In the preceding chapters we considered the dynamics of an individual system. Starting from a smooth state, fine structures develop, which, in general, become unstable at some point. After the onset of instability the structure of the flow is very complex and irregular and, most importantly, the further behavior is unpredictable in the sense that minimal changes would soon lead to a completely different state. Such a behavior is commonly called turbulent. Though a direct view of the continuously changing patterns is certainly most eyecatching and fascinating, a pictorial description of these structures is not very suitable for a quantitative analysis. On the other hand, it is just this chaotic behavior which makes turbulence accessible to a theoretical treatment involving statistical methods. While individual shapes and motions are intricate and volatile, the average properties of the turbulence described by the various correlation functions are, in general, smooth and follow rather simple laws. A well-known paradigm is the turbulent behavior in our atmosphere. We try to predict the short-term changes, called weather, in a deterministic way for as long as is feasible, which, as daily experience shows, is not very long, while predictions of the long-term behavior, called climate, can be made only on a statistical basis.

Dividing the fields into mean and fluctuating parts, we derive equations for the average quantities, the generalized Reynolds equations, which contain second-order moments of the fluctuating parts, the turbulent stresses.