The thermodynamic properties of the earth's lower mantle

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 ρ 0 and K 0, since both K 0 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 ( ρ 0, K 0, 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 P TH. The thermodynamic parameters of the lower mantle are found by iteration. Values of the three uncompressed anharmonic parameters are first arbitrarily selected: α 0 (hot), the coefficient of thermal expansion; γ 0, the Grüneisen parameter; and δ, the second Grüneisen parameter. Using γ 0 and the measured ρ 0 (hot) and K 0 (hot), the values of θ 0 (Debye temperature) and q = dln γ/dln ρ are found from the measured seismic velocities. Then from ( αK T ) 0 and q the thermal pressure P TH at all high temperatures is found. Correlating P TH against T to the geotherm for the lower mantle, P TH is found at all depths Z. The isothermal pressure, along the 0 K isotherm, at every Z is found by subtracting P TH from the measured P given by the seismic model. Using the isothermal pressure at depth Z, the solution for the cold uncompressed density ρ 0C and the cold uncompressed bulk modulus, K T0 is found as a trace in the K T0 − ρ0C plane. A narrow band of solutions is then found for ρ 0C and K T0 at all depths. The thermal expansion at all T is found from [ ρ 0C − ρ 0 (hot)/ ρ 0C. From Suzuki's formula, the best fit to the thermal expansion determines γ 0 and α 0 (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.