Spectral decay of a passive scalar in chaotic mixing

In this paper we take a closer look at the decay phase of a passive, diffus ing, scalar field undergoing steady, three-dimensional chaotic advection. T he energy spectrum of the scalar is obtained by numerical simulation of the advection-diffusion equation at high Peclet number. At large times, the sp ectral decay is found to be exponential and self-similar. It is emphasized that the asymptotic decay-time is an important measure of mixing efficiency , alongside the time required for diffusion to first become effective. The large-wavenumber spectral form, representing the distribution of scalar ene rgy over small scales, is analyzed. Power-law behavior is found at scales i ntermediate between the large ones, comparable in size with the entire flow volume, and the smallest ones, at which diffusion is effective and the spe ctrum falls off exponentially with increasing wavenumber. Fitting of the nu merical results allows the exponent of the power-law to be estimated. It is observed to vary with the parameters of the flow, taking negative values w hich can be either less than or greater than -1. This implies that the domi nant spectral energy at high P may be either at small, large or intermediat e scales, depending on the flow. In consequence, the qualitative nature of the scalar field during the decay phase varies from flow to flow, resulting in differing behavior of the predicted decay times in the large P limit ob tained by asymptotic analysis. The case in which the spectral exponent exce eds -1 is shown to produce more rapid mixing and the corresponding asymptot ic expression for the decay time, independent of P and involving two spectr al parameters, is suggested as a quantitative means for optimizing the flow . (C) 2000 American Institute of Physics. [S1070-6631(00)50011-7].