On thermodynamic description of Mie-Grüneisen materials with Debye specific heats

The entropy of a material obeying a Mie-Grüneisen equation of state and a Debye specific heat is obtained in closed form. Expressions for temperature, isothermal compression, and adiabatic compression are also deduced.     

Self-suppression of phase transitions exemplified by the effects of (de)magnetization of magnetite at the Curie point under high pressure

Flux linkage between coaxial current and pick-up coils wound on a magnetite core, and measurements of the secondary voltage were used as the experimental basis for the determination of permeability. Measurements were made isobarically at pressures of up to 5.6 GPa on polycrystalline samples and single crystals with (111) axis. The Curie temperature, maximum and minimum permeability values, and the temperature coefficient of the permeability were determined as functions of the pressure. The pressure coefficient of the Curie temperature was found to vary from 4.7 to 20 K GPa^?1 for polycrystalline samples and from 17 to 22 k GPa^?1 for single crystals. The ferri-paramagnetic transition, although sharp, occurs over a finite temperature interval. At settings within this interval of rapidly-varying magnetization, the secondary voltage, signifying the permeability, was observed to be unstable and repeatedly self-reversing, thus resulting in oscillations of the values approaching the permeability maximum and, in reverse, the permeability minimum toward the paramagnetic state. Two explanations offer themselves for self-reversals: (1) core temperature changes resulting from changes in eddy current losses; and (2) magnetocaloric oscillations associated with spontaneous (de)magnetization. A case for magnetocaloric oscillations can be made on the grounds that spontaneous magnetization is accompanied by spontaneous strains, change in the specific heat and anomalous thermal expansion. The transition thus provides an example for a situation in which the isentropic values of relative variations of temperature with relative change in volume, which define Grüneisen's gamma, are anomalous.     

The application of CFRP to strengthen buried steel pipelines against subsurface explosion

Multiple explosions in the route of oil and gas transmission pipelines during recent years demonstrate that terrorist attacks and sabotages have unfortunately increased. The present investigation is carried out numerically in order to minimize the amount of damages imposed on steel pipelines under close-in explosions. This research presents a novel concept, using CFRP (Carbon Fiber Reinforced Polymer) to strengthen the wall of steel pipelines against these incidents. For this purpose, a full coupled 3D finite element model developed using a combined Eulerian-Lagrangian method. The simplified Johnson-Cook material model, the JWL equation of state, and the ideal gas equation of state were employed for modeling the pipe material behavior, charge detonation, and air, respectively. Mechanical behavior of the composite wrap was simulated using an anisotropic material model and the damage initiation criteria were based on Hashin's theory. In addition, soil mass behavior was modeled applying a Drucker-Prager strength criterion with piecewise hardening and hydro tensile limit accompanied by Mie-Grüneisen equation of state. Several comparisons carried out between the results from present investigation and those from field and empirical studies and good agreements were observed. The results show that using a proper thickness of CFRP wrap for every particular circumstance can significantly improve the performance of steel pipelines under blast loads. For instance, in the current example, maximum equivalent strains developed in the most of the studied pipelines decreased by over 30% (up to 64%) with the application of 4-mm-thickness CFRP wrap. The present study contributes to protective design of steel pipelines.     

Anhydrous and water-saturated melting experiments of an olivine andesite from Mt Yakushi-Yama, northeastern Shikoku, Japan

Abstract Melting experiments have been carried out on an olivine andesite of Mt Yakushi-Yama from the Miocene Setouchi volcanic belt in northeastern Shikoku, Japan. This andesite has been characterized by a low ratio of FeO*/Mg° (= 0.78). Phase relations have been determined within the pressure range of 2.8 to 19.3 kbar at 1000-1300°C under anhydrous and water-saturated conditions. At pressures less than 8.8 kbar, olivine is a liquidus phase. Orthopyroxene appears on the liquidus at 9.3 kbar under the anhydrous conditions. The multiple saturation point rises up to 17.5 kbar for water-saturated experiments. The andesite melt coexists with olivine and orthopyroxene just below the liquidus at 8.8–9.3 kbar and 1230°C for dry conditions, and at 17.5 kbar and 1060°C under water-saturated conditions. These experimental results indicate that the Yakushi-Yama olivine andesite magma could coexist with a harzburgitic mantle at depths between about 30 and 60 km, and at temperatures between 1060 and 1230°C. Experimental data also suggest a possibility that a high magnesian andesite magma would be generated by a direct partial melting of the uppermost harzburgitic mantle under anhydrous conditions.     

