Study of fractal features of magnetized plasma through an MHD shell model

Abstract

A magnetohydrodynamic (MHD) shell model is used to describe the dissipative events which take place in magnetized plasmas. A scatter plot box-counting fractal dimension D is calculated for the time series of the magnetic energy dissipation rate obtained in the MHD shell model, and the correlation between D and the energy dissipation rate is analyzed. We show that, depending on the values of the viscosity and the diffusivity, the fractal dimension and the occurrence of bursts exhibit correlations similar to those observed in previous studies.

Evolution of fractality in space plasmas of interest to geomagnetic activity

“…Compare, for instance, the different panels in Fig. 16 in Domínguez et al (2017), which show that Pm = 0.2 leads to a very noisy output, unlike simulations with Pm = 1.0 or 2.0, where individual, large peaks can be easily identified from the background. In fact, previous studies have shown that the statistics of bursts follows a power law for Pm = 1 (Boffetta et al, 1999;Lepreti et al, 2004;Carbone et al, 2002), and for this reason we start by taking Pm = ν/η = 1.…”

“…In a previous work (Domínguez et al, 2017), we have applied the box-counting fractal dimension to study the complexity in an MHD shell model, analyzing the correlation between it and the energy dissipation rate, showing that, for certain values of the viscosity and the magnetic diffusivity, the fractal dimension exhibits correlation with the occurrence of bursts, similar to what had been found with geomagnetic data (Domínguez et al, 2014). This suggests that shell models do not only reproduce the power-law statistics of dissipative events in turbulent plasmas but also some features of its fractal behavior.…”

“…Thus, the scatter plot can be built by plotting the ith data in the set, versus the (i +j )th data, with j ≥ 1 in general, and then the fractal dimension calculated depends on j , D j . This was the approach in Domínguez et al (2017), and it is presented here in Sects. 4 and 5.…”

“…However this is not available for the output of the shell model, and our approach was to inspect the data to gain intuition on this. As described in detail in Domínguez et al (2017), we take Pm = ν/η = 1 as fixed, and then a wide range of values of ν, in the interval 10 Domínguez et al, 2017. ) shell model equations using a time step of dt = 10 −4 and for 7×10 8 iterations.…”

“…2-3, and discussed previously in more detail in Domínguez et al, 2014) and the occurrence of dissipative events in an MHD shell model simulation (Sect. 4; see Domínguez et al, 2017 for more details). We also present preliminary results dealing with spacecraft data for the solar wind, related to the appearance of magnetic clouds (Sect.…”