Extreme events and the multifractal butterfly effect
Seminar Room 1, Newton Institute
AbstractScaling 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
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