JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 100, NO. B4, PAGES6351-6365, APRIL 10, 1995 Characterization of mantle convectionexperiments using two-point correlation functions PeterPuster, Thomas H. Jordan, andBradford H. Hager Department of Earth,Atmospheric, andPlanetary Sciences, Massachusetts Institute of Technology, Cambridge Abstract. Snapshots of the temperatureT(r, ½,t), horizontal flow velocity u(r, ½,t), andradial flow velocity w(r, ½,t) obtained from numerical convection experiments of time-dependent flows in annular cylindrical geometry are takento be samples of stationary, rotationally invariant random fields. For such a field f(r, ½,t), the spatio-temporal two-point correlation function, Cff(r, r', A, t* ), isconstructed byaveraging over rotational transformations of this ensemble. To assess the structural differences among mantleconvection experiments we construct three spatial subfunctions of Cff(r,r' ,A,t*): the rms variation, of(r), the radial correlation function, Rf(r,r' ), and the angular correlation function, A .(r,A) R .(r,r') and Af(r,A) are symmetric about the loci r = r and A = 0, respectively, where they achieve their maximum value ofunity. The falloff of Rf and Af away from their symmetry axes can be quantified by a correlation length p .(r) anda correlation angle a -(r), whichwe defineto be the half widths ofthe central peaks at the correlation level 0.75.The behavior of pf isa diagnostic ofradial structure, while af measures average plume width. Wehave used two- pointcorrelation functions of the temperature field (T-diagnostics) andflow velocityfields (V- diagnostics) to quantifysome important aspects of mantle convection experiments. We explore the dependence of differentcorrelation diagnostics on Rayleighnumber, internal heating rate, anddepth-and temperature-dependent viscosity. For isoviscous flows in an annulus, we show howradial averages of or , Pt, and a r scale with Rayleigh number for various internal heating rates. A breakin the power-law relationship at the transition from steady to time- dependent regimes is evident for Or and O[ r butnotfor O' r or theNusselt number. A rapid tenfold to thirtyfold viscosity increase with depth yieldsweakly stratified flows, quantified by o w, which is a measure of radial flux. The horizontal flux diagnostic, o u, reveals thatthe flow organization is sensitive to the depth of the viscosity increase.A jump at middepth induces a significant horizontal return flow at thebase of the upper layer, absent in models with a jump at quarter-depth. We illustrate thatT-diagnostics, which are moreeasilyrelatable to geophysical observables, canserve asproxies for theV-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 mobilizationscheme for the upper boundary layer. Temperature dependence does not appreciably perturb the o-diagnostics or O[ r in theconvecting interior. Changes in theradialcorrelation length aretwofold. First,the greater viscosity of cold downwellings leads to an increase in heightandwidth of the radial correlation maximum nearthe top. Second, the increase in/9 T associated with a viscosity jump is markedly reduced.The lattereffectcanbe explained by weaker,lessstationary hot upwellings, mobilized by the temperature-dependent rheology anddisrupted by the cold,high- viscosity downwellings. Introduction Seismic observations have been a principal tool forprobing Earth's deep interior andaddressing questions regarding the organization of convection currents inthe mantle. Global maps ofmantle shear-wave velocity heterogeneity, 8fl [e.g., Tanimoto, 1990; Masters etal., 1992; Suet al., 1994], provide snapshots of mantle dynamics, assuming that 8fl is proportional to convectively inducedtemperature anomalies ST. (This assumption should bea good working hypothesis in themantle's interior away fromthe chemical boundary layers at the free Copyright 1995 bythe American Geophysical Union. Papernumber 94JB03268. 0148-0227/95/94JB-03268505.00 e,,rfapo •ncl c•c•ro_mantl• hcmndary_) Numerical convection experiments, on the otherhand,give insight into the dynamics of the mantle flow system. Because mantle convection is spatially and temporally chaotic, numerical simulations cannot reproduce the exactgeographical details,but only the grosser aspects of the flow pattern. Thereforequantification schemes that measure the effect of different parameters on convective flow organization and structure are necessary. Examples of flow diagnostics are the angular powerspectrum [e.g.,Jarvisand Peltier, 1986; Tanimoto, 1990] and the root-mean-square (rms) variation on horizontal surfaces [Honda, 1987]. Recently, Jordan et al. [1993] and Puster and Jordan [1994] have introduced two-pointcorrelation diagnostics, the radial and angularcorrelation functions,that are invariant with respectto the temperature coefficient of shear- wave speed, (c•/aT) e, and thus well suited for comparison to seismic observations. Correlation functions were employed in a 6351