Two-point correlation and its eigen-decomposition for optimal characterization of mantle convection

Balachandar, S.

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https://hdl.handle.net/2142/112464 Copy

Description

Title

Two-point correlation and its eigen-decomposition for optimal characterization of mantle convection

Author(s)

Balachandar, S.

Issue Date

1994-10

Keyword(s)

• Two-point Correlation
• Eigen-decomposition
• Optimal Characterization
• Mantle Convection

Abstract

The two-point correlation tensor provides the complete information on mantle convection accurate up to the second order statistics. Unfortunately the two-point spatial correlation tensor in general is a data intensive quantity, with sixteen tensor components, each of which dependent on seven independent variables. Here we consider both the spherical and Cartesian models of mantle convection. In the case of convection in a spherical shell a simplified form of the two-point spatial correlation tensor is obtained by using spherical symmetry about the origin and statistical stationarity. In the Cartesian model, translational invariance along the horizontal directions and axisymmetry about the vertical direction along with stationarity are used to simplify the two-point correlation tensor. In both these cases the general two-point correlation can be expressed in terms of a planar correlation tensor which reduces the two-point correlation's dependence to only three independent variables. Symmetry and incompressibility condition are used to further reduce the number of independent tensor components from sixteen to five. Results of planar correlation for two sample problems: three-dimensional simulation of thermal convection in a Cartesian box (Balachandar et al. 1989) and the tomographic results of Su et al. ( 1994) are presented. The eigenfunctions of the planar correlation tensor provide a rational methodology for further compressing the information content. For example, in the thermal convection problem the first ten eigensolutions of the planar correlation, which constitute nearly a factor of 2,000 reduction in the data, capture the two-point correlation to 95% accuracy in the mean square sense. In comparison, arbitrary subsets of the planar correlation in the form of vertical and horizontal correlations together offer a factor of 33 reduction in data but capture only 6.8% of the two-point correlation. In the case of SH12/WM13 model of Su et al. (1994) the first ten eigenfunctions offer a factor of 20 reduction in the data and capture nearly 86.3% of the informational content of the two-point correlation. The two most energetic eigenfunctions of the tomographic model clearly reveal the signature of the mantle transition zone while the higher order eigenfunctions do not, thus suggesting a partially layered state of the present day mantle.

Publisher

Department of Theoretical and Applied Mechanics. College of Engineering. University of Illinois at Urbana-Champaign

Series/Report Name or Number

• TAM R 770
• 1994-6026

ISSN

0073-5264

Type of Resource

text

Language

eng

Permalink

http://hdl.handle.net/2142/112464

Sponsor(s)/Grant Number(s)

National Science Foundation 94/10

Copyright and License Information

Copyright 1994 Board of Trustees of the University of Illinois

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