Kinematics and Physics of Celestial Bodies

, Volume 34, Issue 5, pp 258–269 | Cite as

Spectra of Turbulence during the Dipolarization of the Magnetic Field

  • L. KozakEmail author
  • B. Petrenko
  • E. Kronberg
  • E. Grigorenko
  • E. Lui
  • S. Cheremnykh


The presence of heterogeneity in turbulent processes has been analyzed, and the spectra of turbulence have been obtained for the regions before and during the dipolarization of the magnetic field in the Earth’s magnetospheric tail from the measurements of four space vehicles of the Cluster-2 mission (the event of September 21, 2005). The spectral and wavelet analysis was supplemented by the investigations of the fluctuation kurtosis for the magnetic field absolute value. In the region of the magnetic field dipolarization in the magnetospheric tail, a decreasing horizontal component of the magnetic field in parts of the tail and an increasing vertical component, kurtosis variations, the presence of strong Pc5 and Pc4 pulsations as well as direct and inverse cascades, a break in the spectra at the frequencies below the proton gyrofrequency, and a change in the character of turbulent motions at different time scales (at large time scales, the turbulent flow corresponds to the homogeneous models of Kolmogorov and Iroshnikov–Kraichnan; at smaller time scales, the turbulent flow is described by the electron magnetohydrodynamic turbulence model) have been detected. Using the measurements from different space vehicles, it was possible to estimate the velocity of the plasma flow in the tail direction.


turbulent processes tail of the Earth’s magnetosphere Pc pulsations turbulence spectra 



The study was performed in accordance with the Targeted Comprehensive Program of the National Academy of Sciences of Ukraine in Plasma Physics. The study was supported by the educational program of the Ministry of Education and Science of Ukraine no. 2201250, “Education and Training of Students, Graduate Students, Academic and Teaching Personnel Abroad” (internship at the Applied Physics Laboratory of Johns Hopkins University, Maryland, United States), grant no. 90312 of the Volkswagen Foundation (VW-Stiftung), and the International Space Science Institute (ISSI-BJ), Beijing, China.


