The Characterization of Seismic Earth Structures and Numerical Mantle Convection Experiments Using Two-point Correlation Functions

We consider a time-dependent random field, f(rZt), defined on a spherical shell [Q=(6,q), O56 , -t<(p it] or cylindrical annulus [92=P, -1E<(P57r]. Examples are the temperature distribution, T(r,2,t), or the radial component of the flow velocity, u(r,2,t), obtained from numerical simulations of high Rayleigh number convection. For such a field the spatio-temporal two-point correlation function, Cff(r,r',A,t*), is constructed by averaging over rotational transformations of this ensemble. To assess the structural differences among mantle convection experiments we construct three spatial subfunctions of Cif(r,r',A,t*): the rms variation, af(r) = Cf(r,r,0,0), the radial correlation function, Rf(r,r?) = Cf(r,r' ,0,0) / af(r) aj(r'), and the angular correlation function, Af(r,A) = Cff(rr,A,0)/ a 2(r). The integral transform of Af(r,A) is the angular power spectrum. Rf(r,r') and Af(r,A) are symmetric about the loci r = r' and A =0, respectively, where they achieve their maximum value of unity. The fall-off of Rf and Af away from their symmetry axes can be quantified by a correlation length pf(r) and a correlation angle af(r), which we define to be the halfwidths of the central peaks at the correlation level 0.75. The behavior of pf is a diagnostic of radial structure, while af measures average plume width. We use two-point correlation functions of the temperature field (T-diagnostics) and flow velocity fields (V-diagnostics) to quantify some important aspects of mantle convection experiments. We explore the dependence of different correlation diagnostics on Rayleigh number, internal heating rate, radial viscosity variations, temperaturedependent rheology, phase changes, and plates. For isoviscous flows in an annulus, we show how radial averages of UT, pr, and aT scale with Rayleigh number for various internal heating rates. A rapid 10-fold to 30-fold viscosity increase with depth yields weakly stratified flows, quantified by q., which is a measure of radial flux. The horizontal flux diagnostic a,, reveals that the flow organization is sensitive to the depth of the viscosity increase. We illustrate that T-diagnostics, which are more easily relatable to geophysical observables, can serve as proxies for the V-diagnostics. A viscosity increase with depth is evident as an increase in the T-diagnostics in the high-viscosity region. For numerical experiments with a temperature-dependent rheology we employ a mobilization scheme for the upper boundary layer. Temperature dependence does not appreciably perturb the a-diagnostics or aT in the convecting interior. Changes in the radial correlation length are two-fold. First, the greater viscosity of cold downwellings leads to an increase in height and width of the radial correlation maximum near the top. Second, the increase in pT associated with a viscosity jump is markedly reduced. An endothermic phase transition manifests itself in the correlation diagnostics as a local minimum in a, and pT and local maxima in UT and aT around the phase transition depth. Temperature-dependent rheology reduces the amount of layering, however, the phase-change induced layering is still apparent in the two-point correlation diagnostics. When the phase change coincides with a rapid viscosity increase the effects of the latter dominate. We investigate the influence of surface plates on the organization of mantle flow. Plates whose geometries evolve with time are modeled by using a temperaturedependent viscosity combined with weak zones (small regions of low viscosity) advected by the flow. The two-point correlation diagnostics obtained from these flows are similar to the temperature-dependent runs with a mobilized upper boundary layer. Differences include an increase in au and aT near the surface, and a shift of the maximum in a. to shallower depths. The main influence of plates is to organize the large-scale flow structure. This is best documented in the angular power spectrum, which has more power concentrated at low wavenumbers. We also quantify some statistics of the plate system, such as plate-size and relative plate-velocity distributions. Average plate velocities decrease nearly monotonically with increasing plate size for cases without a viscosity stratification. Viscously stratified systems exhibit a more uniform average plate-velocity distribution. Comparing plate system statistics from numerical convection calculations to the plate tectonic record for the past 120 Ma favors models with a 30-fold viscosity increase in the lower mantle over those with a viscosity that is constant with depth. We calculate the two-point moment functions for global and regional models of seismic shear velocity heterogeneity, 8#(r,.2). The radial correlation function is least sensitive to the low-pass filtering required when comparing convection experiments to low-resolution seismic images, making it the most useful tool for comparisons between the two. As long as thermal anomalies are predominantly responsible for seismic velocity heterogeneity, a direct comparison between pr and pp is meaningful. We find significant differences between the tomographic models, which frustrate a detailed interpretation of individual features of pp. The overall morphology of the pp-profiles, however, whereas consistent with pT curves for convection models with a 30-fold to 100-fold viscosity increase at 670 km depth, rules out convection models with a viscosity that is constant with depth. We define stratification indices for the radial correlation length, S(p), and average radial flux, S(lul), at 670 km depth. Using stratification values for the seismic models (S(pp) 0.12), we infer S(ul) 0.1, indicating that the presentday style of convection is dominantly whole-mantle. Together with A(ias), a measure of the asymmetry of the radial flux distribution at 670 km depth, S(Iul) furthermore suggests that it is unlikely for the earth to be in an intermittently layered regime. Thesis committee: Dr. Thomas H. Jordan, Professor of Geophysics (thesis supervisor) Dr. Bradford H. Hager, Professor of Geophysics (thesis co-supervisor) Dr. Daniel H. Rothman, Professor of Geophysics Dr. John Grotzinger, Professor of Geology Dr. Richard J. O'Connell, Professor of Geophysics, Harvard University Table of contents Dedication 3 Abstract 5 Table of contents 7 Chapter