The thermodynamic properties of the earth's lower mantle |
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| Authors: | Orson L Anderson Yoshio Sumino |
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| Institution: | 1. Institute of Geophysics and Planetary Physics, University of California, Los Angeles, CA 90024 U.S.A.;2. Department of Earth and Space Sciences, University of California, Los Angeles, CA 90024 U.S.A. |
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| Abstract: | The thermodynamic properties of the lower mantle are determined from the seismic profile, where the primary thermodynamic variables are the bulk modulus K and density ρ. It is shown that the Bullen law (K ∝ P) holds in the lower mantle with a high correlation coefficient for the seismic parametric Earth model (PEM). Using this law produces no ambiguity or trade-off between ρ0 and K0, since both K0 and K′0 are exactly determined by applying a linear K?ρ relationship to the data. On the other hand, extrapolating the velocity data to zero pressure using a Birch-Murnaghan equation of state (EOS) results in an ambiguous answer because there are three unknown adjustable parameters (ρ0, K0, K′0) in the EOS.From the PEM data, K = 232.4 + 3.19 P (GPa). The PEM yields a hot uncompressed density of 3.999 ± 0.0026 g cm?3 for material decompressed from all parts of the lower mantle. Even if the hot uncompressed density were uniform for all depths in the lower mantle, the cold uncompressed mantle would be inhomogeneous because the decompression given by the Bullen law crosses isotherms; for example, the temperature is different at different depths. To calculate the density distribution correctly, an isothermal EOS must be used along an isotherm, and temperature corrections must be placed in the thermal pressure PTH.The thermodynamic parameters of the lower mantle are found by iteration. Values of the three uncompressed anharmonic parameters are first arbitrarily selected: α0 (hot), the coefficient of thermal expansion; γ0, the Grüneisen parameter; and δ, the second Grüneisen parameter. Using γ0 and the measured ρ0 (hot) and K0 (hot), the values of θ0 (Debye temperature) and q = dlnγ/dlnρ are found from the measured seismic velocities. Then from (αKT)0 and q the thermal pressure PTH at all high temperatures is found. Correlating PTH against T to the geotherm for the lower mantle, PTH is found at all depths Z. The isothermal pressure, along the 0 K isotherm, at every Z is found by subtracting PTH from the measured P given by the seismic model. Using the isothermal pressure at depth Z, the solution for the cold uncompressed density ρ0C and the cold uncompressed bulk modulus, KT0 is found as a trace in the KT0?ρ0C plane. A narrow band of solutions is then found for ρ0C and KT0 at all depths.The thermal expansion at all T is found from ρ0C ? ρ0 (hot)/ρ0C. From Suzuki's formula, the best fit to the thermal expansion determines γ0 and α0 (hot). When the values of these two parameters do not agree with the original assumptions, the calculation is repeated until they do agree. In this way all the important thermodynamic parameters are found as a self-consistent set subject only to the assumptions behind the equations used. |
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