Thermal histories of the core and mantle

Recognition that the cooling of the core is accomplished by conduction of heat into a thermal boundary layer (D″) at the base of the mantle, partly decouples calculations of the thermal histories of the core and mantle. Both are controlled by the temperature-dependent rheology of the mantle, but in different ways. Thermal parameters of the Earth are more tightly constrained than hitherto by demanding that they satisfy both core and mantle histories. We require evolution from an early state, in which the temperatures of the top of the core and the base of the mantle were both very close to the mantle solidus, to the present state in which a temperature increment, estimated to be ～ 800 K, has developed across D″. The thermal history is not very dependent upon the assumption of Newtonian or non-Newtonian mantle rheology. The thermal boundary layer at the base of the mantle (i.e., D″) developed within the first few hundred million years and the temperature increment across it is still increasing slowly. In our preferred model the present temperature at the top of the core is 3800 K and the mantle temperature, extrapolated to the core boundary without the thermal boundary layer, is 3000 K. The mantle solidus is 3860 K. These temperatures could be varied within quite wide limits without seriously affecting our conclusions. Core gravitational energy release is found to have been remarkably constant at ～ 3 × 10¹¹ W. nearly 20% of the core heat flux, for the past 3 × 10⁹ y, although the total terrestrial heat flux has decreased by a factor of 2 or 3 in that time. This gravitational energy can power the “chemical” dynamo in spite of a core heat flux that is less than that required by conduction down an adiabatic gradient in the outer core; part of the gravitational energy is used to redistribute the excess heat back into the core, leaving 1.8 × 10¹¹ W to drive the dynamo. At no time was the dynamo thermally driven and the present radioactive heating in the core is negligibly small. The dynamo can persist indefinitely into the future; available power 10¹⁰ y from now is estimated to be 0.3 × 10¹¹ W if linear mantle rheology is assumed or more if mantle rheology is non-linear. The assumption that the gravitational constant decreases with time imposes an implausible rate of decrease in dynamo energy. With conventional thermodynamics it also requires radiogenic heating of the mantle considerably in excess of the likely content of radioactive elements.     

Generation of komatiite magma and gravitational differentiation in the deep upper mantle

The densities of silicate liquids with basic, picritic, and ultrabasic compositions have been estimated from the melting curves of minerals at high pressures. Silicate liquids generated by partial melting of the upper mantle are denser than olivine and pyroxenes at pressures higher than 70 kbar, and garnet is the only phase which is denser than the liquid at pressures from 70 kbar to at least 170 kbar. In this pressure range, garnet and some fraction of liquid separate from ascending partially molten diapirs. It is therefore suggested that aluminium-depleted komatiite with a high Ca/OAl₂O₃ ratio may be derived from diapirs which originated in the deep upper mantle at pressures from 70 kbar to at least 140 kbar (200–400 km in depth), where selective separation of pyropic garnet occurs effectively. On the other hand, aluminium-undepleted komatiite is probably derived from diapirs originating at shallower depths (< 200 km). Enrichment of pyropic garnet is expected at depths greater than 200 km by selective separation of garnet from ascending diapirs. The 200-km discontinuity in the seismic wave velocity profile may be explained by a relatively high concentration of pyropic garnet at depths greater than 200 km.     

Applications of thermodynamics to fundamental earth physics

New fundamental thermodynamic relationships of complete generality and absolute rigour of derivation are not to be expected, because the subject has such a secure and complete basis in classical physics. There is, however, still scope for original, fundamental work based on recognised assumptions and approximations which may be obviously acceptable in particular situations. Clarification of relationships between thermodynamic parameters for materials within the Earth is particularly important because there is so little possibility of measuring them individually. This survey first summarises the established relationships in a very condensed form and then concentrates on some recent developments which have direct bearing on the thermal and mechanical states of the Earth's mantle and core. Considerable use is made of the thermodynamic Grüneisen parameter, which is a dimensionless quantity of order unity for almost all materials, solid, liquid and gaseous, and is directly related to the pressure dependences of elastic constants. This allows its value to be estimated for the different regions of the Earth from seismological data. The thermodynamic (heat engine) efficiency of convection in a homogeneous medium, driving tectonic activity or the geomagnetic dynamo, is found to be the ideal (Carnot) efficiency corresponding to adiabatic temperature differences between the heat source and sink, within the assumption that the thermal expansion coefficient is not strongly temperature dependent. The use of this conclusion to infer tectonic stresses is indicated. The thermodynamic basis for Lindemann's melting law is restated and the reasons for supposing it to be valid for materials at megabar pressures reaffirmed. Application to the inner core boundary gives a fixed point on the Earth's temperature profile. Use of thermodynamic relationships in the interpretation of shock wave compressions is indicated.     

