Scalar decay in two-dimensional chaotic advection and Batchelor-regime turbulence

This paper considers the decay in time of an advected passive scalar in a large-scale flow. The relation between the decay predicted by "Lagrangian stretching theories," which consider evolution of the scalar field within a small fluid element and then average over many such elements, and that observed at large times in numerical simulations, associated with emergence of a "strange eigenmode" is discussed. Qualitative arguments are supported by results from numerical simulations of scalar evolution in two-dimensional spatially periodic, time aperiodic flows, which highlight the differences between the actual behavior and that predicted by the Lagrangian stretching theories. In some cases the decay rate of the scalar variance is different from the theoretical prediction and determined globally and in other cases it apparently matches the theoretical prediction. An updated theory for the wavenumber spectrum of the scalar field and a theory for the probability distribution of the scalar concentration are presented. The wavenumber spectrum and the probability density function both depend on the decay rate of the variance, but can otherwise be calculated from the statistics of the Lagrangian stretching history. In cases where the variance decay rate is not determined by the Lagrangian stretching theory, the wavenumber spectrum for scales that are much smaller than the length scale of the flow but much larger than the diffusive scale is argued to vary as k(-1+rho), where k is wavenumber, and rho is a positive number which depends on the decay rate of the variance gamma(2) and on the Lagrangian stretching statistics. The probability density function for the scalar concentration is argued to have algebraic tails, with exponent roughly -3 and with a cutoff that is determined by diffusivity kappa and scales roughly as kappa(-1/2) and these predictions are shown to be in good agreement with numerical simulations. (C) American Institute of Physics.