The methods and approaches to determine characteristics of turbulent environment

1Kozak, LV
1Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
Space Sci.&Technol. 2016, 22 ;(2):60-77
Publication Language: Ukrainian
The methods and approaches that can be used to analyze the hydrodynamic and magneto-hydrodynamic turbulent flows are selected. It was determined that the methods of statistical physics are most suitable to characterize the type of turbulent processes. Within the statistical approach we consider the fractal analysis (determination of fractal length and height of the maximum of the probability density fluctuations of the studied parameters), and multifractal analysis (study of a power dependence of high order statistical moments and construction of multifractal spectrum). It is indicated that statistical analysis of the properties of turbulent processes can be supplemented with the spectral studies: Fourier and wavelet analysis. In order to test the methods and  approaches discussed in the work the fero-probe measurements of magnetic field fluctuations obtained by the spacecraft “Samba” of the mission “Cluster 2” with discreetness of 22.5 Hz in the transition regions of Earth’s magnetosphere and solar wind were used.
We get a good agreement between different approaches and their mutual complementing to provide a general view of the turbulence.
Keywords: fractal analysis of satellite measurements, multifractal analysis, solar wind plasma, statistical analysis, the Earth’s magnetosphere, the fluctuations of the magnetic field, turbulence
 1. Astaf’eva N. M. Wavelet analysis: basic theory and some applications. Phys. Usp. 166(11), 1145—1170 (1996) [in Russian].
2. Zagorodnij A. G., Cheremnykh O. K. Introduction to the plasma physics, 696 p. (Nauk. dumka, Kiev, 2014) [in Russian].
3. Zaslavskij G. M., Sagdeev R. Z. Introduction to the nonlinear physics. From the pendulum to turbulence and chaos, 368 p. (Nauka, Moscow, 1988) [in Russian].
4. Kadomcev B. B. Turbulence of plasma Questions of plasma theory. Is.4, Ed. by M. A. Leontovich, 188—339 (Atomizdat, Moscow, 1964) [in Russian].
5. Kadomcev B. B. Collective phenomena in plasma, 303 p. (Nauka, Moscow, 1988) [in Russian].
6. Kozak L.V. A statistical approach for turbulent processes in the Earth ’s magnetosphere from measurements of the satellite Interball. Kosm. nauka tehnol, 16 (1), 28—39 (2010) [in Russian].