The high-temperature acoustic Grüneisen parameter in the earth's interior

The value of the acoustic Grüneisen parameter, γ_a, in the earth's interior has been computed using data from recent models obtained by inversion of normal data. In this paper we emphasize the data from the PEM model of the earth because there has been sufficient smoothing of the seismic data so that the derivatives d ln ν_s/d ? and d ln ν_p/d ? can be well defined at all depths.The results for the lower mantle show that γ_a decreases exponentially from 1.3 to 1.0, and there are several consistent cross-checks of the limiting values. We find γ_a is about 1.5 for the inner core and outer core. These results confirm, in broad outline, the results of others who computed γ for the core by entirely different methods. They also confirm a higher value of γ in the inner core. The value of γ_a in the lower mantle follows a ρ^?1.35 law, which is reminiscent of the expirical law γρ = constant, commonly used in shock-wave analyses.     

Melting relations of a magnesian abyssal tholeiite and the origin of MORBs

Melting relations of a glassy magnesian olivine tholeiite from the FAMOUS area have been studied within the pressure range 1 atm to 15 kbar. From 1 atm to 10 kbar, olivine is the liquidus phase, followed by plagioclase and Ca-rich clinopyroxene. Above 10 kbar, Ca-rich clinopyroxene appears on the liquidus, followed by orthopyroxene and spinel. Near 10 kbar, olivine, orthopyroxene, clinopyroxene, spinel and plagioclase crystallize within 10°C of the liquidus. This indicates that a liquid of this magnesian olivine tholeiite composition could coexist with mantle peridotite at about 10 kbar. This result is in agreement with the geochemistry of Ni; the Ni concentration of the studied sample corresponds to the theoretical concentration in a primary magma [14,15].These data suggest that at least some magnesian mid-oceanic ridge basalts (MORBs) could be primary melts segregated from the mantle at depths near the transition zone between plagioclase lherzolite and spinel lherzolite (about 10 kbar). Based on this model, the residual mantle after extraction of MORBs should be lherzolite, not harzburgite.High-pressure (7–10 kbar) fractionation models involving olivine, plagioclase and clinopyroxene, which have been proposed by several workers (e.g. [36]) to explain the varieties of MORBs, were re-emphasized based on this melting study. The rare occurrence of clinopyroxene as a phenocryst phase in MORBs is explained by precipitation in a magma chamber at high pressure, or by dissolution of clinopyroxene formed earlier at high pressure.     

On lithospheric flexure seaward of the Bonin and Mariana trenches

More than 25 bathymetry profiles have been used to examine the flexure of the Pacific lithosphere seaward of the Izu-Bonin and Mariana trenches. Selected bathymetry profiles have been corrected for the effects of sediment loading and compared to simple elastic and elastic-plastic models for lithospheric flexure seaward of these trenches. Profiles of the northern Mariana trench, where the seaward wall is relatively gentle, can be explained by a simple elastic model without an applied horizontal load. Profiles of the Izu-Bonin and southern Mariana trenches, where the seaward wall is relatively steep, can be explained by an elastic-plastic model with an applied load of 4.0–6.0 kbar, depending on the uniform yield stress assumed. If it is assumed the yield stress varies with depth the horizontal load required is significantly reduced (?2.5kbar). The magnitude of the horizontal load cannot be determined with certainty, however, since it is not known how the yield stress may vary with depth. The elastic-plastic models examined all required significant differences (～1.0kbar) in the horizontal load along the Izu-Bonin and Mariana trenches. These differences, which reach a maximum between the northern Izu-Bonin and northern Mariana trenches, appear to correlate with changes in the pattern of seismicity and tectonics landward of these trenches.     

