Lagrangian chaos and small scale structure of passive scalars

Abstract

We revise the classical theory of Batchelor, which gives a k−1 law for the power spectrum of a passive scalar at wavenumbers k, for which the molecular diffusion is unimportant and 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. Moreover our approach permits showing that the k−1 regime is present in fluids which are not turbulent and it is valid for all dimensionalities d⩾2.