Review article. — Rev. Mod. Phys., 2001, Vol. 73, No. 4, p. 913–975.The understanding of fluid turbulence has considerably progressed in recent years. The application of the methods of statistical mechanics to the description of the motion of fluid particles, i.e. to the Lagrangian dynamics, has led to a new quantitative theory of intermittency in turbulent transport. The first analytical description of anomalous scaling laws in turbulence has been obtained. The underlying physical mechanism reveals the role of statistical integrals of motion in non-equilibrium systems. For turbulent transport, the statistical conservation laws are hidden in the evolution of groups of fluid particles and arise from the competition between the expansion of a group and the change of its geometry. By breaking the scaleinvariance symmetry, the statistically conserved quantities lead to the observed anomalous scaling of transported elds. Lagrangian methods also shed new light on some practical issues, such as mixing and turbulent magnetic dynamo.Contents
Particles in fluid turbulence
Single-particle diffusion
Two-particle dispersion in a spatially smooth velocity
Two-particle dispersion in a nonsmooth incompressible flow
Two-particle dispersion in a compressible flow
Multiparticle dynamics, statistical conservation laws and breakdown of scale invariance
Passive Fields
Unforced evolution of passive scalar and vector Fields
Cascades of a passive scalar
Passive fields in the inertial interval of turbulence
Lagrangian numerics
Inverse cascade in the compressible Kraichnan model
Lessons for general scalar turbulence
Burgers and Navier-Stokes equations
Burgers turbulence
Incompressible turbulence from a Lagrangian viewpoint
