Inverse statistics of smooth signals: the case of two dimensional turbulence

The problem of inverse statistics (statistics of distances for which the signal fluctuations are larger than a certain threshold) in differentiable signals with power law spectrum, $E(k)\sim k^{-\alpha }$, $3\le \alpha <5$, is discussed. We show that for these signals, with random phases, exit-distance moments follow a bi-fractal distribution. We also investigate two dimensional turbulent flows in the direct cascade regime, which display a more complex behavior. We give numerical evidences that the inverse statistics of 2d turbulent flows is described by a multi-fractal probability distribution, i.e. the statistics of laminar events is not simply captured by the exponent $\alpha$ characterizing the spectrum.