Predictability and the Butterfly Effect in Turbulent Flows

We discuss the dynamical growth of a disturbance in turbulent flows based on numerical calculations of an approximative model of the Navier–Stokes equations, a so-called shell model. A disturbance at small length scales is observed to propagate (and increase) towards large length scales by an inverse cascade of duration T, the predictability time. At increasing Reynolds number Re, the mean predictability time Tt is found to decrease proportionally to Re^−0.47. Moreover, the probability distribution of (T − Tt)/Tt changes its shape as Re increases: at relatively small values of Re it has an almost Gaussian shape, while at large Re it gets an exponential tail, indicating the possibility of large excursions for Tt.