  1. 1.
    G. I. Barenblatt, “Turbulent boundary layers at very large Reynolds numbers,” Russ. Math. Surv. 59, 47–64 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    L. V. Kozak, S. P. Savin, V. P. Budaev, L. A. Lezhen, and V. A. Pilipenko, “Character of turbulence in the boundary regions of the Earth’s magnetosphere,” Geomagn. Aeron. (Engl. Transl.) 52, 445–455 (2012).Google Scholar
  3. 3.
    L. V. Kozak, “The methods and approaches to determine characteristics of turbulent environment,” Kosm. Nauka Tekhnol. 22, 60–77 (2016).CrossRefGoogle Scholar
  4. 4.
    L. V. Kozak, V. A. Pilipenko, O. M. Chugunova, and P. N. Kozak, “Statistical analysis of turbulence in the foreshock region and in the Earth’s magnetosheath,” Cosmic Res.49, 194–204 (2011).ADSCrossRefGoogle Scholar
  5. 5.
    A. N. Kolmogorov, “Local structure of turbulence in an incompressible viscous fluid at very high Reynolds numbers,” Sov. Phys. Usp. 10, 734–746 (1968).ADSCrossRefGoogle Scholar
  6. 6.
    Space Geophysics, Ed. by L. M. Zelenyi and I. S. Veselovskii (Fizmatlit, Moscow, 2008), Vol. 1 [in Russian].Google Scholar
  7. 7.
    A. Nishida, Geomagnetic Diagnosis of the Magnetosphere (Springer-Verlag, New York, 1978; Mir, Moscow, 1980).Google Scholar
  8. 8.
    P. G. Frik, Turbulence: Models and Approaches. A Course of Lectures (Permsk. Gos. Tekh. Univ., Perm’, 1999), Vol. II [in Russian].Google Scholar
  9. 9.
    U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge Univ. Press, Cambridge, 1995; Fazis, Moscow, 1998) [in Russian].Google Scholar
  10. 10.
    J. W. Bieber, E. Stone, E. W. Hones, et al., “Plasma behavior during energetic electron streaming events: Further evidence for substorm-associated magnetic reconnection,” Geophys. Res. Lett. 9, 664–667 (1982).ADSCrossRefGoogle Scholar
  11. 11.
    D. Biskamp, E. Schwarz, and J. F. Drake, “Two-dimensional electron magnetohydrodynamic turbulence,” Phys. Rev. Lett. 76, 1264–1272 (1996).ADSCrossRefGoogle Scholar
  12. 12.
    M. N. Caan and R. L. McPherron, ‘The statistical magnetic signatures of magnetospheric substorms,” Planet. Space Sci. 26, 269–279 (1978).ADSCrossRefGoogle Scholar
  13. 13.
    T. Chang, “Self-organized criticality, multi-fractal spectra, sporadic localized reconnections and intermittent turbulence in the magnetotail,” Phys. Plasmas 6, 4137–4149 (1999).ADSCrossRefGoogle Scholar
  14. 14.
    G. Consolini, M. Kretzschmar, A. T. Y. Lui, G. Zimbardo, W. M. Macek, “On the magnetic field fluctuations during magnetospheric tail current disruption: A statistical approach,” J. Geophys. Res.: Space Phys. 110, A07202 (2005).ADSCrossRefGoogle Scholar
  15. 15.
    M. Farge, “Wavelet transforms and their applications to turbulence,” Annu. Rev. Fluid Mech. 24, 395–458.Google Scholar
  16. 16.
    D. H. Fairfield, T. Mukai, M. Brittnacher, et al., “Earthward flow bursts in the inner magnetosphere and their relation to auroral brightenings, AKR intensifications, geosynchronous particle injections and magnetic activity,” J. Geophys. Res. Space Phys. 104, 355–370 (1999).ADSCrossRefGoogle Scholar
  17. 17.
    L. A. Frank, W. R. Paterson, J. Sigwarth, and S. Kokubun, “Observations of magnetic field dipolarization during auroral substorm onset,” J. Geophys. Res.: Space Phys. 105, 15897–15912 (2000).ADSCrossRefGoogle Scholar
  18. 18.
    E. E. Grigorenko, E. A. Kronberg, P. W. Daly, N. Y. Ganushkina, J.-A. Sauvaud, and L. M. Zelenyi, “Origin of low proton-to-electron temperature ratio in the Earth’s plasma sheet,” J. Geophys. Res.: Space Phys. 121 (10) (2016). doi 10.1002/2016JA022874Google Scholar
  19. 19.
    A. Grinsted, J. C. Moore, and S. Jevrejeva, “Application of the cross wavelet transform and wavelet coherence to geophysical time series,” Nonlinear Process. Geophys. 11, 561–566 (2004).ADSCrossRefGoogle Scholar
  20. 20.
    Handbook of the Solar-Terrestrial Environment, Ed. by Y. Kamide and A. Chian (Springer-Verlag, Berlin, 2007).Google Scholar
  21. 21.
    S. Jevrejeva, J. C. Moore, and A. Grinsted, “Influence of the Arctic Oscillation and El Niño-Southern Oscillation (ENSO) on ice conditions in the Baltic Sea: The wavelet approach,” J. Geophys. Res.: Atmos. 108, 4677–4708 (2003).ADSCrossRefGoogle Scholar
  22. 22.
    Rae I. Jonathan, I. R. Mann, V. Angelopoulos, K. R. Murphy, D. K. Milling, A. Kale, H. U. Frey, G. Rostoker, M. J. Engebretson, M. Moldwin, S. Mende, H. J. Singer, and E. F. Donovan, “Near-Earth initiation of a terrestrial substorm,” J. Geophys. Res.: Space Phys. 114, 2156–2202 (2009).Google Scholar
  23. 23.
    L. V. Kozak, A. S. Prokhorenkov, and S. P. Savin, “Statistical analysis of the magnetic fluctuations in boundary layers of Earth’s magnetosphere,” Adv. Space Res. 56, 2091–2096 (2015).ADSCrossRefGoogle Scholar
