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## A test of an alternative finite-strain equation-of-state for the lower mantle

I. Jackson

It has long been recognised that there is no unique choice of the strain measure to be used in finite-strain equations of state. However, experimentally determined compression curves for standard materials such as MgO provide an opportunity to test the performance of equations of state based on the alternative Eulerian, Lagrangian and Hencky (or natural) strain measures. In this way it was demonstrated conclusively in the early 1970's that the Eulerian strain measure is to be preferred over the Lagrangian if the Taylor expansion of Helmholz free energy in powers of strain is to be truncated at third order. The natural strain measure eH = (1/3) ln (V/V0) recently proposed by Poirier and Tarantola needs to be tested in the same way. Accordingly, shock compression curves (Hugoniots) for MgO were calculated from 3rd-order Eulerian (Birch-Murnaghan) and Poirier-Tarantola isentropes with ultrasonically determined K0 and K'0 along with a common Mie-Grüneisen-Debye treatment of the additional thermal pressure. The Hugoniot based on the Birch-Murnaghan isentrope accurately reproduces the shock compression data whereas the Hugoniot based on the 3rd-order Poirier-Tarantola isentrope is clearly systematically too compressible at very high pressure (Figure 9). Pending further testing, the Eulerian strain measure is therefore preferred.

 Figure 9 Comparison of calculated Hugoniots for MgO based on the alternative Eulerian and Poirier-Tarantola isentropes. The Hugoniot based on the 3rd-order Poirier-Tarantola isentrope is clearly too compressible at very high pressure.

Nevertheless, the implications have been explored of fitting the Poirier-Tarantola equation of state (at either 3rd or 4th order) to the prem model of Dziewonski and Anderson for the pressure dependence of the seismic parameter f and the density r. The curves labelled 'IIIP' and 'III$&phi$' in the uppermost panel of Figure 10 define the values of the zero-pressure bulk modulus K0, that for each trial value of the zero-pressure density r0, provide optimal fits to P(eH) and f(eH), respectively. The intersection of these curves defines a unique (0, K0) combination that simultaneously fits both datasets very well. The associated values of the higher derivatives K'0 and K0K"0 can be read off the lower panels. The optimal 3rd-order Poirier-Tarantola fit to the prem lower mantle is given by (r0, K0, K'0, K0K"0) = (3.965, 193.5, 4.79, -11.6), where K0K0" = - 3 - K'0(K'0 - 3). Relative to the corresponding 3rd-order Eulerian fit, 0 is marginally (0.5%) lower. K0 is substantially (9%) lower resulting in more compressible behaviour at low P offset at higher P through a markedly higher (+23%) value for K'0. The consequences of such a high value of K'0 are tempered by a ~3-fold increase relative to the Eulerian fit in the magnitude of K0K"0. Such differences would have profound implications for interpretation of the elasticity of the lower mantle in terms of chemical composition and temperature.

 Figure 10. Adiabatic decompression of the prem lower mantle model with the Poirier-Tarantola finite-strain equation of state. The vertical line spanning all three panels links the intersections of the IIIP and IIIf covariance curves, thereby serving to identify the parameters of the optimal fit to both P(eH) and r(eH).
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