L\'evy Dynamics of Stretching in 2-Dimensional Steady Random Flow Fields

Stretching and compression of material fluid elements is key for the understanding and quantification of the mixing dynamics. For 2-dimensional steady random flows the elongation of a material strip $\rho(t)$ grows algebraically as $\rho(t) \propto t^{\gamma}$. The stretching exponent $\gamma$ depends on the heterogeneity strength. While the Poincar\'e-Bendixson theorem explains the absence of exponential stretching in steady $2d$ flows, the mechanisms of the algebraic stretching behavior and its relation to the flow statistics are not known. Here we formulate the deformation of a material fluid element in streamline coordinates, which unravels the dynamics of the stretching provess as a L\'evy walk. We provide an explicit relation between the stretching process and the flow heterogeneity and derive the scaling behavior of elongation with time. We find for the stretching exponent $\gamma$ is bounded between $1/2$ and $2$, where $\gamma = 1/2$ corresponds to weak heterogeneity and $\gamma = 2$ to strongly heterogeneous flow fields.