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Spectral Analysis and Computation of Effective Diffusivities for Steady Random Flows

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Technical Report

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University of California at Irvine Irvine United States

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Advective diffusion of passive tracers by fluid flow plays a key role in the transport of salt, heat, buoys and markers in geophysical flows, in the dispersion of pollutants and trace gases in the atmosphere, and even in the motion of sea ice floes influenced by winds and ocean currents. The long time, large scale behavior of such systems is equivalent to an enhanced diffusion process with an effective diffusivity tensor D. Over twenty five years ago, a Stieltjes integral representation for the homogenized tensor was derived, involving the spectral measure of a self-adjoint operator that depends on the flow characteristics. However, analytical calculations of D have been obtained for only a few simple flows, and numerical computations of the effective behavior based on this powerful representation have apparently not been attempted. We overcome these limitations by providing a mathematical foundation for rigorous computation of D. We explore two different approaches and for each approach we derive Stieltjes integral representations for the symmetric and antisymmetric parts of D, involving a spectral measure of a self-adjoint random operator. In discrete formulations of each approach, we express the spectral measure explicitly in terms of standard or generalized eigenvalues and eigenvectors of Hermitian matrices. We develop and implement a numerically efficient projection method that significantly reduces the dimension of the random matrix in computations of spectral statistics. We use this method to compute the effective behavior for several model flows and relate spectral characteristics to flow geometry and transport properties.

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