C D ] 1 1 Ju n 20 08 Stochastic processes in turbulent transport ∗

This is a set of four lectures devoted to simple ideas about turbulent transport, a ubiquitous nonequilibrium phenomenon. In the course similar to that given by the author in 2006 in Warwick [45], we discuss lessons which have been learned from naive models of turbulent mixing that employ simple random velocity ensembles and study related stochastic processes. In the first lecture, after a brief reminder of the turbulence phenomenology, we describe the intimate relation between the passive advection of particles and fields by hydrodynamical flows. The second lecture is devoted to some useful tools of the multiplicative ergodic theory for random dynamical systems. In the third lecture, we apply these tools to the example of particle flows in the Kraichnan ensemble of smooth velocities that mimics turbulence at intermediate Reynolds numbers. In the fourth lecture, we extends the discussion of particle flows to the case of non-smooth Kraichnan velocities that model fully developed turbulence. We stress the unconventional aspects of particle flows that appear in this regime and lead to phase transitions in the presence of compressibility. The intermittency of scalar fields advected by fully turbulent velocities and the scenario linking it to hidden statistical conservation laws of multi-particle flows are briefly explained. 1 Turbulence and turbulent transport Hydrodynamical turbulence, with its involved flow patterns of interwoven webs of eddies changing erratically in time [68], is a fascinating phenomenon occurring in nature from microscopic scales to astronomical ones and exposed to our scrutiny in everyday experience. Despite the ubiquity of turbulence, an understanding of this far-from-equilibrium phenomenon from first principles is a long-standing open problem, one of the few remaining theoretical challenges of classical physics. Although it seems still a too farfetched goal, there has been a constant progress over years in the theoretical and practical knowledge of hydrodynamical turbulence and, even more so, of other non-equilibrium phenomena of a similar nature. This progress has been due to developments of experimental techniques and computer power, but also to new theoretical ideas inspired by studies of simple models of turbulence-related systems. The present lectures will discuss one circle of such ideas more relevant to the problem of transport properties in turbulent flows than to turbulence itself. They concern properties of stochastic processes underlying the turbulent mixing, see also [75, 39]. In refs. [60, 78, 73, 79, 76] one may find discussion of other theoretical, experimental, numerical and practical aspects of turbulent transport. For the use of stochastic processes in modeling of non-linear fluid dynamics itself, see e.g. [32]. It is believed that realistic turbulent flows are described by hydrodynamical equations, like the NavierStokes ones, that govern the evolution of the velocity field and, eventually, of other relevant fields like fluid density, temperature, etc. As any structure whose dynamics is governed by evolution equations, the turbulent flow may be viewed from a theoretical point as a dynamical system. Many attempts were made then to apply to turbulence ideas from the dynamical systems theory, see e.g. [22]. These ideas, developed for low-dimensional dynamical systems, although often successful in explaining the onset of turbulence, seem to miss important features when confronted with the fully developed turbulence. Some of them, however, prove very useful in the description of transport in turbulent flows, as we shall see in the sequel. ∗ Text of lectures at the School on Stochastic Geometry, the Stochastic Loewner Evolution, and Non-Equilibrium Growth Processes, Trieste, Italy, July 7-18, 2008