The statistics of stretching and stirring in chaotic flows

Abstract

The statistics of stretching and stirring in time‐periodic chaotic flows is studied numerically by following the evolution of stretching of O(10⁵) points. The ratio between stretchings accumulated by each point at successive periods is referred to as a multiplier, and the total stretching is the product of multipliers. As expected, the mean stretching of the population increases exponentially whereas the probability density function of multipliers converges—in just two periods or so—to a time‐invariant distribution. There is, however, a considerable degree of order in the spatial distribution of stretching in spite of conditions of global chaos. The self‐correlation of multipliers shows as well considerable structure and often there are segregated populations of points: the largest population consists of points that experience extensive stretching, efficient stirring, and have a distribution of stretching values that evolves asymptotically—in about ten periods—into a limiting time‐invariant scaling distribution. The remaining points experience slow stretching and, although they also exhibit scaling behavior, are effectively segregated from the rest of the system in the time scale of our simulations.