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).     

How strong is the magnetic field in the Earth's liquid core?

A crucial step in the investigation of the energetics of motions in the Earth's core and the generation of the geomagnetic field by the hydromagnetic dynamo process is the estimation of the average strength $\text{B}$ of the magnetic field B = B_p + B_T in the core. Owing to the probability that the toroidal field B_T in the core, which has no radial component, is a good deal stronger than the poloidal field B_p, direct downward extrapolation of the surface field to the core-mantle interface gives no more than an extreme lower limit to $\text{B}$. This paper outlines the indirect methods by which $\text{B}$ can be estimated, arguing that $\text{B}$ is probably about 10^?2 T (100 Γ) but might be as low as 10^?3 T (10 Γ) or as high as 5 × 10^?2 T (500 Γ).     

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.     

Earth's magnetic field modelling and earth's structure and evolution

The advantages of the approximation of the Earth's magnetic field by means of the field of the so-called natural magnetic sources are discussed. The shifting of these natural magnetic sources, determined for different epochs, is used to forecast the Earth's magnetic field and to draw conclusions about the motion of the corresponding part of the Earth. On the basis of the representation of the Earth's magnetic field from several past geological epochs as a field of one optimum dipole a new theory about the Earth's evolution is proposed.     

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.     

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.     

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.     

Upwelling flow in Earth's mantle and sea-floor spreading

Upwelling flows in the Earth's mantle are accompanied by mass, momentum and energy transports from deep to upper layers. Those flows beneath the mid-ocean ridges give rise to sea-floor spreading. Mantle plumes, on the other hand, cause hot spots to be formed on the Earth's surface. Using the basic equations of fluid dynamics, temperature and velocity distributions in two-dimensional upwelling and cylindrical plumes can be obtained by an integral-relation method. Then the mass, momentum and energy transported to the lithosphere by these upwelling flows can readily be calculated. Based on those results we can more thoroughly discuss problems of plate dynamics, such as the driving mechanism of plate motion, the causes of formation of rift valleys over mid-ocean ridges, and the effect of mantle plumes on sea-floor spreading.     

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.     

Tomographic inversion for three-dimensional anisotropy of Earth’s inner core

Seismological studies generally suggest that the Earth’s inner core is anisotropic and the anisotropic structure changes significantly both laterally and with depth. Previous body-wave studies of the inner core have relied on ray tracing or waveform modeling using one-dimensional (1D) models. Here we present non-linear tomographic inversions of the inner core anisotropy using three-dimensional (3D) ray tracing, spline parameterization, and a large collection of PKP differential travel times. We adapt a pseudo-bending ray tracing (PBR) method in spherical coordinates for seismic rays that traverse the inner core (PKP(DF) phase). The method iteratively perturbs each discontinuity point and continuous segment of the ray through 3D earth structure so that its travel time is minimum. The 3D anisotropic structure of the inner core is approximated to the first order as 3D heterogeneous (but isotropic) structure for a given ray. The data are corrected using a scaled mantle tomographic model. The inner core anisotropy model obtained has the following major features. (1) The model has strong hemispherical and depth variation. The isotropic velocity in the topmost inner core is greater in quasi-eastern hemisphere (QEH) (40–160°E) than in quasi-western hemisphere (QWH) (other longitudes). The anisotropy is weak in QEH to the depth of 600–700 km below the inner core boundary (ICB), while in QWH, the anisotropy increases at much shallower depth (about 100–200 km below the ICB) to about 3–4%, then remains at about 2–4% throughout the rest of the inner core. (2) The anisotropy form changes abruptly (over a depth range of about 150 km) at the radius of about 600 km, slightly less than half of the inner core radius, forming a distinct inner inner core (IIC). The velocity in the IIC has maximums at equatorial and polar directions and minimum at an angle of about 40° from the equatorial plane. The velocity in the outer inner core (OIC), however, changes little for ray directions 0–40° from the equatorial plane. (3) Despite large variation of the anisotropy, the isotropic velocity (Voigt average) throughout the inner core is nearly uniform. The results suggest that the OIC is likely composed of the same type of iron crystals with uniform chemistry, but the IIC may be composed of a different type of crystal alignment, a different iron phase, or a different chemical composition. Our tests on model parameterization, mantle correction, and linear and non-linear inversion suggest the main features of our model are very robust. However, fine scale structures are likely to differ, particularly in the major transition zones, e.g., in the topmost QWH (isotropy to anisotropy), between OIC and IIC (change in the form of anisotropy), and between QEH and QWH in OIC (difference in anisotropy strength). Searches for possible waveform complications from these boundaries need to be aware of the directional dependence and geographical variation to be successful.     

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.     

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.     

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.     

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.     

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.     

Thermal regime in the earth's core and lower mantle

Current views favour the presence of sulphur in the core, giving a composition of Fe + FeS. It is argued that the core composition is close to the eutectic and that this eutectic composition is Fe₂S. The consequences for the thermal regime in the core are examined in terms of the most likely properties of the Fe₂S eutectic. This leads to much lower temperatures than would be expected for an iron or FeSi core.Consideration of the thermal regime in the mantle and the probable thermal properties of lower-mantle assemblages leads to a similar low temperature for the core-mantle boundary. These temperatures require a temperature gradient near the adiabatic in the mantle, implying a convective thermal history.     

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.