G. Boffetta

Predictability in the large: an extension of the concept of Lyapunov exponent

We investigate the predictability problem in dynamical systems with many degrees of freedom and a wide spectrum of temporal scales. In particular, we study the case of three-dimensional turbulence at high Reynolds numbers by introducing a finite-size Lyapunov exponent which measures the growth rate of finite-size perturbations. For sufficiently small perturbations this quantity coincides with the usual Lyapunov exponent. When the perturbation is still small compared to large-scale fluctuations, but large compared to fluctuations at the smallest dynamically active scales, the finite-size Lyapunov exponent is inversely proportional to the square of the perturbation size. Our results are supported by numerical experiments on shell models. We find that intermittency corrections do not change the scaling law of predictability. We also discuss the relation between the finite-size Lyapunov exponent and information entropy.

Predictability: a way to characterize complexity

Abstract Different aspects of the predictability problem in dynamical systems are reviewed. The deep relation among Lyapunov exponents, Kolmogorov–Sinai entropy, Shannon entropy and algorithmic complexity is discussed. In particular, we emphasize how a characterization of the unpredictability of a system gives a measure of its complexity. Adopting this point of view, we review some developments in the characterization of the predictability of systems showing different kinds of complexity: from low-dimensional systems to high-dimensional ones with spatio-temporal chaos and to fully developed turbulence. A special attention is devoted to finite-time and finite-resolution effects on predictability, which can be accounted with suitable generalization of the standard indicators. The problems involved in systems with intrinsic randomness is discussed, with emphasis on the important problems of distinguishing chaos from noise and of modeling the system. The characterization of irregular behavior in systems with discrete phase space is also considered.

DISPERSION OF PASSIVE TRACERS IN CLOSED BASINS : BEYOND THE DIFFUSION COEFFICIENT

We investigate the spreading of passive tracers in closed basins. If the characteristic length scale of the Eulerian velocities is not very small compared with the size of the basin the usual diffusion coefficient does not give any relevant information about the mechanism of spreading. We introduce a finite size characteristic time τ(δ) which describes the diffusive process at scale δ. When δ is small compared with the typical length of the velocity field one has τ(δ)∼λ−1, where λ is the maximum Lyapunov exponent of the Lagrangian motion. At large δ the behavior of τ(δ) depends on the details of the system, in particular the presence of boundaries, and in this limit we have found a universal behavior for a large class of system under rather general hypothesis. The method of working at fixed scale δ makes more physical sense than the traditional way of looking at the relative diffusion at fixed delay times. This technique is displayed in a series of numerical experiments in simple flows.

Dynamics and statistics of heavy particles in turbulent flows

We present the results of direct numerical simulations (DNS) of turbulent flows seeded with millions of passive inertial particles. The maximum Reynolds number is Re λ∼ 200. We consider particles much heavier than the carrier flow in the limit when the Stokes drag force dominates their dynamical evolution. We discuss both the transient and the stationary regimes. In the transient regime, we study the growth of inhomogeneities in the particle spatial distribution driven by the preferential concentration out of intense vortex filaments. In the stationary regime, we study the acceleration fluctuations as a function of the Stokes number in the range St ∈ [0.16:3.3]. We also compare our results with those of pure fluid tracers (St = 0) and we find a critical behavior of inertia for small Stokes values. Starting from the pure monodisperse statistics we also characterize polydisperse suspensions with a given mean Stokes, .

Detecting barriers to transport: A review of different techniques

Abstract 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), to the Okubo–Weiss (OW) and Hua–Klein (HK) criteria. We show that the OW and HK are the 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 model of a meandering jet, and Lagrangian tracers advected by stratospheric circulation.

Acceleration statistics of heavy particles in turbulence

We present the results of direct numerical simulations of heavy particle transport in homogeneous, isotropic, fully developed turbulence, up to resolution

GROWTH OF NONINFINITESIMAL PERTURBATIONS IN TURBULENCE

512^3

Chaotic advection in point vortex models and two-dimensional turbulence

(

Lagrangian statistics of particle pairs in homogeneous isotropic turbulence

R_\lambda\approx 185

Evidence for the double cascade scenario in two-dimensional turbulence.

). Following the trajectories of up to 120 million particles with Stokes numbers, St , in the range from 0.16 to 3.5 we are able to characterize in full detail the statistics of particle acceleration. We show that: (i) the root-mean-squared acceleration