Dariusz Wójcik

In our previous studies we have examined solar wind and magnetospheric plasma turbulence, including Markovian character on large inertial magnetohydrodynamic scales. Here we present the results of the statistical analysis of magnetic field fluctuations in the Earth’s magnetosheath, based on the Magnetospheric Multiscale mission at much smaller kinetic scales. Following our results on spectral analysis with very large slopes of about −16/3, we apply a Markov-process approach to turbulence in this kinetic regime. It is shown that the Chapman–Kolmogorov equation is satisfied and that the lowest-order Kramers–Moyal coefficients describing drift and diffusion with a power-law dependence are consistent with a generalized Ornstein–Uhlenbeck process. The solutions of the Fokker–Planck equation agree with experimental probability density functions, which exhibit a universal global scale invariance through the kinetic domain. In particular, for moderate scales we have the kappa distribution described by various peaked shapes with heavy tails, which, with large values of the kappa parameter, are reduced to the Gaussian distribution for large inertial scales. This shows that the turbulence cascade can be described by the Markov processes also on very small scales. The obtained results on kinetic scales may be useful for a better understanding of the physical mechanisms governing turbulence.

We apply statistical analysis to search for processes responsible for turbulence in physical systems. In our previous studies, we have shown that solar wind turbulence in the inertial range of large magnetohydrodynamic scales exhibits Markov properties. We have recently extended this approach on much smaller kinetic scales. Here we are testing for the Markovian character of stochastic processes in a kinetic regime based on magnetic field and velocity fluctuations in the solar wind, measured onboard the Magnetospheric Multiscale (MMS) mission: behind the bow shock, inside the magnetosheath, and near the magnetopause. We have verified that the Chapman-Kolmogorov necessary conditions for Markov processes is satisfied for local transfer of energy between the magnetic and velocity fields also on kinetic scales. We have confirmed that for magnetic fluctuations, the first Kramers-Moyal coefficient is linear, while the second term is quadratic, corresponding to drift and diffusion processes in the resulting Fokker-Planck equation. It means that magnetic self-similar turbulence is described by generalized Ornstein-Uhlenbeck processes. We show that for the magnetic case, the Fokker-Planck equation leads to the probability density functions of the kappa distributions, which exhibit global universal scale invariance with a linear scaling and lack of intermittency. On the contrary, for velocity fluctuations, higher order Kramers-Moyal coefficients should be taken into account and hence scale invariance is not observed. However, the nonextensity parameter in Tsallis entropy provides a robust measure of the departure of the system from equilibrium. The obtained results are important for a better understanding of the physical mechanism governing turbulent systems in space and laboratory.

We have recently argued that a simple nonlinear law could possibly be important for the origin of the Universe resulting in fractal or multifractal features. Various fractal scaling models of the large-scale mass distribution have already been proposed. The expected universal multifractal function for galaxies is similar to that identi ed by NASA’s Voyager mission in the Solar System. Hence we now apply the similar method for determination of the reliable multifractal spectrum of distribution of galaxies on cosmological scales, based on selected observations from a million of galaxies in the Redshift Catalog updated in June 2008. We show that the observed spectrum is consistent with the weighted one- or two-scale Cantor set models characteristic for turbulence in laboratory and inside the Sun’s heliosphere immersed in the very local interstellar medium. However, the total degree of multifractality Delta=0.2 is smaller than that inside the heliosphere. This would be characteristic for a simple linear fractal scaling of galaxy distribution, but somewhat varying for nearby (Delta=0.1) and the most remote galaxies (Delta=0.2) receding from our Solar System. The parameters p=0.45 and lambda<=1/2 for one-scale model are apparently related to some voids in the large-scale distribution of matter. A possible asymmetry (A 3/4) of the total spectrum for the two-scale weighted Cantor set (A != 1) could admittedly be attributed to some deviations from the Hubble’s law for the ideal uniform expansion of the Universe.