Incipient Turbulence and Chaos

Abstract

Analysing the fundamental problems of the NS equation has, in particular, brought up clearly the lack of an adequate algorithm, i.e. convergent and constructive, for its solution. Furthermore even if we knew that the fluid equations had unique and regular solutions, for regular initial data (for the NS equation this is true if d = 2 and likely if d = 3, but false if d >4) this would not help much in the understanding of the physical properties of such solutions at large times.

Lack of periodicity is very common in natural systems, and is one of the distinguishing features of turbulent flow. Because instantaneous flow patterns are so irregular, attention is often confined to the statistics of turbulence, which, in contrast to the details of turbulence, often behave in a regular well-organized manner. The short-range weather forecaster, however, is forced willy-nilly to predict the details of the large scale turbulent eddies — the cyclones and anticyclones — which continually arrange themselves into new patterns. Thus there are occasions when more than the statistics of irregular flow are of very real concern.

(E.N. Lorenz, 1963, [Lo63] p. 131.)