High-temperature anelasticity and elasticity of mantle peridotite

An apparatus has been devised which allows precise creep and relaxation measurements to be made on minerals and rocks at temperatures up to 1600°C and at very low deviatoric stresses (1 < σ < 300 bar). This paper is concerned with measurements on mantle peridotite (lherzolite) from Balmuccia (Zone of Ivrea, Italy).The reaction of the sample to a step-like increase in stress is called its “creep function”. It is shown that the creep function contains all the necessary information to derive the spectra of the quality factor Q(ω) and of Young's modulus E(ω), within the seismic range of frequencies, provided the material behaves as a linear system. This has been proven up to a strain of 5 × 10^?5.The Q^?1-spectra at 1200 and 1300°C, obtained by Fourier inversion from the creep function, show no pronounced peak in the frequency band 0.01 < tf < 1 Hz and exhibit a general tendency to decrease slightly with frequency. The creep function: ?(t) = ?_u · [1 + 3.7 · q · {(1 + 50t)^0.27 ? }], where q is related to Q, satisfactorily describes the data at high temperatures and leads to $\text{Q}{}^{\mathrm{?1}}\text{(ω, T) = 3 × 10}{}^3\text{· ω}{}^{\mathrm{<mtext>?0.27\; ·\; </mtext><mtext>exp</mtext>}}\text{(}\text{?30}\text{RT}\text{)}$E(ω) is related to Q(ω) by the material dispersion equation. Above 1100°C the unrelaxed Young's modulus decreases rapidly with temperature according to an activation energy of about 20 kcal/mole. A lowering of short period S-wave velocity by 40% and P-wave velocity by 10% occurs below the solidus. Therefore, no partial melting is required in the asthenosphere.Steady-state creep at low axial stresses (20 < σ < 100 bar), obtained from the same rock, follows the relation $\text{?}\text{?}\text{= 3 × 10}{}^7\text{· δ}{}^{1.4}\text{·}\text{exp}\text{(}\text{?125}\text{RT}\text{)}$ indicative of grain boundary diffusion or superplasticity. At higher stresses a power law $\text{?}\text{?}\text{= 45 · δ}{}^4\text{·}\text{exp}\text{(}\text{?125}\text{RT}\text{)}$ typical of dislocation creep, is found.The frequency dependence of Q and the ratio of the activation energies of Q and are indicative of so called “high-temperature background absorption”, as the dominant mechanism, and of a diffusion-controlled dislocation mobility common to both absorption and creep. From a, b, and c, relations between the effective viscosity η_f and Q of the form: $\text{log}\text{η}{}_{\mathrm{e??}}\text{=}\text{1}\text{α}\text{·}\text{log}\text{Q ? (n ? 1) ·}\text{log}\text{ω +}\text{log}\text{D}$ are derived, where α ～ 0.25, n is the power of σ, and D is a constant.     

Upper mantle plasticity from laboratory experiments

On the basis of two assumptions i.e. (1) plastic and anelastic behaviour of the upper mantle can be approximated by the behaviour of the dominant mineral olivine, and (2) the behaviour of natural olivine and synthetic forsterite are similar, we have investigated the flow laws and the flow microstructures of forsterite single crystals. The results obtained between 1400–1650°C and 10–100 MPa suggest a model of climb controlled creep in which the a edge dislocations are dominant. The activation energy measured in that regime is 4.7 eV, close to that of Si self-diffusion and the flow law is $\text{?}\text{?}\text{=10}{}^6\text{σ}{}^{2.6}\text{exp}\text{(?4.7}\text{eV/k}\text{T)}$, where σ is in MPa. Extrapolation of these results to the upper mantle would imply very low stresses (i.e. ?10 MPa) in the asthenosphere. However the effect of pressure and grain size are unknown and extrapolation to very low stresses is not straightforward.     

