Mixing Distributions Produced by Multiplicative Stretching in Chaotic Flows

Chaotically advected fluids are both a visual demonstration of stretching and folding leading to chaos and a prototypical example of a multiplicative process with weakly correlated steps. Complementary aspects of the process are studied by means of stretching calculations for different flows under both globally and partially chaotic conditions. Stretching is examined in two different ways: (i) as stretching plots, focusing primarily on stretching at small scales and on the comparison of the spatial distribution of stretching with dye structures and with unstable manifolds, and (ii) as time evolving probability density functions, analyzed using scaling techniques that renormalize the distributions by means of their moments. The first approach leads to the conclusion that the manifold structure generates the striation pattern observed in dye deformation experiments, and that both manifolds and dye patterns agree with stretching plots even at small scales. The second approach demonstrates that the multiplicative nature of stretching generates universal statistics which are reflected in self-similar scaling distributions.