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Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier

  • Benoit B. Mandelbrot (a1)
  • DOI: http://dx.doi.org/10.1017/S0022112074000711
  • Published online: 01 March 2006
Abstract

Kolmogorov's ‘third hypothesis’ asserts that in intermittent turbulence the average ε of the dissipation ε, taken over any domain D, is ruled by the lognormal probability distribution. This hypothesis will be shown to be inconsistent, save under assumptions that are extreme and unlikely. Further, a widely used justification of lognormality, due to Yaglom and based on probabilistic argument involving a self-similar cascade, will be discussed. In this model, lognormality indeed applies strictly when D is ‘an eddy’, typically a three-dimensional box embedded in a self-similar hierarchy, and may perhaps remain a reasonable approximation when D consists of a few such eddies. On the other hand, the experimental situation is described better by considering averages taken over essentially one-dimensional domains D. The first purpose of this paper is to carry out Yaglom's cascade argument, labelled as ‘microcanonical’, for such averaging domains. The second is to replace Yaglom's model by a different, less constrained one, based upon the concept of ‘canonical cascade’. It will be shown, both for one-dimensional domains in a microcanonical cascade, and for all domains in canonical cascades, that in every non-degenerate caJe the distribution of ε differs from the lognormal distribution. Depending upon various parameters, the discrepancy may be either moderate, or considerable, or even extreme. In the latter two cases, high-order moments of E turn out to be infinite. This avoids various paradoxes (to be explored) that are present in Kolmogorov's and Yaglom's approaches. The third purpose is to note that high-order moments become infinite only when the number of levels of the cascade tends to infinity, meaning that the internal scale η tends to zero. Granted the usual value of η, this number of levels is actually small, so the representativity of the limit is questionable. This issue was investigated through computer simulation. The results bear upon the question of the extent to which Kolmogorov's second hypothesis applies in the face of intermittency. The fourth purpose is as follows. Yaglom noted that the cascade model predicts that dissipation only occurs in a portion of space of very small total volume. In order to describe the structure of this portion of space, it will be shown useful to introduce the concept of the ‘intrinsic fractional dimension’ A of the carrier of intermittent turbulence. The fifth purpose is to study the relations between the parameters ruling the distribution of η, and those ruling its spectral and dimensional properties. Both conceptually and numerically, these various parameters turn out to be distinct, which opens up several problems for empirical study.

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