Creep response of the lunar crust in mare regions from an analysis of crater deformation

The settling trends of 318 lunar mare craters are compared with predictions of numerical finite-element models in order to determine the creep response of the upper lunar mare crust. No settling is evident in craters smaller than 5 km in diameter. Settling rates of larger craters increase as function of crater size in a manner suggesting a non-linear lunar creep response corresponding to the power law $\text{ε}\text{?}\text{= 8.3 · 10}{}^{\mathrm{?34}}\text{σ}{}^2$ where ε&#x0301; is the strain rate and σ is the differential stress. However, the observed nonlinearity is probably an apparent nonlinearity resulting from the temperature induced viscosity decrease with depth due to a lunar crustal temperature gradient of 3° C/km and a creep activation energy of 20 kcal/mole. It is concluded that creep in the lunar medium is essentially Newtonian, and that the effective viscosity of the upper lunar mare crust is (1.6 ± 0.3) · 10²⁵ poise.     

High-pressure recovery of olivine: implications for creep mechanisms and creep activation volume

High-temperature and high-pressure recovery experiments were made on experimentally deformed olivines at temperatures of 1613–1788 K and pressures of 0.1 MPa to 2.0 GPa. In the high-pressure experiments, a piston cylinder apparatus was used with BN and NaCl powder as the pressure medium, and the hydrostatic condition of the pressure was checked by test runs with low dislocation density samples. No dislocation multiplication was observed. The kinetics of the dislocation annihilation process were examined by different initial dislocation density runs and shown to be of second order, i.e.

$\text{d}\text{ρ}\text{d}\text{t}\text{= ?p}{}^2\text{K}{}_0\text{exp}\text{[?(}\text{E}{}^1\text{+PV}{}^1\text{RT}\text{]}$

where ρ is the dislocation density, k₀ is a constant, $\text{E}{}^1\text{and}\text{V}{}^1$ are the activation energy and volume respectively, and P, R and T are pressure, gas constant and temperature, respectively. Activation energy and volume were estimated from the temperature and pressure dependence of the dislocation annihilation rate as $\text{E}{}^1\text{=389±59}\text{kJ mol}{}^{\mathrm{?1}}$ and $\text{V}{}^1\text{=14±2}\text{cm}{}^3\text{mol}{}^{\mathrm{?1}}$, respectively.The diffusion constants relevant to the dislocation annihilation process were estimated from a theoretical relation k=αD where $\text{k=k}{}_0\text{exp}\text{[?(E}{}^1\text{+ PV}{}^1\text{)/RT], D}$ is the diffusion constant and α is a non-dimensional constant of ca. 300. The results agree well with the self-diffusion constant of oxygen in olivine. This suggests that the dislocation annihilation is rate-controlled by the (oxygen) diffusion-controlled dislocation climb.The mechanisms of creep in olivine and dry dunite are examined by using the experimental data of static recovery. It is suggested that the creep of dry dunite is rate-controlled by recovery at cell walls or at grain boundaries which is rate-controlled by oxygen diffusion. Creep activation volume is estimated to be 16±3 cm³ mol^?1.     

Spatial power spectra of the crustal geomagnetic field and core geomagnetic field