  24. 24.
    L. V. Kozak, A. T. Y. Lui, E. A. Kronberg, and A. S. Prokhorenkov, “Turbulent processes in Earth’s magnetosheath by Cluster mission measurements,” J. Atmos. Sol.-Terr. Phys. 154, 115–126 (2017).ADSCrossRefGoogle Scholar
  25. 25.
    R. H. Kraichnan, “Convergents to turbulence functions,” J. Fluid Mech. 41, 189–217 (1970).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    R. H. Kraichnan, “The structure of isotropic turbulence at very high Reynolds numbers,” J. Fluid Mech. 5, 497–543 (1959).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    E. A. Kronberg, E. E. Grigorenko, D. L. Turner, P. W. Daly, Y. Khotyaintsev, L. Kozak, “Comparing and contrasting dispersionless injections at geosynchronous orbit during a substorm event,” J. Geophys. Res.: Space Phys. 122 (10) (2017). doi 10.1002/2016JA023551Google Scholar
  28. 28.
    R. E. Lopez, “Magnetospheric substorms,” Johns Hopkins APL Tech. Dig. 11, 264–271 (1990).ADSGoogle Scholar
  29. 29.
    Lui, A.T.Y., “Extended consideration of a synthesis model for magnetospheric substorms,” in Magnetospheric Substorms, Ed. by J. R. Kan, T. A. Potemra, S. Kokobun, and T. Iijima (AGU, Washington, DC, 1991), in Ser: Geophysical Monograph, Vol. 64.Google Scholar
  30. 30.
    A. T. Y. Lui, “Inferring global characteristics of current sheet from local measurements,” J. Geophys. Res.: Space Phys. 98, 13423–13427 (1993).ADSCrossRefGoogle Scholar
  31. 31.
    A. T. Y. Lui, “Current disruption in the Earth’s magnetosphere: Observations and models,” J. Geophys. Res. : Space Phys. 101, 13067–13088 (1996).ADSCrossRefGoogle Scholar
  32. 32.
    A. T. Y. Lui, R. E. Lopez, B. J. Anderson, et al., “Current disruptions in the near-Earth neutral sheet region,” J. Geophys. Res.: Space Phys. 97, 1461–1480 (1992).ADSCrossRefGoogle Scholar
  33. 33.
    A. T. Y. Lui, Multiscale phenomena in the near-Earth magnetosphere,” J. Atmos. Sol.-Terr. Phys. 64, 125–143 (2002).ADSCrossRefGoogle Scholar
  34. 34.
    A. T. Y. Lui, “Potential plasma instabilities for substorm expansion onsets,” Space Sci. Rev. 113, 127–206 (2004).ADSCrossRefGoogle Scholar
  35. 35.
    A. T. Y. Lui, Y. Zheng, Y. Zhang, S. Livi, H. Reme, M. W. Dunlop, G. Gustafsson, S. Mende, C. Mouikis, and L. M. Kistler, “Cluster observation of plasma flow reversal in the magnetotail during a substorm.” Ann. Geophys. 24, 2005–2013 (2006).ADSCrossRefGoogle Scholar
  36. 36.
    R. L. McPherron, “Substorm related changes in the geomagnetic tail: The growth phase,” Planet. Space Sci. 20, 1521–1539 (1972).ADSCrossRefGoogle Scholar
  37. 37.
    S. Ohtani, M. A. Shay, and T. Mukai, “Temporal structure of the fast convective flow in the plasma sheet: Comparison between observations and two-fluid simulations,” J. Geophys. Res.: Space Phys. 109, A03210 (2004). doi 10. 1029/2003JA010002Google Scholar
  38. 38.
    G. Paschmann and P. W. Daly, Spectral Analysis, Reprinted from Analysis Methods for Multi-Spacecraft Data, ISSI Scientific Report No. SR-001, ed. 1.1 (2000).Google Scholar
  39. 39.
    V. A. Sergeev, D. G. Mitchell, and D. J. Williams, “Structure of the tail plasma/current sheet at ~11 Re and its changes in the course of a substorm,” J. Geophys. Res.: Space Phys. 98, 17345–17365 (1993).ADSCrossRefGoogle Scholar
  40. 40.
    THOR Exploring Plasma Energization in Space Turbulence, ESA/SRE Assessment Study Report (2017).Google Scholar
  41. 41.
    C. Torrence and G. P. Compo, “A practical guide to wavelet analysis,” Bull. Am. Meteorol. Soc. 79, 61–78 (1998).ADSCrossRefGoogle Scholar
  42. 42.
    A. G. Yahnin, I. V. Despirak, A. A. Lubchich, et al., “Indirect mapping of the source of the oppositely directed fast plasma flows in the plasma sheet onto the auroral display,” Ann. Geophys. 24, 679–687 (2006).ADSCrossRefGoogle Scholar
  43. 43.
    S. Zacks, Theory of Statistical Inference (John Wiley & Sons, New York, 1971), in Ser.: Probability & Mathematical Statistics, Vol. 582.Google Scholar

Copyright information

© Allerton Press, Inc. 2018

Authors and Affiliations

  • L. Kozak
    • 1
    • 2
    Email author
  • B. Petrenko
    • 1
  • E. Kronberg
    • 3
  • E. Grigorenko
    • 4
  • E. Lui
    • 5
  • S. Cheremnykh
    • 2
  1. 1.Taras Shevchenko National University of KyivKyivUkraine
  2. 2.Space Research Institute, National Academy of Sciences of Ukraine and State Space Agency of UkraineKyivUkraine
  3. 3.Max Planck InstituteGöttingenGermany
  4. 4.Space Research Institute, Russian Academy of SciencesMoscowRussia
  5. 5.Johns Hopkins University, LaurelMarylandUnited States

Personalised recommendations