Averaging versus Chaos in Turbulent Transport?

Abstract

In this paper we analyze the transport of passive tracers by deterministic stationary incompressible flows which can be decomposed over an infinite number of spatial scales without separation between them. It appears that a low order dynamical system related to local Peclet numbers can be extracted from these flows and it controls their transport properties. Its analysis shows that these flows are strongly self-averaging and super-diffusive: the delay $\tau(r)$ for any finite number of passive tracers initially close to separate till a distance $r$ is almost surely anomalously fast ($\tau(r)\sim r^{2-\nu}$, with $\nu>0$). This strong self-averaging property is such that the dissipative power of the flow compensates its convective power at every scale. However as the circulation increase in the eddies the transport behavior of the flow may (discontinuously) bifurcate and become ruled by deterministic chaos: the self-averaging property collapses and advection dominates dissipation. When the flow is anisotropic a new formula describing turbulent conductivity is identified.