Fluctuation-response relation in turbulent systems

We address the problem of measuring time properties of response functions (Green functions,) in Gaussian models (Orszag-McLaughin) and strongly non-Gaussian models (shell models for turbulence), We introduce the concept of halving-time statistics to have a statistically stable tool to quantify the time decay of response functions and generalized response functions of high order. We show numerically that in shell models for three-dimensional turbulence response functions are inertial range quantities. This is a strong indication that the invariant measure describing the shell-velocity fluctuations is characterized by short range interactions between neighboring shells.