristics of Al-depleted komatiites. Alternatively, these two types of komatiites could have been derived from different layers in a fractionated mantle. A limited amount of Hf isotope data for Archaean komatiites seems to suggest that both mechanisms are important. This chemically and minerallogically layered mantle, if it existed, was homogenized by mantle convection after early Archaean times.Constant P₂O₅/TiO₂, Ni/Mg, Co/Mg, Fe/Mg ratios (siderophile/lithophile) and PGE abundances, estimated for the mantle sources of komatiites from Archaean to modern times, strongly argue against continuous growth of the Earth's core since the early Archaean.Extensive crustal contamination might have been involved in the generation of Archaean-early Proterozoic siliceous high magnesian basalts with “boninite affinity”. However, involvement of chemically modified ancient continental lithosphere may also be important in the generation of these basalts.     

An asymptotic solution of hydrodynamic equations at the mid-plane of a plume in the Earth's mantle

From the partial differential equations of hydrodynamics governing the movements in the Earth's mantle of a Newtonian fluid with a pressure- and temperature-dependent viscosity, considering the bilateral symmetry of velocity and temperature distributions at the mid-plane of the plume, an analytical solution of the governing equations near the mid-plane of the plume was found by the method of asymptotic analysis. The vertical distribution of the upward velocity, viscosity and temperature at the mid-plane, and the temperature excess at the centre of the plume above the ambient mantle temperature were then calculated for two sets of Newtonian rheological parameters. The results obtained show that the temperature at the mid-plane and the temperature excess are nearly independent of the rheological parameters. The upward velocity at the mid-plane, however, is strongly dependent on the rheological parameters.     

Hafnium/rare earth element fractionation in the sedimentary system and crustal recycling into the Earth's mantle

Among long-lived radioactive parent-daughter element pairs, the ratio Lu/Hf is strongly fractionated relative to constant Sm/Nd in the Earth's sedimentary system. This is caused by high resistance to chemical weathering of the mineral zircon (Zr,Hf)SiO₄. Zircon-bearing sandy sediments on and near continents have very low Lu/Hf, while deep-sea clays have up to three times the chondritic Lu/Hf ratio. Turbidity currents mechanically carry the low-Lu/Hf sandy material onto the ocean floor. The results are important for the crust-to-mantle recycling discussion, where most recycled materials would be subducted oceanic sediments. Such sediment should be capable of explaining the HfNd mantle isotopic variation by mixing with peridotite, but in fact any average pelagic sediment has Nd/Hf and Lu/Hf too high to allow mixing curves to pass through the Hf/Nd isotopic array. The array could only be reproduced by subduction of turbidite sandstone with pelagic sediment in the approximate ratio 1.2 to 1, and by maintaining a good mixture between the two components. At least today, turbidites are available for subduction only at locations quite different and distant from those where pelagic sediments may be recycled; furthermore, mantle isotopic variation shows that the mantle often cannot mix itself well enough to homogenize these widely-separated sedimentary components to the degree required. The Lu/Hf fractionations place a severe restriction on the ability of recycled sediments to explain mantle isotopic patterns.     

Nagssugtoqidian mobile belt of West Greenland: a cryptic 1850 Ma suture between two Archaean continents—chemical and isotopic evidence

New chemical and isotopic data permit the recognition of a cryptic suture zone between two Archaean continental masses within the Nagssugtoqidian mobile belt of West Greenland. This discovery has important implications for Precambrian crustal evolution: suture zones may not always be identifiable from geological field observations, with the consequence that mobile belts in which undetected sutures exist may be mis-identified as ensialic, and thought to require special non-plate tectonic models to account for their development.The Nagssugtoqidian belt consists mainly of Archaean gneisses reworked during the Proterozoic, with metamorphic grade and degree of isotopic disturbance increasing towards the centre of the belt. At the centre of the belt the Nagssugtoqidian includes metasediments and calc-alkaline volcanic and plutonic rocks of Proterozoic age, almost always strongly deformed and metamorphosed. From isotopic evidence (Sr_i ca. 0.703; model μ₁ values ca. 8.0; initial ε_Nd ca. 0) it is clear that the Proterozoic igneous rocks do not include any significant contributions derived from the Archaean crust, and the chemistry of the rocks, together with the isotope data, suggests that they were formed at a destructive plate margin. The Proterozoic rocks are found in a narrow zone (up to 30 km wide) between the Archaean gneisses to the north and south of Nordre Strømfjord, and are interpreted as reflecting the existence of a suture between two Archaean continental blocks. Zircon UPb data and other isotope evidence show that subduction started before ca. 1920 Ma ago, and lasted until ca. 1850 Ma when collision occurred, with consequent crustal thickening, high-grade metamorphism and local anatexis. Given the time-span for the operation of subduction, the existence of a wide Nagssugtoqidian ocean can be inferred, even for slow rates of plate motion.The Proterozoic and Archaean gneisses in the Nagssugtoqidian belt are very similar lithologically and chemically, and it has only been possible to distinguish between them using isotopic criteria. Suture zones of this kind are very difficult to detect, and may be present elsewhere within the reworked Archaean terrains of northern Greenland and Canada.