7.4 Fragmentation models for the energy transfer rate

Cascade models can be organized as a collection of fragments at a given scale $l$ , which results from the fragmentation of structures at the scale $l'> l$ , down to the dissipative scale (Novikov, 1969 ). Sophisticated statistics are applied to obtain scaling exponents $zp$ for the $p$ -th order structure function.

The random- $b$ model (Benzi et al., 1984 ) can be derived by invoking that the space-filling factor for the fragments at a given scale in the energy cascade is given by a random variable $b$ . The probability of occurrence of a given $b$ is assumed to be a bimodal distribution where the eddies fragmentation process generates either space-filling eddies with probability $q$ or planar sheets with probability $(1 - q)$ (for conservation $0 < q < 1$ ). It can be found that

where the free parameter $q$ can be fixed through a fit on the data.

The $p$ -model (Meneveau, 1991 ; Carbone, 1993 ) consists in an eddies fragmentation process described by a two-scale Cantor set with equal partition intervals. An eddy at the scale $l$ , with an energy derived from the transfer rate $er$ , breaks down into two eddies at the scale $l/2$ , with energies $mer$ and $(1 - m)er$ . The parameter $0.5 < m < 1$ is not defined by the model, but is fixed from the experimental data. The model gives

In the model by She and Leveque (see, e.g., She and Leveque, 1994 ; Politano and Pouquet, 1998 ) one assumes an infinite hierarchy for the moments of the energy transfer rates, leading to $e(pr+1) ~ [e(rp)]b[e(r oo )]1- b$ , and a divergent scaling law for the infinite-order moment $e(r oo ) ~ r- x$ , which describes the most singular structures within the flow. The model reads

The parameter $C = x/(1 - b)$ is identified as the codimension of the most singular structures. In the standard MHD case (Politano and Pouquet, 1995 ) $x = b = 1/2$ , so that $C = 1$ , that is, the most singular dissipative structures are planar sheets. On the contrary, in fluid flows $C = 2$ and the most dissipative structures are filaments. The large $p$ behavior of the $p$ -model is given by $zp ~ (p/m) log2(1/m) + 1$ , so that Equations (40

, 41

) give the same results providing $-x m -~ 2$ . As shown by Carbone et al. (1996b ) models are able to capture intermittency of fluctuations in the solar wind. The agreement between the curves $zp$ and normalized scaling exponents is excellent, and this means that we realistically cannot discriminate between the models we reported above.