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 three-dimensional and two-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 three- and two-dimensional turbulence are eventually affected by the inverse error cascade. Nevertheless the predictability of these flows, measured with the relative error, is increased (both in 3D and 2D) by a factor of about 40?50%, compared with a calculation with the same initial conditions and involving stationary energy spectra. Possible applications to atmospheric and oceanic situations are discussed. Finally, we examine the consequences of these results concerning the coherence of quasi two-dimensional structures in turbulent flows submitted to a barotropic instability, a rotation, or a magnetic field (in the magneto-hydrodynamic case).