Statistical predictability of decaying turbulence.

Abstract

We use statistical models of turbulence with 'eddy damping' (EDQNM) in order to study the problem of predictability of freely evolving two- and three-dimensional isotropic turbulent flows. The application of statistical theories to this problem necessitates taking into account long-range interactions between very different scales ('nonlocal' interactions) intervening in the evolution of the error spectrum. We have therefore developed an analytical and numerical modeling of the nonlocal interactions enabling us to ensure the 'realizability' of the error spectrum. First, we validate our numerical codes by retrieving, in the case of a stationary turbulence, the results of Leith and Kraichnan. Second, the calculations carried out in the case of freely evolving 3-dimensional and 2-dimensional turbulence allow for the determination of temporal evolution laws of quantities characterizing the inverse error cascade. Various inertial ranges are displayed for the spectra, and some analogies with the passive scalar problem are discussed. Different determinations of a predictability time are proposed. The major conclusion of this work is that the growing large scales of the 3- and 2-dimensional turbulence are eventually affected by the inverse error cascade. Nevertheless, the predictability of these flows, measured with the relative error, is increased by a factor of about 40-50%.-from Authors