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Extreme events and the multifractal butterfly effect

Presented by: 
S Lovejoy McGill University
Thursday 31st October 2013 - 15:50 to 16:25
INI Seminar Room 1
Scaling processes abound in geophysics and this has important consequences for the probability distributions of the corresponding intensive and extensive geophysical variables. Classical scaling processes – such as in classical turbulence – are self-similar, they are characterized by exponents which are invariant under isotropic scale changes. However, the atmosphere and lithosphere are strongly stratified so that we must generalize the notion of scale allowing for invariance under anisotropic zooms. When this is done, it is often found that scaling can apply over huge ranges, up to planetary in extent. It is now clear that the generic scaling process is the multifractal cascade in which a scale invariant dynamical mechanism repeats (multiplicatively) from scale to scale; anisotropic scaling – and multifractal universality classes - imply that multifractals are widely relevant in the earth sciences. General (canonical) multifractal processes developed over finite ranges of scale and analyzed at their smallest scale (the “bare” process), have “long-tailed” distributions (e.g. the lognormal). However the small scale cascade limit is singular so that the integration/averaging of cascades developed down to their small scale limits leads to “dressed” properties characterized notably by “fat-tailed” power law probability distributions Pr(x>s)=s**-qD where x is a random value, s a threshold and qD the critical exponent implying that the moments for q>qD diverge. For cascades averaged over scales larger than the inne r cascade scale, the moments q>qD are no longer determined by the large scale finite by the small scale details: the “multifractal butterfly effect”. The sampling properties of such processes can be understood with “multifractal phase transitions”; we review this as well as evidence for the divergence of moments in laboratory, atmospheric and climatological series, and in data from the solid earth and discuss implications (abrupt changes, etc.).
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Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons