### 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 , at a given scale, are built up by a convolution of
Gaussian distributions of width , whose distribution is given by , namely
In a purely self-similar situation, where the energy cascade generates only a trivial variation of with
scales, the width of the distribution is zero and, invariably, we recover a Gaussian distribution for
. On the contrary, when the cascade is not strictly self-similar, the width of is different
from zero and the scaling behavior of the width of can be used to characterize
intermittency.