Characterization of mantle convection experiments using two-point correlation functions

Snapshots of the temperature T(r,ϕ,t), horizontal flow velocity u(r,ϕ,t), and radial flow velocity w(r, ϕ, t) obtained from numerical convection experiments of time-dependent flows in annular cylindrical geometry are taken to be samples of stationary, rotationally invariant random fields. For such a field ƒ(r,r2 ϕ, t), the spatio-temporal two-point correlation function, Cƒƒ (r, r1, Δ, 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 Cƒƒ (r, r1, Δ, t*) : the rms variation, σƒ (r), the radial correlation function, Rƒ (r,r1), and the angular correlation function, Aƒ(r, Δ). Rƒ (r,r1) and Aƒ(r,Δ) are symmetric about the loci r = r1 and Δ=0, respectively, where they achieve their maximum value of unity. The falloff of Rƒ and Aƒ away from their symmetry axes can be quantified by a correlation length ρƒ(r) and a correlation angle αƒ(r), which we define to be the half widths of the central peaks at the correlation level 0.75. The behavior of ρƒ is a diagnostic of radial structure, while αƒ measures average plume width. We have used 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, and depth- and temperature-dependent viscosity. For isoviscous flows in an annulus, we show how radial averages of σT, ρT, and αT scale with Rayleigh number for various internal heating rates. A break in the power-law relationship at the transition from steady to time-dependent regimes is evident for ρT and αT but not for σT or the Nusselt number. A rapid tenfold to thirtyfold viscosity increase with depth yields weakly stratified flows, quantified by aw, which is a measure of radial flux. The horizontal flux diagnostic, σu, reveals that the flow organization is sensitive to the depth of the viscosity increase. A jump at middepth induces a significant horizontal return flow at the base of the upper layer, absent in models with a jump at quarter-depth. 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 σ-diagnostics or αT in the convecting interior. Changes in the radial correlation length are twofold. 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 ρT associated with a viscosity jump is markedly reduced. The latter effect can be explained by weaker, less stationary hot upwellings, mobilized by the temperature-dependent rheology and disrupted by the cold, high-viscosity downwellings.