7. Kozak L. V., Pilipenko V. A., Chugunova O. M., Kozak P. N. Statistical analysis of turbulence in the foreshock region and in the Earth’s magnetosheath. Cosmic Research, 49 (3), 202—212 (2011) [in Russian].
8. Kozak L. V., Savin S. P., Budaev V. P., Pilipenko V. A., Lezhen L. A. Character of turbulence in the boundary regions of the Earth’s magnetosphere. Geomagnetism and Aeronomy, 52 (4), 470—481 (2012) [in Russian].
9. Kolmogorov A. N. The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds’ Numbers. Dokl. Academy of Sciences of the USSR, 30(4), 299—303 (1941) [in Russian].
10. Zelenyj L. M., Veselovskij I. S. (Eds.) Space geoheliophysics [Kosmicheskaja geogeliofizika]. Vol. 1, 624 p. (Vol. 1-2; Vol. 1) (Fizmatlit, Moscow, 2008) [in Russian].
11. Landau L. D., Lifshic E. M. Hydrodynamics [Gidrodinamika], 736 p. (Nauka, Moscow, 1988) [in Russian].
12. Monin S., Jaglom A. M. Statistical Fluid Mechanics. Part 2. Turbulence Mechanics [Statisticheskaja gidromehanika. Pt. 2. Mehanika turbulentnosti], 720 p. (Gidrometeoizdat, Leningrad, 1967) [in Russian].
13. Novikov E. A., Stjuart R. U. The intermittency of turbulence and a range of energy dissipation fluctuations [Peremezhaemost’ turbulentnosti i spektr fljuktuacij dissipacii jenergii]. Bulletin of the Academy of Sciences of the USSR. Geophysics Series, N 3, 408—413 (1964) [in Russian].
14. Savin S. P., Zelenyi L. M., Amata E., et al. Dynamic interaction of plasma flow with the hot boundary layer of a geomagnetic trap. Journal of Experimental and Theoretical Physics Letters, 79 (8), 452—456 (2004) [in Russian].
15. Frik P. G. Wavelet analysis and hierarchical model of turbulence, 40 p. (IMSS UrO RAN, Perm’, 1992) [in Russian].
16. Frik P. G. Turbulence: Models and Approaches. Lection Course. Pt. II, 136 p. (Perm State Techn. Univ., Perm’, 1999) [in Russian].
17. Frisch U. Turbulence. The Legacy of A. N. Kolmogorov, 343 p. (Fazis, Moscow, 1998) [in Russian].
18. Bacry E., Muzy J. F., Arneodo A. Singularity spectrum of fractal signals from wavelet analysis: exact results. J. Statistical Phys. 70, 635—654 (1993).
19. Benzi R., Ciliberto S., Tripiccione R., Baudet C., Massaioli F., Succi S. Extended self-similarity in turbulent flows. Phys. Rev. E. 48, R29—R32 (1993).
20. Budaev V. P., Savin S., Zelenyi L., et al. Intermittency and extended self—similarity in space and fusion plasma: boundary effects. Plasma Phys. Control Fusion, 50, 074014—074023 (2008).
21. Consolini G., Kretzschmar M., Lui A. T. Y., Zimbardo G., Macek W. M. On the magnetic field fluctuations during magnetospheric tail current disruption: A statistical approach. J. Geophys. Res., 110, A07202 (2005).
22. Consolini G., Lui A. T. Y. Symmetry breaking and nonlinear wave-wave interaction in current disruption: possible evidence for a dynamical phase transition. Magnetospheric current systems, Eds S.-I. Ohtani, R. Fuijii, M. Hesse, R. L. Lysak, 118, 395 p. (AUG, Washington, 2000).
23. Dubrulle B. Intermittency in fully developed turbulence: Log-Poisson statistics and generalized scale covariance. Phys. Rev. Lett., 73, 959—962 (1994).
24. Feder J. Fractals, 12 p. (Plenum Press, New York, 1988).
25. Grossmann A., Morlet J. Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM J. Mathematical Analysis, 15, 723—731 (1984).
26. Kozak L., Lui A., Savin S. Statistical analysis of the magnetic field measurements. Odessa Astron. Publs., 26 (2), 268—271 (2014).
27. Kraichnan R. H. The structure of isotropic turbulence at very high Reynolds numbers. J. Fluid Mech., 5, 497—543 (1959).
28. Kraichnan R. H. Convergents to turbulence functions. J. Fluid Mech., 41, 189—217 (1970).
29. Lauwerier H. A. Fractals — images of chaos, 240 p. (Princetion Univ. Press, London, 1991).
30. Lovejoy S., Schertzer D., Silas P. Diffusion in one dimensional multifractal porous media. Water Resources Research, 34, 3283—3291 (1998).
31. Mallat S., Hwang W. L. Singularity detection and processing with wavelets. IEEE Transactions on Inform. Theory, 32 (2), 617—643 (1992). 
32. Ohtani S., Higuchi T., Lui A. T. Y., Takahashi K. Magnetic uctuations associated with tail current disruption: fractal analysis. J. Geophys. Res., 100, 19135—19147 (1995). 
33. Politano H., Pouquet A., Carbone V. Determination of anomalous exponents of structure functions in two-dimensional magnetohydrodynamic turbulence. Europhys. Lett. 43, 516—521 (1998). 

34. Savin S. P., Borodkova N. L., Budnik E.Yu., et al. Interball tail probe measurements in outer cusp and boundary layers. Geospace Mass and Energy Flow: Results from the International Solar-Terrestrial Physics Program, Eds J. L. Horwitz, et al., 104, 25—44 (AGU, Washington, 1998). 
35. Schertzer D., Lovejoy S., Hubert P. An introduction to stochastic multifractal fields. Mathematical problems in environmental science and engineering, Eds A. Ern, L. Weiping. 4. Series in contemporary applied mathematics, 106—179 (Higher Education Press, Beijing, 2002).

36. Schroter E. H., Soltau D., Wiehr E. The German solar telescopes at the Observatorio del Teide. Vistas in Astron., 28, 519—525 (1985).

37. She Z., Leveque E. Universal scaling laws in fully developed turbulence. Phys. Rev. Lett., 72, 336—339 (1994).
38. Yordanova E., Grzesiak M., Wernik A. W., Popielawska B., K. Stasiewicz K. Multifractal structure of turbulence in the magnetospheric cusp. Ann. geophys., 22, 2431—2440 (2004).