Dynamical Systems Approach to Turbulence

Abstract

This book, first published in 1998, treats turbulence from the point of view of dynamical systems. The exposition centres around a number of important simplified models for turbulent behaviour in systems ranging from fluid motion (classical turbulence) to chemical reactions and interfaces in disordered systems.The modern theory of fractals and multifractals now plays a major role in turbulence research, and turbulent states are being studied as important dynamical states of matter occurring also in systems outside the realm of hydrodynamics, i.e. chemical reactions or front propagation. The presentation relies heavily on simplified models of turbulent behaviour, notably shell models, coupled map lattices, amplitude equations and interface models, and the focus is primarily on fundamental concepts such as the differences between large and small systems, the nature of correlations and the origin of fractals and of scaling behaviour. This book will be of interest to graduate students and researchers interested in turbulence, from physics and applied mathematics backgrounds.

Multi-time, multi-scale correlation functions in turbulence and in turbulent models

“…where τ nN τ n − τ N represents the time-delay needed for an energy burst to travel from shell n to shell N and the functions F p,q (x) and f p,q (x) are defined in the same way as done for (11). Let us observe that the sum in the above expression goes only up to the index of the largest scale n: this is because only for time larger than τ n the correlation is a true multi-time correlation.…”

“…In order to check the representations (11) and (16) we have performed numerical investigations in a class of dynamical models of turbulence (shell models) [11]. Within this modeling the approximation of local-interactions among velocity fluctuations at different scales is exact and therefore no-sweeping effects are present.…”

“…As a consequence of having eliminated the sweeping effects from the many time statistics, every phenomenological description that applies to fully developed Navier-Stokes turbulence can be straightforwardly abridged to make it applicable to shell models of turbulence (for a recent review see [11]). Shell models in this sense can be viewed as a shorthand of Navier-Stokes equations in the Quasi-Lagrangian frame of reference.…”

The Lorenz–Fermi–Pasta–Ulam experiment

“…The scaling of the characteristic properties with the number of subsystems is indicative that the weakly coupled lattice is an example of an "extensive" system [55].…”

Periodic orbit theory of two coupled Tchebyscheff maps

“…Refs. [1,2,3,4] give a good overview of early work to 1998 including the above applications; recent developments include advances in the mathematics of the relevant Banach spaces and transfer operators [5,6] as well as ever widening applications; examples include ecology [7,8,9], neurons [10,11], traffic flow [12] cryptography [13,14] and field theory [15,16]. From the point of view of understanding the dynamics of coupled chaotic maps, the most common CML studied are where the local dynamics f is quadratic (the "logistic" map); this class of one dimensional maps is now reasonably well understood.…”