Lowes (1966, 1974) has introduced the function R_n defined by $\text{R}{}^{\mathrm{n}}\text{=(n + 1)}\text{∑}\text{m=0}\text{∞}\text{[(g}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{+ (h}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{]}\text{where}\text{g}{}_{\mathrm{n}}{}^{\mathrm{m}}\text{and}\text{h}{}_{\mathrm{n}}{}^{\mathrm{m}}$ are the coefficients of a spherical harmonic expansion of the scalar potential of the geomagnetic field at the Earth's surface. The mean squared value of the magnetic field B = ??V on a sphere of radius r > α is given by $\text{〈}\text{B}\text{·〉 =}\text{∑}\text{n=1}\text{∞}\text{R}{}^{\mathrm{n}}\text{(}\text{a/}\text{r}\text{)}{}^{\mathrm{2n=4}}\text{where}\text{a}$ is the Earth's radius. We refer to R_n as the spherical harmonic spatial power spectrum of the geomagnetic field.In this paper it is shown that Rⁿ = R^M_n = R^C_n where the components R_n^M due to the main (or core) field and R_n^C due to the crustal field are given approximately by $\text{R}{}^{\mathrm{M}}{}_{\mathrm{n}}\text{= [}\text{(n =1)/}\text{(n + 2)}\text{](1.142 × 10}{}^9\text{)(0.288}{}^{\mathrm{n}}\text{Λ}{}^2\text{R}{}^{\mathrm{C}}{}_{\mathrm{n}}\text{= [}\text{(n =1){[1 — exp(-n/290)]/}\text{(n/290)} 0.52 Λ}{}^2\text{where}\text{Iγ = 1 nT}$. The two components are approximately equal for n = 15.Lowes has given equations for the core and crustal field spectra. His equation for the crustal field spectrum is significantly different from the one given here. The equation given in this paper is in better agreement with data obtained on the POGO spacecraft and with data for the crustal field given by Alldredge et al. (1963).The equations for the main and crustal geomagnetic field spectra are consistent with data for the core field given by Peddie and Fabiano (1976) and data for the crustal field given by Alldredge et al. The equations are based on a statistical model that makes use of the principle of equipartition of energy and predicts the shape of both the crustal and core spectra. The model also predicts the core radius accurately. The numerical values given by the equations are not strongly dependent on the model.Equations relating average great circle power spectra of the geomagnetic field components to R_n are derived. The three field components are in the radial direction, along the great circle track, and perpendicular to the first two. These equations can, in principle, be inverted to compute the R_n for celestial bodies from average great circle power spectra of the magnetic field components.     

Orthogonality of spherical harmonic coefficients

An essentially arbitrary function V(θ, λ) defined on the surface of a sphere can be expressed in terms of spherical harmonics $\text{V(θ, Λ) = a}\text{∑}\text{n=1}\text{∞}\text{∑}\text{m=0}\text{n}\text{p}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{(cos θ) (g}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{cos mΛ + h}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{sin mΛ)}$ where the P_n^m are the seminormalized associated Legendre polynomials used in geomagnetism, normalized so that $\text{〈[P}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{(cos θ) cos mΛ]〉}{}^2\text{=1/}\text{(2n+1)}$ The angular brackets denote an average over the sphere. The class of functions V(θ, λ) under consideration is that normally of interest in physics and engineering. If we consider an ensemble of all possible orientations of our coordinate system relative to the sphere, then the coefficients g_n^m and h_n^m will be functions of the particular coordinate system orientation, but $\text{〈:(g}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{〉) = 〈(h}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{=}\text{S}{}_{\mathrm{n}}\text{/}\text{(2n=1)}$ where $\text{S}{}_{\mathrm{n}}\text{=}\text{∑}\text{m=0}\text{n}\text{[(g}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{+ (h}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{]}$ for any orientation of the coordinate system (S_n is invariant under rotation of the coordinate system). The averages are over all orientations of the system relative to the sphere. It is also shown that 〈g^m_ng^l_p〉 = 〈h^m_nh^l_p〉 = 0 for l ≠ m or p ≠ n and 〈g^m_nh^l_p〉 = 0 fro all n, m, p, l.     

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.     

Effect of oxygen partial pressure on the dislocation recovery in olivine: a new constraint on creep mechanisms

The dislocation annihilation rate in experimentally deformed olivine single crystals was measured as a function of oxygen partial pressure (P_O2). It was shown that the dislocation annihilation rate decreased with increasing P_O2. This result is inconsistent with the reported P_O2 dependence of creep rate () in single olivine crystals, thus indicating that the creep in single olivine crystals is not rate-controlled by recovery, under the experimentally investigated conditions.     

