Introduction to chaos and diffusion

This contribution is relative to the opening lectures of the ISSAOS 2001 summer school and it has the aim to provide the reader with some concepts and techniques concerning chaotic dynamics and transport processes in fluids. Our intention is twofold: to give a self-consistent introduction to chaos and diffusion, and to offer a guide for the reading of the rest of this volume.Comment: 39 page

The predictability problem in systems with an uncertainty in the evolution law

The problem of error growth due to the incomplete knowledge of the evolution law which rules the dynamics of a given physical system is addressed. Major interest is devoted to the analysis of error amplification in systems with many characteristic times and scales. The importance of a proper parameterization of fast scales in systems with many strongly interacting degrees of freedom is highlighted and its consequences for the modelization of geophysical systems are discussed.Comment: 20 pages RevTeX, 6 eps figures (included

Relaxation of finite perturbations: Beyond the Fluctuation-Response relation

We study the response of dynamical systems to finite amplitude perturbation. A generalized Fluctuation-Response relation is derived, which links the average relaxation toward equilibrium to the invariant measure of the system and points out the relevance of the amplitude of the initial perturbation. Numerical computations on systems with many characteristic times show the relevance of the above relation in realistic cases.Comment: 7 pages, 5 figure

Non Asymptotic Properties of Transport and Mixing

We study relative dispersion of passive scalar in non-ideal cases, i.e. in situations in which asymptotic techniques cannot be applied; typically when the characteristic length scale of the Eulerian velocity field is not much smaller than the domain size. Of course, in such a situation usual asymptotic quantities (the diffusion coefficients) do not give any relevant information about the transport mechanisms. On the other hand, we shall show that the Finite Size Lyapunov Exponent, originally introduced for the predictability problem, appears to be rather powerful in approaching the non-asymptotic transport properties. This technique is applied in a series of numerical experiments in simple flows with chaotic behaviors, in experimental data analysis of drifter and to study relative dispersion in fully developed turbulence.Comment: 19 RevTeX pages + 8 figures included, submitted on Chaos special issue on Transport and Mixin

Drifter dispersion in the Adriatic Sea: Lagrangian data and chaotic model

International audienceWe analyze characteristics of drifter trajectories from the Adriatic Sea with recently introduced nonlinear dynamics techniques. We discuss how in quasi-enclosed basins, relative dispersion as a function of time, a standard analysis tool in this context, may give a distorted picture of the dynamics. We further show that useful information may be obtained by using two related non-asymptotic indicators, the Finite-Scale Lyapunov Exponent (FSLE) and the Lagrangian Structure Function (LSF), which both describe intrinsic physical properties at a given scale. We introduce a simple chaotic model for drifter motion in this system, and show by comparison with the model that Lagrangian dispersion is mainly driven by advection at sub-basin scales until saturation sets in

Experimental evidence of chaotic advection in a convective flow

Lagrangian chaos is experimentally investigated in a convective flow by means of Particle Tracking Velocimetry. The Fnite Size Lyapunov Exponent analysis is applied to quantify dispersion properties at different scales. In the range of parameters of the experiment, Lagrangian motion is found to be chaotic. Moreover, the Lyapunov depends on the Rayleigh number as ${\cal R}a^{1/2}$. A simple dimensional argument for explaining the observed power law scaling is proposed.Comment: 7 pages, 3 figur

The Richardson's Law in Large-Eddy Simulations of Boundary Layer flows

Relative dispersion in a neutrally stratified planetary boundary layer (PBL) is investigated by means of Large-Eddy Simulations (LES). Despite the small extension of the inertial range of scales in the simulated PBL, our Lagrangian statistics turns out to be compatible with the Richardson $t^3$ law for the average of square particle separation. This emerges from the application of nonstandard methods of analysis through which a precise measure of the Richardson constant was also possible. Its values is estimated as $C_2\sim 0.5$ in close agreement with recent experiments and three-dimensional direct numerical simulations.Comment: 15 LaTex pages, 4 PS figure

Detecting barriers to transport: A review of different techniques

We review and discuss some different techniques for describing local dispersion properties in fluids. A recent Lagrangian diagnostics, based on the Finite Scale Lyapunov Exponent (FSLE), is presented and compared to the Finite Time Lyapunov Exponent (FTLE), and to the Okubo-Weiss (OW) and Hua-Klein (HK) criteria. We show that the OW and HK are a limiting case of the FTLE, and that the FSLE is the most efficient method for detecting the presence of cross-stream barriers. We illustrate our findings by considering two examples of geophysical interest: a kinematic meandering jet model, and Lagrangian tracers advected by stratospheric circulation.Comment: 15 pages, 9 figures, submitted to Physica

Characterization of a periodically driven chaotic dynamical system

We discuss how to characterize the behavior of a chaotic dynamical system depending on a parameter that varies periodically in time. In particular, we study the predictability time, the correlations and the mean responses, by defining a local--in--time version of these quantities. In systems where the time scale related to the time periodic variation of the parameter is much larger than the ``internal'' time scale, one has that the local quantities strongly depend on the phase of the cycle. In this case, the standard global quantities can give misleading information.Comment: 15 pages, Revtex 2.0, 8 figures, included. All files packed with uufile