Formation,history and energetics of cores in the terrestrial planets

The cores of the terrestrial planets Earth, Moon, Mercury, Venus and Mars differ substantially in size and in history. Though no planet other than the Earth has a conclusively demonstrated core, the probable cores in Mercury and Mars and Earth's core show a decrease in relative core size with solar distance. The Moon does not fit this sequence and Venus may not. Core formation must have been early (prior to ～4 · 10⁹ yr. ago) in the Earth, by virtue of the existence of ancient rock units and ancient paleomagnetism and from UPb partitioning arguments, and in Mercury, because the consequences of core infall would have included extensional tectonic features which are not observed even on Mercury's oldest terrain. If a small core exists in the Moon, still an open question, completion of core formation may be placed several hundred million years after the end of heavy bombardment on tectonic and thermal grounds. Core formation time on Mars is loosely constrained, but may have been substantially later than for the other terrestrial planets. The magnitude and extent of early heating to drive global differentiation appear to have decreased with distance from the sun for at least the smaller bodies Mercury, Moon and Mars.Energy sources to maintain a molten state and to fuel convection and magnetic dynamos in the cores of the terrestrial planets include principally gravitational energy, heat of fusion, and long-lived radioactivity. The gravitational energy of core infall is quantifiable and substantial for all bodies but the Moon, but was likely spent too early in the history of most planets to prove a significant residual heat source to drive a present dynamo. The energy from inner core freezing in the Earth and in Mercury is at best marginally able to match even the conductive heat loss along an outer core adiabat. Radioactive decay in the core offers an attractive but unproven energy source to maintain core convection.     

Equation of state of gold and its application to the phase boundaries near 660 km depth in Earth’s mantle

The pressure-volume-temperature equation of state (EOS) of gold is fundamental to high-pressure science because of its widespread use as an internal pressure standard. In particular, the EOS of gold has been used in recent in situ multi-anvil press studies for determination of phase boundaries related to the 660-km seismic discontinuity. These studies show that the boundaries are lower by 2 GPa than expected from the depth of the 660-km discontinuity. Here we report a new P-V-T EOS of gold based on the inversion of quasi-hydrostatic compression and shock wave data using the Mie-Grüneisen relation and the Birch-Murnaghan-Debye equation. The previously poorly constrained pressure derivative of isothermal bulk modulus and the volume dependence of Grüneisen parameter (q=d lnγ/d ln V) are determined by including both phonon and electron effects implicitly: K′_0T=5.0±0.2 and q=1.0±0.1. This combined with other accurately measured parameters enables us to calculate pressure at a given volume and temperature. At 660-km depth conditions, this new EOS yields 1.0±0.2 GPa higher pressure than Anderson et al.’s EOS which has been used in the multi-anvil experiments. However, after the correction, there still exists a 1.5-GPa discrepancy between the post-spinel boundary measured by multi-anvil studies and the 660-km discontinuity. Other potential error sources, such as thermocouple emf dependence on pressure or systematic errors in spectroradiometry, should be investigated. Theoretical and experimental studies to better understand electronic and anharmonic effects in gold at high P-T are also needed.     

Models for the origin and composition of the earth,and the hypothesis of potassium in the Earth's core

Progress in understanding the condensation of planetary constituents from a solar nebula necessitates a re-examination of models for the origin and composition of the Earth. All models which appear to be viable require the Earth to have an Fe–FeS core and the full, or nearly full, solar (i.e. chondritic) K/Si ratio. The crust and upper mantle do not contain the requisite potassium for the entire Earth to have the solar K/Si ratio. Therefore, these models require that much of the Earth's potassium, about 80–90%, must be in the deep interior—in the lower mantle or in the core.The hypothesis that a substantial fraction of the Earth's potassium is in the Fe–FeS core is based on the chalcophilic behavior of potassium. Data including the stability of K₂S, the occurrence of potassium in sulfide phases in meteorites and in metallurgical systems, and most importantly, experiments on potassium partitioning between solid silicates and Fe–FeS melts support this hypothesis. The present data appear to require at least several percent of the Earth's total potassium to be in the core. Incorporation of much larger amounts of potassium into the core, possibly most of the 80–90% of the Earth's potassium which is postulated to be in the deep interior, is not contradicted by the present data. Additional experimental data, at high pressures, are required before quantitative estimates of the core's potassium content can be made.It is likely that⁴⁰K is a significant heat source in the core. Decay of⁴⁰K is a plausible energy source to drive core convection to maintain the geomagnetic field, and to drive mantle convection and seafloor spreading.     

The onset of convection in a rotating layer of viscous fluid with an imposed magnetic field: the dependence on the prandlt numbers

The onset of Boussinesq convection in a horizontal layer of an electrically conducting incompressible fluid is considered. The layer rotating about a vertical axis is heated from below; a vertical magnetic field is imposed. Rigid electrically insulating boundaries are assumed. The loss of stability of the trivial steady state, which occurs as the Rayleigh numbers increase, can be accompanied by the development of a monotonic or an oscillatory instability, depending on the parameter values of the problem at hand (the Taylor number, the Chandrasekhar number, the kinematic and the magnetic Prandtl numbers). When the instability is monotonic, the emerging convective rolls themselves are also unstable if the Taylor number is sufficiently large (the so-called Küppers-Lortz instability takes place). In the present work it is studied how the critical value of the Rayleigh number, the type of the trivial steady state instability, and the critical value of the Taylor number for the Küppers-Lortz instability depend on the kinematic and the magnetic Prandtl numbers. We consider the values of the Prandtl number not exceeding 1, which is typical for the outer core of the Earth.     