The electric field produced in the mantle by the dynamo in the core

A rigorous singular perturbation theory is developed to estimate the electric field E produced in the mantle M by the core dynamo when the electrical conductivity σ in M depends only on radius r, and when |r?_rln σ| ? 1 in most of M. It is assumed that σ has only one local minimum in M, either (a) at the Earth's surface ?V, or (b) at a radius b inside the mantle, or (c) at the core-mantle boundary ?K. In all three cases, the region where σ is no more than e times its minimum value constitutes a thin critical layer; in case (a), the radial electric field E_r ≈ 0 there, while in cases (b) and (c), E_r is very large there. Outside the critical layer, E_r ≈ 0 in all three cases. In no case is the tangential electric field E_S small, nearly toroidal, or nearly calculable from the magnetic vector potential A as ??_tA_S. The defects in Muth's (1979) argument which led him to contrary conclusions are identified. Benton (1979) cited Muth's work to argue that the core-fluid velocity u just below ?K can be estimated from measurements on ?V of the magnetic field B and its time derivative $\text{?}{}_{\mathrm{t}}\text{B}$. A simple model for westward drift is discussed which shows that Benton's conclusion is also wrong.In case (a), it is shown that knowledge of σ in M is unnecessary for estimating E_S on ?K with a relative error |r?_r 1nσ|^?1from measurements of E_S on ?V and knowledge of ?_tB in M (calculable from ?_tB on ?V if σ is small). Then, in case (a), u just below ?K can be estimated as $\text{?}\text{r}\text{×}\text{E}{}_{\mathrm{S}}\text{/B}{}_{\mathrm{r}}$. The method is impractical unless the contribution to E_S on ?V from ocean currents can be removed.The perturbation theory appropriate when σ in M is small is considered briefly; smallness of σ and of |r?_r ln σ|^?1 a independent questions. It is found that as σ → 0, B approaches the vacuum field in M but E does not; the explanation lies in the hydromagnetic approximation, which is certainly valid in M but fails as σ → 0. It is also found that the singular perturbation theory for |r?_r ln σ|^?1 is a useful tool in the perturbation calculations for σ when both σ and |r?_r ln σ|^?1 are small.     

General relationships among sound speeds: II. Theory and discussion

The dependence of bulk sound speed V_φ upon mean atomic weight $\text{m}$ and density ρ can be expressed in a single equation:

$\text{V}{}_{\mathrm{\phi }}\text{=Bρ}{}^{\mathrm{\lambda }}\text{(}\text{m}{}_0\text{m}{}^{\mathrm{<mtext>[</mtext><mtext>1</mtext><mtext>2</mtext><mtext>+\lambda (1?c)]</mtext>}}\text{(}\text{km/sec}\text{)}$

Here B is an empirically determined “universal” parameter equal to 1.42, $\text{m}{}_0\text{= 20.2}$, a reference mean atomic weight for which well-determined elastic properties exist, and λ = 1.25 is a semi empirical parameter equal to $\text{γ ?}\text{1}\text{3}$ where γ is a Grüneisen parameter. The constant c = (? ln V_M/? ln $\text{m}\text{)}{}_{\mathrm{X}}$, where V_M is molar volume, is in general different for different crystal structure series and different cation substitutions. However, it is possible to use c_Fe = 0.14 for Fe²⁺Mg²⁺ and GeSi substitutions and c_Ca ? 1.3 for CaMg substitutional series. With these values it is pos to deduce from the above equation Birch's law, its modifications introduced by Simmons to account for Ca-bearing minerals, variations in the seismic equation of state observed by D.L. Anderson, and the apparent proportionality of bulk modulus K to V_M^?4.     

Anelastic degradation of acoustic pulses in rock

Measurements on acoustic pulses propagating in massive rock lead to a simple empirical relationship between the pulse rise time, τ and the time of propagation of a pulse, t:

$\text{τ=τ}{}_0\text{+C}\text{∫}\text{)}\text{T}\text{Q}{}^{\mathrm{?1}}\text{d}\text{t}$

where τ₀ is the initial rise time (at t = 0), Q is the anelastic parameter which may be expressed in terms of the fractional loss of energy per cycle of a sinusoidal wave, Q = 2π(ΔE/E)^?1, and is assumed to be essentially independent of frequency, and C is a constant whose value we estimate experimentally to be 0.53 ± 0.04. Of the linear theories of seismic pulse attenuation, model 2 of Azimi et al. (1968) is favoured. Pulse shapes computed from equations of Futterman (1962) also give C = 0.5, but the pulse arrives earlier than in a non-attenuating medium with the same elasticity and density. Pulse shapes calculated using Strick's (1967, 1970, 1971) theory give values of C incompatible with our results. The observations suggest that a method of estimating the Q-structure of the earth from seismic pulse rise times may have a particular advantage over the spectral ratio method.     

