and the -th order structure function is immediately written through the integral over all (continuous) values of weighted by a smooth function , i.e.,
A moment of reflection allows us to realize that in the limit the integral is dominated by the minimum value (over ) of the exponent and, as shown by Frisch (1995), the integral can be formally solved using the usual saddle-point method. The scaling exponents of the structure function can then be written as
In this way, the departure of from the linear Kolmogorov scaling and then intermittency, can be characterized by the continuous changing of as varies. That is, as varies we are probing regions of fluid where even more rare and intense events exist. These regions are characterized by small values of , that is, by stronger singularities of the gradient of the field.
Owing to the famous Landau footnote on the fact that fluctuations of the energy transfer rate must be taken into account in determining the statistics of turbulence, people tried to interpret the non-linear energy cascade typical of turbulence theory, within a geometrical framework. The old Richardson’s picture of the turbulent behavior as the result of a hierarchy of eddies at different scales has been modified and, as realized by Kraichnan (1974), once we leave the idea of a constant energy cascade rate we open a “Pandora’s box” of possibilities for modeling the energy cascade. By looking at scaling laws for and introducing the scaling exponents for the energy transfer rate , it can be found that (being when the Kolmogorov-like phenomenology is taken into account, or when the fluid is magnetically dominated). In this way the intermittency correction are determined by a cascade model for the energy transfer rate. When is a non-linear function of , the energy transfer rate can be described within the multifractal geometry (see, e.g., Meneveau, 1991 and references therein) characterized by the generalized dimensions (Hentschel and Procaccia, 1983). The scaling exponents of the structure functions are then related to by
The correction to the linear scaling is positive for , negative for , and zero for . A fractal behavior where gives a linear correction with a slope different from .
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