Multifractal two-scale Cantor set model for slow solar wind turbulence in the outer heliosphere during solar maximum

Abstract

Abstract. To quantify solar wind turbulence, we consider a generalized two-scale weighted Cantor set with two different scales describing nonuniform distribution of the kinetic energy flux between cascading eddies of various sizes. We examine generalized dimensions and the corresponding multifractal singularity spectrum depending on one probability measure parameter and two rescaling parameters. In particular, we analyse time series of velocities of the slow speed streams of the solar wind measured in situ by Voyager 2 spacecraft in the outer heliosphere during solar maximum at various distances from the Sun: 10, 30, and 65 AU. This allows us to look at the evolution of multifractal intermittent scaling of the solar wind in the distant heliosphere. Namely, it appears that while the degree of multifractality for the solar wind during solar maximum is only weakly correlated with the heliospheric distance, but the multifractal spectrum could substantially be asymmetric in a very distant heliosphere beyond the planetary orbits. Therefore, one could expect that this scaling near the frontiers of the heliosphere should rather be asymmetric. It is worth noting that for the model with two different scaling parameters a better agreement with the solar wind data is obtained, especially for the negative index of the generalized dimensions. Therefore we argue that there is a need to use a two-scale cascade model. Hence we propose this model as a useful tool for analysis of intermittent turbulence in various environments and we hope that our general asymmetric multifractal model could shed more light on the nature of turbulence.

New insights and best practices for the successful use of Empirical Mode Decomposition, Iterative Filtering and derived algorithms

“…Moreover, the choice of the parameter p, with which we defined the weights w 11 and w 12 in 3, tunes the degree of intermittency of the final result u n (x) , as it regulates how uneven the partition of the amount E is from the (n − 1)-th to the n-th generation: p = 1 2 reproduces Kolmogorov's non-intermittent K41 theory 119 , while p → 0 and p → 1 gives rise to more and more intermittent distributions, with a mirror symmetry between p ∈ 0, 1 2 and p ∈ 1 2 , 1 . Last, but not least, the p-model distribution shows a well known multi-fractal singularity spectrum 122,124,125 . All those reasons render the p-model series a suitable test bed for multiscale analysis techniques, as EMD and IF, because the ground truth is an intermittent signal of perfectly known characteristics, and the extent to which those are retrieved by the various analysis tools clearly emerges.…”

Fractal density and singularity analysis of heat flow over ocean ridges

“…These two parameters (ρ α and α) can be estimated from observed data by measuring the intercept and slope of the log-log plot of m against ε , which plots as a straight line. Singularity is observed in non-linear processes including cloud formation 53 , rainfall 54 , hurricanes 55 , flooding 56 , landslides 57 , earthquakes 58 , mineralisation 59 60 , and solar wind turbulence 61 .…”

Multifractal Turbulence in the Heliosphere

“…In this way we have looked at the inhomogeneous rate of the transfer of the energy flux indicating multifractal and intermittent behavior of solar wind turbulence. In particular, we have studied in detail fluctuations of the velocity of the flow of the solar wind, as measured in the inner heliosphere by Helios and ACE , and Voyager in the outer heliosphere Macek & Wawrzaszek, 2011b), including Ulysses observations at high heliospheric latitudes .…”

“…The same analysis has been repeated for the Voyager 2 data for the slow solar wind during the solar maxima : 1981, 1989, 2001, at 10, 30, and 65 AU, correspondingly. This has allowed us to investigate the dependence of the multifractal spectra on the phase of the solar cycle (Macek & Wawrzaszek, 2011b).…”