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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

p- [ p/m -1] zp = m - log2 1 - q + q2 , (39)
where the free parameter q can be fixed through a fit on the data.

The p-model (Meneveau, 1991Jump To The Next Citation PointCarbone, 1993Jump To The Next Citation Point) 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

z = 1 - log [mp/m + (1 - m)p/m]. (40) p 2

In the model by She and Leveque (see, e.g., She and Leveque, 1994Politano 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

[ ] -p ( x)p/m zp = m (1 - x) + C 1 - 1 - C . (41)
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 (40View Equation, 41View Equation) 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.
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