A multiscale measure for mixing

We present a multiscale measure for mixing that is based on the concept of weak convergence and averages the “mixedness” of an advected scalar field at various scales. This new measure, referred to as the Mix-Norm, resolves the inability of the L2 variance of the scalar density field to capture small-scale variations when advected 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 scalar field, wherein Lyapunov-exponent based measures fail to do so. We relate the Mix-Norm to the classical ergodic theoretic notion of mixing and present its formulation in terms of the power spectrum of the scalar field. 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. In particular, we show that the Mix-Norm can capture known exponential and algebraic mixing properties of certain maps. We also analyze numerically the behaviour of scalar fields evolved by the Standard Map using the Mix-Norm.