The traditional description of turbulence (as summarized in the monograph by Monin and Yaglom [1971, 1975]) employs statistical methods, truncation schemes in the form of approximate closure theories and phenomenological models (e.g. Kolmogorov's theories of 1941 and 1962). The complementary point of view guiding our description of turbulence is to regard the Navier–Stokes equations, or other partial differential equations describing turbulent systems, as a deterministic dynamical system and to regard the turbulence as a manifestation of deterministic chaos.
In the case of fully developed turbulence the direct simulation of the Navier–Stokes equations is prohibitively difficult owing to the large range of relevant length scales. It is thus important to study simplified models, and a large part of this book is devoted to the introduction and investigation of such models. We shall give an introduction to the dynamical systems approach to turbulence and show the applicability of methods borrowed from dynamical systems to a wide class of dynamical states in spatially extended systems, for which we shall use the general term turbulence.
It is important to note that the dynamical models employed to describe turbulent states are not low-dimensional. In flows with high Reynolds numbers or in chaotic systems of large spatial extent, the number of relevant degrees of freedom is very large, and our primary interest is to explore properties that are well defined in the ‘thermodynamic limit’, where the system size (or Reynolds number) becomes very large.