Spectral analysis and computation for homogenization of advection diffusion processes in steady flows

Abstract

Advective diffusion 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 dynamics 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 matrix D*. Three decades ago, a Stieltjes integral representation for the homogenized matrix, involving a spectral measure of a self-adjoint operator, was developed. However, analytical calculations of D* have been obtained for only a few simple flows, and numerical computations of the effective behavior based on this spectral representation have apparently not been attempted. We overcome these limitations by providing a mathematical foundation for the computation of Stieltjes integral representations of D*. We explore two different approaches and for each approach we derive new Stieltjes integral representations and rigorous bounds for the symmetric and antisymmetric parts of D*, involving the molecular diffusivity and a spectral measure μ of a self-adjoint operator that depends on the characteristics of a randomly perturbed periodic flow. In discrete formulations of each approach, we express μ explicitly in terms of standard (or generalized) eigenvalues and eigenvectors of Hermitian matrices. We develop and implement an efficient numerical algorithm that combines beneficial numerical attributes of each approach. We use this method to compute the effective behavior for model flows and relate spectral characteristics to flow geometry and transport properties.