The cooling Earth: A reappraisal

By treating the lithosphere as a diffusive boundary layer to mantle convection, the convective speed or mantle creep rate, $\text{?}\text{?}$, can be related to the mantle-derived heat flux, $\text{Q}\text{?}$. If cell size is independent of $\text{Q}\text{?}{}^2$ then $\text{?}\text{?}\text{∝}\text{Q}\text{?}$. (If cell size varies with $\text{Q}\text{?}$, then a different power law prevails, but the essential conclusions are unaffected.) Then the factthat for constant thermodynamic efficiency of mantle convection, the mechanical power dissipation is proportionalto $\text{Q}\text{?}$, gives convective stress $\text{σ ∝}\text{Q}\text{?}{}^{\mathrm{?1}}$, i.e. the stress increases as the convection slows. This means an increasing viscosityor stiffness of the mantle which can be identified with a cooling rate in terms of a temperature-dependent creep law. If we suppose that the mantle was at or close to its melting point within 1 or 2 × 10⁸ years of accretionof the Earth, the whole scale of cooling is fixed. The present rate of cooling is estimated to be about 4.6 × 10^?8 deg y^?1 for the average mantle temperature, assumed to be 2500 K, but this very slow cooling rate represents a loss ofresidual mantle heat of 7 × 10¹² W, about 30% of the total mantle-derived heat flux. This conclusion requires theEarth to be deficient in radioactive heat, relative to carbonaceous chondrites. A consideration of mantle outgassing and atmospheric argon leads to the conclusion that the deficiency is due to depletion of potassium, and that the K/U ratio of the mantle is only about 2500, much less than either the crustal or carbonaceous chondritic values. Thetotal terrestrial potassium is estimated to be about 6 × 10²⁰ kg. Acceptance of the cooling of the Earth removes the necessity for potassium in the core.     

Variation of the magnetic properties in a basaltic dyke with concentric cooling zones

The effects of the variation of magnetic grain size on the magnetic properties of rocks have been studied throughout a reversely magnetized basaltic dyke with concentric cooling zones.Except in a few tachylites in which the magnetic mineral is a Ti-rich titanomagnetite, in the bulk of the dyke the magnetization is carried by almost pure magnetite grains. Although the percentage p of these magnetic oxides varies slightly, the large changes in the various magnetic parameters observed across the dyke are essentially attributable to large variations in the grain size of the magnetic particles.From the outer scoria region, where the magnetic grains are a mixture of single-domain (SD) and superparamagnetic (SP) grains, to the tachylite zone with finely crystallized basaltic glass containing interacting elongated SD particles, one observes an increase of both the ratio of the saturation remanent magnetization and the saturation induced magnetization J_rs/J_is, the bulk coercive force H_c, the median destructive field MDF, the intensity of the remanent magnetization J_r, and the Koenigsberger ratio Q. In the tachylites these parameters reach unusually high values, for subaerial basalts:

$\text{〈}\text{J}{}_{\mathrm{rs}}\text{J}{}_{\mathrm{is}}\text{〉 = 0.3, 〈H}{}_{\mathrm{c}}\text{〉 = 460 O}{}_{\mathrm{e}}\text{, 〈MDF〉 = 620 O}{}_{\mathrm{e}}\text{r.m.s., 〈J}{}_{\mathrm{r}}\text{〉 = 2.7 · 10}{}^{\mathrm{?2}}\text{e.m.u. cm}{}^{\mathrm{?3}}\text{〈Q〉 = 24}$

These parameters decrease in the basalt toward the centre of the dyke where pseudo-single-domain (pseudo-SD) particles coexist together with multidomain (MD) grains. The susceptibility remains approximately constant from the inner basalt to the tachylite, but increases in the scoria up to values 10 times higher owing to the presence of SP particles. The magnetic viscosity increases also drastically toward the margin of the dyke due to an increase of the fraction of the SD particles just above the superparamagnetic threshold.     

