✎ multi-fractal_stuff [mist]

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In the literature (mainly covered by Uriel Frisch's book), you will find three ways to characterise intermittency:
  - PDFs of increments x_l: P(x_l), which display fatter wings at small lags l.
  - Scaling of structure functions: <x_l^0p> ~ C_p x_L^p l^tau(p) at small l, or at every l depending on the source. The function tau(p) is said to characterise intermittency.
  - Multifractal spectrum: P( ln|x_l|) ~ c(alpha) (l/L)^f(alpha) at small lags l. f(alpha) is called the multi-fractal spectrum: if you interpret alpha as the scaling, x_l=x_L (l/L)^alpha ==> alpha = ln (x_l/x_L) / ln (l/L). If we interpret P(ln|x_l|) as a volume, the previous scaling can be interpreted as places where x_l scales as l^alpha have a fractal co-dimension f(alpha).

  The functions tau(p) and f(alpha) are related by a Legendre transform.
  If you know the scalings C_p or c(alpha), and you assume the scalings hold at every lags (or for a continuous range of lags, in the inertial range or in the dissipative range of scales), you can build the PDFs. So technically, these three views are nearly equivalent, and many authors in addition assume either c(alpha) is constant or C_p is constant.

Here are a few papers to take you through some of the technical introductions to Multi-fractals adn intermittency related stuff.

  - Generalised fractal dimensions: a slightly mathematical paper, but enlightening, which links Hausdorff dimension, information dimension and others under a common formula for the dimension.
  - Meneveau & Srinivasan (1987): uses the generalised fractal dimension to link it to the structure function scalings. Measures it on data assuming a bi-fractal model.
  - HDR of Chevillard, and our recent paper Durrive, Lesaffre, Ferrière (2020) which proposes physically motivated constructions for velocity (and B) fields in HD or MHD turbulence which have known statistical properties of turbulence.
  - Muzy(2019). Very technical paper, but I'm happy I went deeply through it. This paper links the scalings to wavelet coefficients, and to the theory of Lévy-Khintchine (see Muzy+Bacry 2010 for more details on this). I still have to make my mind about this work, but this may be the way to link 3D to plane of sky projections.