Claudio Viotti, University of North Carolina, Chapel Hill


Space-time scales in shear-enhanced diffusion

Abstract: The spectrum of spatial and temporal scales that characterize the complete time evolution of a passive scalar advected by parallel shear flows is studied in the asymptotic limit of large Peclet (Pe) numbers. A well know feature of the problem is to display intermediate stages, and mathematically the occurrence of these transients can be viewed as a competition in the asymptotic dominance between long/short scale aspect ratios. We provide analytical predictions of the associated time scales by a modal analysis of the eigenvalue problem arising in the separation of variables of the governing advection-diffusion equation. At the short scales (high wavenumber limit) a WKBJ mathematical structure emerges. This part of the analysis is technical, as the corresponding spectrum is dominated by asymptotically coalescing turning points in the limit of large Pe numbers. The range of scales characterized by WKBJ modes fades for large wavelengths into one where Taylor diffusivity emerges. This second limit is studied by a regular perturbation expansion of the spectrum. The connection of these analytical results with full time evolution are illustrated by highly resolved numerical simulations of the evolving scalar distribution in the particular case of point-like initial data, stressing the relation between different spectral bands and superdiffusive regimes.

Mentors: Rich McLaughlin, Roberto Camassa (University of North Carolina, Chapel Hill)