Electrical conductivity of ultramafic rocks to 1820 kelvin

Electrical conductivity, σ, of six ultramafic rocks (garnet-bearing peridotites and an eclogite) has been investigated in the temperature range 670–1820 K under known environment. Between 670 and ～1320 K σ increases with 1/T monotonically, first slowly, by 1–1.5 orders of magnitude. Above 1320 K σ increases sharply; an increase of 3–4 orders of magnitude is observed between 1320 and 1670 K. In the same temperature range the σ values for all the six rocks fall within 1.5 orders of magnitude, the lowest conductivity being for a spinel lherzolite. The range of σ values is narrowed at higher temperatures. The differences in σ values above 1470 K may be explained on the basis of varying degrees of partial melting in the rocks. Over the entire range of temperature, the σ values for the ultramafic rocks are lower than those reported for basalts but higher (by 1–2 orders of magnitude) than those for single-crystal olivines.     

An experimental study of magnetite-titanomagnetite interdiffusion

Interdiffusion experiments were performed between Fe₃O₄ (single crystal) and Fe_2.8Ti_0.2O₄ (powder), under self-buffering conditions (temperature range 600–1034°C), and for various oxygen potentials at 1400°C. Profiles of Fe and Ti were obtained by electronprobe microanalysis, and the interdiffusion coefficient $\text{D}$ was calculated by the Boltzmann-Matano method. Low-temperature data at 3 mole% Ti could be described by $\text{D}\text{= (3.85}{}_{\mathrm{?1.11}}{}^{\mathrm{+1.68}}\text{) × 10}{}^{\mathrm{?3}}\text{exp}\text{(2.23 ± 0.04}\text{eV}\text{/kT)}\text{cm}{}^2\text{/}\text{s}$. An estimate is given for the time to interdiffuse 2μm at various temperatures, and the results compared with recent experiments.     

Melting temperature distribution and fractionation in the lower mantle

The melting curve of perovskite MgSiO₃ and the liquidus and solidus curves of the lower mantle were estimated from thermodynamic data and the results of experiments on phase changes and melting in silicates.The initial slope of the melting curve of perovskite MgSiO₃ was obtained as $\text{d}\text{T}{}_{\mathrm{<mtext>m</mtext>}}\text{/}\text{d}\text{P?77}\text{K}\text{GPa}{}^{\mathrm{?1}}$ at 23 GPa. The melting curve of perovskite was expressed by the Kraut-Kennedy equation as $\text{T}{}_{\mathrm{<mtext>m</mtext>}}\text{(}\text{K}\text{)=917(}\text{1+29.6ΔV}\text{V}{}_0\text{)}$, where T_m?2900 K and P?23 GPa; and by the Simon equation, $\text{P(}\text{GPa}\text{)?23=21.2[}\text{(T}{}_{\mathrm{<mtext>m</mtext>}}\text{(}\text{K}\text{)}\text{2900)}{}^{1.75}\text{?1}\text{]}$.The liquidus curve of the lower mantle was estimated as $\text{T}{}_{\mathrm{<mtext>liq</mtext>}}\text{? 0.9 T}{}_{\mathrm{<mtext>m</mtext>}}$ (perovskite) and this gives the liquidus temperature T_liq=7000 ±500 K at the mantle-core boundary. The solidus curve of the lower mantle was also estimated by extrapolating the solidus curve of dry peridotite using the slope of the solidus curve of magnesiowüstite at high pressures. The solidus temperature is ～ 5000 K at the base of the lower mantle. If the temperature distribution of the mantle was 1.5 times higher than that given by the present geotherm in the early stage of the Earth's history, partial melting would have proceeded into the deep interior of the lower mantle.Estimation of the density of melts in the MgOFeOSiO₂ system for lower mantle conditions indicates that the initial melt formed by partial fusion of the lower mantle would be denser than the residual solid because of high concentration of iron into the melt. Thus, the melt generated in the lower mantle would tend to move downward toward the mantle-core boundary. This downward transportation of the melt in the lower mantle might have affected the chemistry of the lower mantle, such as in the D″ layer, and the distribution of the radioactive elements between mantle and core.     