The electronic thermodynamics of iron under Earth core conditions

The LMTO method is used to calculate the electronic band structure of iron in the ϵ-phase (hcp) and in the γ-phase (fcc) for seven compressions from 4 to 980 GPa. The electronic specific heat c_υe(T) is calculated for each phase by numerical integration from the resultant density of states. Previous work is thus supported for γ-iron and extended to ϵ-iron, the most likely inner core component. A simple parameterization of c_υe is given for use in making geophysical estimates. Other thermodynamic parameters which are calculated are the electronic free energy, the thermal electronic pressure, and an electronic Gruneisen parameter, γ_e.Recent studies of liquid iron and iron alloys indicate that the density of states at the Fermi level does not differ much from that calculated for pure crystalline iron. We cautiously apply our results to the outer core and find that c_υe = 1.7 ± 0.7R and γ_e = 1.3 ± 0.4. This indicates that the total heat capacity of the core is one-quarter that of the entire Earth.     

Phase transformations in MSiO4 compounds at high pressures and their geophysical implications

The phase behaviour of MSiO₄ compounds (MHf, Zr, U and Th0 has been investigated at high pressures and temperatures in a diamond-anvil press coupled with laser heating. All of these compounds have been found to undergo two or perhaps three phase transformations at pressures below 300 kbar. The high-pressure phase transformations of these compounds differ from one another, with the exception of HfSiO₄ and ZrSiO₄, which undergo identical phase transformations. The ultimate phase assemblages of these compounds are of dense component dioxides (although this is yet to be confirmed in ThSiO₄). It is suggested that the heat-producing elements U and Th would exist as dioxide solid solutions rather than silicates in the deep interior of the earth. Moreover, the densities of these dioxides are more than twice as great as mantle silicates and even slightly greater than pure iron under similar P, T conditions. Gravitational separation due to mandle convection may transport these dioxides to the deep interior of the earth to form deep heat sources. It is also suggested, however, that these deep heat sources are located in the inner-outer core boundary instead of in the lower mantle.     

Using numerical modeling for assessing the recoverable reserves of a geothermal steam field: The Pauzhetka geothermal field

A conceptual hydrogeological model has been created and a corresponding 3D numerical, thermal hydrodynamic model developed for the Pauzhetka geothermal field; the model covers an area of 13.6 km² and includes three layers: a basement with conduits that supply the heat carrier, a hydrothermal reservoir, and an upper aquifer with percolation “windows.” Inversion is handled by the iTOUGH2 program; the model was calibrated using the 1960-2006 data on the natural state and extraction at 13675 calibration points. The inversion simulation has made it possible to identify and evaluate the key parameters of the model and to identify the sources that generate the recoverable reserves. Forecasting modeling for the period from 2007 to 2032 shows a sustainable extraction of 29 kg/s steam, provided five additional wells have been put into operation, which will provide 6.8 MWs of production by the geothermal power plant. The results of forecasting modeling, in combination with observations on long-term operation, allow an evaluation of the recoverable reserves in the industrial categories.     

Geothermal gradient of the upper mantle beneath Jeju Island, Korea: Evidence from mantle xenoliths

Abstract Ultramafic xenoliths found in alkali basalts from Jeju Island, Korea are mostly spinel lherzolites accompanied by subordinate amount of spinel harzburgites and pyroxenites. The combination of results from a two-pyroxene geothermometer and Ca-in-olivine geobarometer yields temperature–pressure (T–P) estimates for spinel peridotites that fall in experimentally determined spinel lherzolite field in CaO-Fe-MgO-Al₂O₃-SiO₂-Cr₂O₃ (CFMASCr) system. These T–P data sets have been used to construct the Quaternary Jeju Island geotherm, which defines a locus from about 13 kbar at 880°C to 26 kbar at 1040°C. The geothermal gradient of Jeju Island is greater than that of the conventional conductive models, and may be as a result of a thermal perturbation by the heat input into the lithospheric mantle via the passage and emplacement of magma. Spinel–lherzolite is the main constituent rock-type of the lithospheric mantle beneath Jeju Island. Pyroxenites may be intercalated in peridotites at similar depth and temperature as re-equilibrated veins or lenses.