Intermittency and Multifractal Transport in Geography, Geosciences and Networks

Daniel Schertzer
École Nationale des Ponts-et-Chaussées (ENPC)

Fundamental steps have been achieved in the recent years to unify scaling laws in geography, geosciences and networks Dodds and Rothman, 2000), in particular by pointing out their possible common source as being turbulence (Birnir, 2007). In this presentation, we will focus on the fundamental feature of turbulent transport to be intermittent.

Intermittent transport means that its intensity/efficiency to exceed a given threshold concentrate over only a small fraction of the available space-time space. Furthermore, this concentration is stronger and stronger for higher and higher intensity levels. This concentration can be so drastic that it will corresponds to embedded fractions of the space with smaller and smaller fractal dimensions.

Physically, it was shown to be the generic outcome of cascade processes (Schertzer and Lovejoy, 1984) and it corresponds to the notion of multifractal (Parisi and Frisch,1985). Mathematically, it corresponds to have a (stochastic) multi-singular invariant measure, therefore to have an infinite hierarchy of singularity orders coupled with that of the corresponding supports. However, we will show that under fairly general conditions this hierarchy can be defined with the help of a few exponents, which not only have a strong physical significance, but can be determined either theoretically or empirically (Schertzer and Lovejoy, 1997).

These theoretical considerations will be illustrated with the help of a recent multifractal analysis of streamflows at 14000 hydrological stations of US and Canada in thframework of a CEATI research project (Tchiguirinskaia et al.)
and with the help of atmospheric turbulence analyses (Lovejoy et al., 2007).

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