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7.5 A model for the departure from self-similarity

Besides the idea of self-similarity underlying the process of energy cascade in turbulence, a different point of view can be introduced. That is a model that tries to characterize the behavior of the PDFs through the scaling laws of the parameters, which describe how the shape of the PDFs changes going towards small scales. In its simplest form the model can be introduced by saying that PDFs of increments ± dZ l, at a given scale, are built up by a convolution of Gaussian distributions of width ± 2 1/2 s = <(dZ l ) >, whose distribution is given by Gc(s), namely
integral ( ± 2 ) P (dZ± ) = -1- oo G (s)exp - (dZ-l-)- ds. (42) l 2p 0 c 2s2 s
In a purely self-similar situation, where the energy cascade generates only a trivial variation of s with scales, the width of the distribution Gc(s) is zero and, invariably, we recover a Gaussian distribution for P (dZl±). On the contrary, when the cascade is not strictly self-similar, the width of Gc(s) is different from zero and the scaling behavior of the width c2 of G (s) c can be used to characterize intermittency.
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