Lagrangian Chaos and Transport in Fluids

摘要

We revise the classical theory of Batchelor, which gives the k-1 law for the power spectrum of a passive scalar at wavenumbers k, for which the molecular diffusion is unimportant and is much smaller than the fluid viscosity. Using some ideas borrowed from the theory of dynamical systems, we show that this power law is related to the chaotic motion of marker particles (Lagrangian chaos) and to the incompressibility constraint. Our approach permits of showing that the k~(-1) regime is also present in fluids which are not turbulent and it is valid for all dimensionalities d >= 2. Moreover one can show that, in the kinematic dynamo approximation, the magnetic field is of a multifractal nature as a consequence of the spatial fluctuations in the exponential growth of the field.