Introduction

In the earlier chapters of this book the emphasis has been on a study of the existence and stability of nonlinear waves and solitons, that is of coherent structures. Such structures are found in Nature and thus certainly deserve our attention. However, a much more universal type of behaviour is described under the umbrella of turbulence. One envisages turbulence as a phenomenon where some measurable quantity has a rapid space and/or time dependence. For example, in the case of water passing over a weir, the complicated behaviour is apparent in the local velocity of the water. One sees eddies (or vortices) of a range of sizes. They not only move with some background velocity but also interact with one another to produce a continually changing picture.

For another example, consider turbulence in the wake of a cylinder if the water flow is of very high Reynolds number (see Fig. 1.5(c), (d)).

The problem of trying to understand turbulence has been with us for centuries but it still remains a basic unresolved problem. (The beauty and complexity of turbulence was well appreciated by Leonardo da Vinci as is evidenced in his drawings of vortices in water, Fig. 11.1.)

Somewhat ironically, the study of turbulence in plasmas, which themselves are much more complicated media, is more tractable than in water and considerable progress has been made in the last twenty years. However, most theories to date are restricted in that they assume that the energy in the fluctuations is small compared to the kinetic energy of the particles.