Role of the Lagrangian chaoticity on the small scale structure of passive scalars in fluids

We study the small scale (viscous convective subrange) structure of convective quantities in incompressible fluids. 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. We stress that the k^-1 regime is also present in fluids which are not turbulent. Moreover our approach permits showing that Batchelor's law is valid for all dimensionalities d⩾2. We consider in particular the case of fully developed turbulence in two and three dimensions. We show that when d = 2, the k^-1 power law is obeyed even in the inertial range, in contrast with the d = 3 case where one has approximately a k^-5/3 power law in the inertial range, and a k^-1 power law in the viscous convective subrange.