On the nature of turbulence

The definition of turbulence and the differences between turbulence and random wave motions of liquids or gases are discussed. The Landau scheme of the generation of turbulence with increasing Reynolds number as a result of a sequence of normal bifurcations that creates a quasiperiodic motion is considered; several examples are discussed, including flow between rotating cylinders, convection at small Prandtl numbers, and the boundary layer at a flat plate. Results obtained in recent years from the ergodic theory and associated with the discovery of strange attractors in the phase spaces of typical dynamic systems are described. Flows with inverse bifurcations are considered, including the plane Poiseuille flow and Lorenz's example with idealized three-mode convection at large Prandtl numbers. In the latter case, the results of numerical calculations are analyzed and point to the existence of a strange attractor with the structure of a Cantor discontinuum; other examples of systems with strange attractors are also considered. It becomes clear that strange attractors in the phase spaces of systems with few modes may explain their nonperiodic behavior, but cannot explain why turbulence has a continuous spatial spectrum.