A Multiscale Measure for Quantifying Mixing of Passive Scalar Fields

In spite of a large amount of recent research on the problem of fluid mixing and its control, there is no consensus on a proper measure for quantifying mixing. We present a measure of mixing that is based on the concept of weak convergence and is capable of probing the “mixedness” of the advected passive scalar field at various scales. This new measure, called the Mix-Norm, resolves the inability of the L variance of the scalar density field to capture small-scale variations generated by chaotic maps or flows. In addition, the Mix-Norm succeeds in capturing the efficiency of a mixing protocol in the context of a particular initial density field, wherein Lyapunov-exponent based measures fail to do so. The Mix-Norm is a pseudo-norm for checking weak convergence on the space of scalar density fields, which turns out to be a critical link in justifying its validity as a measure for mixing. We demonstrate the utility of the Mix-Norm by showing how it measures the efficiency of mixing due to various discrete dynamical systems and to diffusion.