Self-similar continuous cascades supported by random Cantor sets: Application to rainfall data.

We introduce a variant of continuous random cascade models that extends former constructions introduced by Barral-Mandelbrot and Bacry-Muzy in the sense that they can be supported by sets of arbitrary fractal dimension. The so-introduced sets are exactly self-similar stationary versions of random Cantor sets formerly introduced by Mandelbrot as "random cutouts." We discuss the main mathematical properties of our construction and compute its scaling properties. We then illustrate our purpose on several numerical examples and we consider a possible application to rainfall data. We notably show that our model allows us to reproduce remarkably the distribution of dry period durations.

Random cascade model in the limit of infinite integral scale as the exponential of a nonstationary 1/f noise: application to volatility fluctuations in stock markets.

In this paper we propose a new model for volatility fluctuations in financial time series. This model relies on a nonstationary Gaussian process that exhibits aging behavior. It turns out that its properties, over any finite time interval, are very close to continuous cascade models. These latter models are indeed well known to reproduce faithfully the main stylized facts of financial time series. However, it involves a large-scale parameter (the so-called "integral scale" where the cascade is initiated) that is hard to interpret in finance. Moreover, the empirical value of the integral scale is in general deeply correlated to the overall length of the sample. This feature is precisely predicted by our model, which, as illustrated by various examples from daily stock index data, quantitatively reproduces the empirical observations.

Intermittency of surface-layer wind velocity series in the mesoscale range.

We study various time series of surface-layer wind velocity at different locations and provide evidences for the intermittent nature of the wind fluctuations in mesoscale to large-scale range. By means of the magnitude covariance analysis, which is shown to be a more efficient tool to study intermittency than classical scaling analysis, we find that all wind series exhibit similar features than those observed for laboratory turbulence. Our findings suggest the existence of a "universal" cascade mechanism associated with the energy transfer between synoptic motions and turbulent microscales in the atmospheric boundary layer.

Spatial intermittency of surface layer wind fluctuations at mesoscale range.

We study various hourly surface layer wind series recorded at different sites in the Netherlands by the "Royal Netherlands Meteorological Institute." By reporting all velocity magnitude correlation coefficients, associated with the available couples of locations, as a function of their spatial distance, we find that they fall on a single curve. This curve turns out to be remarkably well described by a logarithmic shape, characteristic of continuous cascades with an intermittency coefficient λ2 ≃ 0.04 and an integral scale L ≃ 600 km. Along the same line, we study the scaling properties of spatial velocity increment structure functions. This allows one to estimate the ζ(q) spectrum and to confirm an intermittent nature of mesoscale fluctuations similar to the one observed in fully developed turbulence.