A note on pulse broadening and anelastic attenuation in near-surface rocks

A pulse rise-time method has been used to study pulse broadening on seismograms generated by a weight drop source at distances up to 600 m. Both source and receiver were placed on glacial overburden overlying a gneiss-monzonite rock body. One of two data sets showed a significant increase in pulse rise time, τ, as a function of travel time, T. This increase, if due to anelastic attenuation in the uppermost part of the rock body, implies a Q value of 243 ± 53, assuming a linear relationship between τ and T. The data were not capable of discriminating between the models of pulse broadening of Gladwin and Stacey (τ ∞ T) and Ricker ($\text{τ ∞ T}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$).     

Splitting of dislocations in olivine,cross-slip-controlled creep and mantle rheology

The splitting of the [100] dislocations of forsterite Mg₂SiO₄ is investigated in a hard-sphere model. Glissile splittings exist in (001) and the most energetically favorable is the one that does not involve cutting of SiO₄ tetrahedra: $\text{[100] → [}\text{1}\text{6}\text{,}\text{1}\text{9}\text{,}\text{1}\text{6}\text{+ [}\text{2}\text{3}\text{, 0, 0] + [}\text{1}\text{6}\text{,}\text{1}\text{9}\text{,}\text{1}\text{6}$ [100] screw dislocations are shown to be able to split simultaneously on (001) and (010) according to the reaction: $\text{[100] = [}\text{1}\text{6}\text{,}\text{1}\text{36}\text{,}\text{1}\text{4}\text{] + [}\text{1}\text{6}\text{,}\text{1}\text{9}\text{,}\text{1}\text{6}\text{] + [}\text{1}\text{3}\text{, 0, θ] + [}\text{1}\text{6}\text{,}\text{1}\text{36}\text{,}\text{1}\text{4}\text{] + [}\text{1}\text{6}\text{,}\text{1}\text{6}\text{,}\text{1}\text{6}\text{]}$ This sessile configuration is analogous to the one found for screw dislocations of body-centered cubic metals. It accounts forthe long rectilinear sessile screw segments commonly observed, in experimentally and naturally deformed olivine. A new creep model for the upper mantle is proposed, where recovery is controlled by cross-slip of screw dislocations instead of climb of edge dislocations. The creep law, fitted on the experimental results of the literature is: $\text{?}\text{?}\text{= 1.2 · 10}{}^4\text{σ}{}^2\text{exp}\text{? [(125000 ? 11.7σ + PV}{}^1\text{/RT]>}$ (σ in bars) the activation volume for cross-slip is estimated and viscosity-depth curves are plotted. The proposed creep mechanism is grain-size independent and less non-Newtonian than the climb-controlled one; it is found to be dominant over the latter at stresses smaller than 100 bar.     

Are anharmonicity corrections needed for temperature-profile calculations of interiors of terrestrial planets?

It is shown that there is linearity between the thermal pressure P_TH and T between the Debye temperature θ and some high temperature $\text{T}{}^1$. $\text{T}{}^1$ has been measured at 1 atm and is reported for several minerals including, for example, MgO (1300 K) and forsterite (1200 K). The change in thermal pressure from room temperature for five solids, so far measured, indicate striking linearity with T at high temperatures.It is further shown that the value of $\text{T}{}^1$ increases greatly as the pressure increases. It is therefore concluded that P_TH is probably linear with T for mantle minerals under mantle conditions. The proportionality constant is derived from the measurements of thermal expansivity and bulk modulus at high temperature and zero pressure.The argument is then reversed. Assuming that the thermal pressure is in fact linear with T for the various shells in a planet, the resulting density and temperature profile of the planet is derived. The resulting density profile of the Earth compares favorably with corresponding values of recent seismic profiles.