Line Stretching in Random Flows

An \emph{ab initio} analysis of stretching of the length $l(t)$ of material lines in steady and unsteady random chaotic flows is performed under Fickian and non-Fickian (anomalous) transport. We show that fluid stretching is an autocatalytic process with finite sampling that is governed by a competition between ensemble and temporal averaging processes. The topological entropy $h\equiv \lim_{t\rightarrow\infty} \ln l(t)/t$ converges to the Lyapunov exponent $\lambda_\infty$ in Fickian flows, whereas in non-Fickian flows $h$ aconverges to $\lambda_\infty$ plus a contribution from the variance $\sigma_\lambda^2$ of the finite time Lyapunov exponent. This study uncovers the rich dynamics of deformation in random flows, provides methods to characterize deformation from macroscopic observations, corrects classical fluid stretching models and calls for a reassessment of experimental data and fluid stretching models in turbulent and chaotic flows.