Turbulent transport and integrable chaos

Many basic features of transport by turbulent flows may be captured by simple models where turbulent velocities are described by an imposed random ensemble and the transported matter (pollutant) or fields (temperature, magnetic field) are assumed to be carried by the flow without influencing it. From the mathematical point of view, such models are random dynamical systems where one studies the flow equations $\ d{\bf R}={\bf v}(t,{\bf R})\ $ with a random right hand side. Such systems come in two sorts: the more standard one, with typical velocities smooth in space, used to model flows at moderate Reynolds numbers, and a less standard one, with spatially rough velocities, that applies to high Reynolds number flows. In the simplest model proposed by Robert Kraichnan almost 40 years ago, the velocities are assumed to form a white noise in time and many of the interesting questions, some traditionally asked for dynamical systems, some going beyond, find analytic answers. As an illustration, I shall sketch how some known integrable models of quantum mechanics provide the control of large deviations of finite-time Lyapunov exponents of the Kraichnan model. Such large deviations determine the decay of temperature fluctuations and the growth of pollutant or magnetic field inhomogeneities as well as the multifractal properties of particle suspensions.