Research School of Earth Sciences


A test of an alternative finitestrain equationofstate for the lower mantleI. Jackson It has long been recognised that there is no unique choice of the strain measure to be used in finitestrain 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 e_{H} = (1/3) ln (V/V_{0}) recently proposed by Poirier and Tarantola needs to be tested in the same way. Accordingly, shock compression curves (Hugoniots) for MgO were calculated from 3rdorder Eulerian (BirchMurnaghan) and PoirierTarantola isentropes with ultrasonically determined K_{0} and K'_{0} along with a common MieGrüneisenDebye treatment of the additional thermal pressure. The Hugoniot based on the BirchMurnaghan isentrope accurately reproduces the shock compression data whereas the Hugoniot based on the 3rdorder PoirierTarantola isentrope is clearly systematically too compressible at very high pressure (Figure 9). Pending further testing, the Eulerian strain measure is therefore preferred.
Nevertheless, the implications have been explored of fitting the PoirierTarantola 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 zeropressure bulk modulus K_{0}, that for each trial value of the zeropressure density r_{0}, provide optimal fits to P(e_{H}) and f(e_{H}), respectively. The intersection of these curves defines a unique (_{0}, K_{0}) combination that simultaneously fits both datasets very well. The associated values of the higher derivatives K'_{0} and K_{0}K"_{0} can be read off the lower panels. The optimal 3rdorder PoirierTarantola fit to the prem lower mantle is given by (r_{0}, K_{0}, K'_{0}, K_{0}K"_{0}) = (3.965, 193.5, 4.79, 11.6), where K_{0}K_{0}" =  3  K'_{0}(K'_{0}  3). Relative to the corresponding 3rdorder Eulerian fit, _{0} is marginally (0.5%) lower. K_{0} 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 ~3fold increase relative to the Eulerian fit in the magnitude of K_{0}K"_{0}. Such differences would have profound implications for interpretation of the elasticity of the lower mantle in terms of chemical composition and temperature.

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