A simple theoretical model is presented for the statistical distribution of small-scale, passive heterogeneity in a convecting mantle. The important new variables used to characterize the anomaly field are the autocorrelation function R and its Fourier transform, the energy spectrum E. The anomaly variance, the sample variance (i.e., the anomaly variance as filtered by the sampling process) and the mixing time scale are expressed in terms of E. Equations governing the time evolution of R and E are derived, starting with the transport-diffusion equation. The energy spectrum equation is solved for deformation models with homogeneous normal and shear strain components. The predictions from these models are compared with results of numerical experiments on mixing of slab-shaped heterogeneities by Bénard convection. Both the numerical experiments and the theory indicate that compositional heterogeneity subjected to convective flow becomes drawn into thin sheets oriented subparallel to the transport direction. Re-mixing of subducted slabs produces a laminated mantle. The energy spectrum of the laminate is highly anisotropic. Where the transport direction is predominantly horizontal, the energy at high wave numbers is in the vertical; the low wave number energy is in the horizontal plane. The shape of the energy spectrum contains information on mean mantle strain rates, age of the anomalies, and solid state diffusion rates.
Bibliographical noteFunding Information:
This work has been generously supported by NSF Grants EAR-81 17439, EAR-81 15828 and EAR-84 07805. We are indebted to Paul Silver, George Backus and Don Anderson for providing information on polarization anisotropy.