Theory of thermal and elastic properties of the lower mantle and core

Direct observations of material properties at pressures in the megabar range have increased dramatically in both number and quality in the past decade, stimulating the demand for a matching improvement in theoretical understanding. Central to this understanding are thermodynamic studies. The familiar first derivative identities (Maxwell relations) are no longer adequate as a basic tool. A comprehensive table of second derivatives is assembled here. Most of the relationships in the table include temperature and volume dependences of specific heat. At the high temperatures of the Earth's interior the classical assumption of constant specific heat is a reasonable approximation for insulators, including the mantle, but is clearly invalid in the core, where conduction electrons account for about 30% of the heat capacity. A crucial parameter in Earth dynamics is thermal expansion coefficient, which is related to other properties by the Grüneisen parameter, γ. Recent work has resolved a long-standing disagreement between alternative theories of γ so that we can now be more confident of numerical values in the deep interior. The significance and usefulness of a simple linear relationship between pressure, P, incompressibility, K, and rigidity modulus, μ, of close-packed crystals, is emphasized. It leads directly to the asymptotic limit K′∞ = ( ∂K ∂P)P → ∞ and provides a fixed point for equations of state beyond the observable range. It also gives a more direct estimate than previously available of the temperature dependences of elastic moduli in the lower mantle, confirming the difficulty with a thermal interpretation of tomographically observed velocity anomalies. A new approach to finite strain theory is advocated, based on the variation of dK dP with P K. It is shown that values of derivative quantities, such as dK dP, given by models such as the Preliminary Reference Earth Model (PREM), must be interpreted with caution. Suggested minor revisions of PREM avoid some equation-of-state difficulties