On Compressibility and Chemical Inhomogeneity in the Earth's Core

Summary A critical examination has been made of features of the writer's formula η= dk/dp‐g ‐1 dφ/dz (i) for estimating density gradients in chemically inhomogeneous regions of the Earth's deep interior, where k and p denote the incompressibility and pressure at depth z , and φ is a function of the P and S seismic velocities. In particular, implications of the variation of k with composition, and of the deviation of dk/dp from ( δk/δp ) n , have been examined in some detail and some important new relations derived. (The subscripts n and p denote constant composition and constant pressure, respectively; other symbols are defined in the text.) One relation expresses η in terms of dφ/dz , ( δk/δp ) n and the function ψ, where ψ= 1‐ φ ‐1 ( δk/δp ) p . A second relation connects the seismically determined quantity dφ/dz with ψdn/dz , where n is a variable representing the chemical composition. Attention is drawn to the significance of the variable ψ in estimating values of η and dp/dz. If ρ could be adequately estimated from experimental data, it would be possible to narrow the uncertainties in the determination of density gradients in the Earth's deep interior. The variable ψ is important in another way. Should it become small compared with unity for a range of values of ρ, certain difficulties could arise in estimating η. In a region where the P and S seismic velocity gradients are “normal”, the inference of approximate chemical homogeneity could be open to question. In a region where the velocity gradients are not normal, unexpectedly large differences between dk/dp and ( δk/δp ) n could arise, but these would be accompanied by abnormally high density gradients. Specimen numerical calculations suggest that these difficulties are probably not serious for the materials of the Earth's deep interior, and that while cautions are needed in applying (i), the results of earlier applications of (i) are unlikely to be inaccurate beyond the stated uncertainties. The theory also indicates how estimates of the effective variation of the representative atomic number with depth in the core might be arrived at from knowledge of values of the seismic function dφ/dz. It is likely that various equations developed in the theory will be of general value, even where numerical inferences cannot be drawn with precision at present.