Modeling the bulk- and shear-elasticity of a multi-phase mantle

In the 50 years since the classic paper of Birch, two major advances have transformed our view of the earth's elasticity and constitution: the imaging of the three-dimensional structure of the mantle, and the discovery and mapping of multiple high-pressure polymorphic phase transformations in mantle compositions. These breakthroughs have revolutionized our understanding of the composition, mineralogy, thermal state, and dynamics of the mantle. In order to explore the relationship between phase transformations and three-dimensional earth structure, we have been developing a unified description of the petrology and thermoelasticity of mantle assemblages. This approach allows us to construct models of the mantle that specify the variation of mineralogy and seismic wave velocities with pressure and temperature, and which are directly comparable to seismological observations. Our method begins with the concept of fundamental thermodynamic relations, which permits straightforward computation of phase equilibria, and, at the same time, all isotropic thermodynamic properties through volume and temperature derivatives of the appropriate thermodynamic potential. The integrated form of the Mie-Gruneisen equation of state provides an excellent starting point for an account of what is currently known of mantle phase equilibria and elastic properties. We have now generalized our approach to account for anisotropic properties. By generalizing the Gruneisen parameter and its volume derivative, q, to their appropriate tensorial forms, we are able to describe the pressure and temperature dependence of the shear (S) and longitudinal (compressional, P) wave velocities as well as the bulk sound velocity and density of mantle assemblages. We have applied our method to modeling of one-dimensional seismic wave velocity profiles of the mantle in various tectonic regimes, including old and young ocean, and shields. We compute the equilibrium phase assemglage and its seismic wave velocities along geotherms that account for lithospheric cooling and, in the case of shields, radioactive heat production. Seismologically determined shear velocity profiles are characterized by a distinct minimum in velocity at depths of 100-200 km. Comparison with our computed seismic wave profiles quantify the influence of attenuation and dispersion of seismic wave velocities in this depth range, which is found to be large and comparable to the influence of mantle composition (difference between basalt and harzburgite). Isochemical mantle compositions do not show a G discontinuity, as observed seismically in oceanic regions. We explore the possibility that this feature is caused by variations with depth in bulk composition, possibly due to basalt extraction.