Solnechno-Zemnaya Fizika Solnechno-Zemnaya Fizika Солнечно-земная физика 2712-9640 85194 10.12737/szf-111202503 Результаты исследований Results of current research Результаты исследований Dependence of eddy diffusion coefficient on plasma parameter β in Earth’s magnetotail Зависимость коэффициента вихревой диффузии от плазменного параметра β в хвосте магнитосферы Земли https://orcid.org/0009-0000-1955-8176 Найко Даниил Юрьевич Naiko Daniil Yurievich https://orcid.org/0000-0003-2509-8166 Овчинников Илья Львович Ovchinnikov Ilya L'vovich ilya@psn.ru

кандидат физико-математических наук;

candidate of physical and mathematical sciences;

 https://orcid.org/0000-0001-6980-3993 Антонова Елизавета Евгеньевна Antonova Elizaveta Evgenievna

доктор физико-математических наук;

doctor of physical and mathematical sciences;

 Московский государственный университет имени М.В.Ломоносова Lomonosov Moscow State University Институт космических исследований Российской академии наук Institute of Space Researches of the Russian Academy of Sciences 26 03 2025 26 03 2025 11 1 31 40 09 07 2024 23 10 2024 https://zh-szf.ru/en/nauka/article/85194/view

В работе проведен анализ зависимостей коэффициентов вихревой диффузии в направлениях X, Y и Z системы координат GSM от плазменного параметра β с учетом расстояния от Земли, направления межпланетного магнитного поля и условий геомагнитной активности в хвосте магнитосферы по данным Magnetospheric Multiscale Mission (MMS). Данные параметры определяются величинами среднеквадратичных скоростей ионов и их автокорреляционными временами. Коэффициенты вихревой диффузии характеризуют величину турбулентного транспорта в хвосте магнитосферы и являются параметрами модели турбулентного плазменного слоя. Анализировалось более 20000 12-минутных интервалов, во время которых спутники MMS находились внутри области с плотностью плазмы более 0.1 см–3 и средней энергией ионов более 0.5 кэВ. Показано, что с возрастанием плазменного параметра растут и коэффициенты вихревой диффузии. Данный рост прекращается при β~1. Анализ относительного вклада изменений среднеквадратичной скорости и автокорреляционного времени в коэффициент вихревой диффузии показал отсутствие существенной зависимости от автокорреляционного времени.

The work analyzes dependences of eddy diffusion coefficients in the X, Y, and Z directions of the GSM coordinate system on the plasma parameter β, taking into account the distance from Earth, the direction of the interplanetary magnetic field, and conditions of geomagnetic activity in the magnetotail according to MMS mission data. These parameters are determined by root-mean-square velocities of ions and their autocorrelation time. Eddy diffusion coefficients characterize the magnitude of turbulent transport in the magnetotail and are the parameters of the model of turbulent plasma sheet. We have analyzed more than 20000 12-min intervals during which the MMS satellites were located within a region with plasma density more than 0.1 cm–3 and average ion energy more than 0.5 keV. It is shown that as the plasma parameter increases, the eddy diffusion coefficients increase as well. This increase stops at β~1. Analysis of the relative contribution of changes in root-mean-square velocity and autocorrelation time to the eddy diffusion coefficient has revealed that there is no significant dependence on autocorrelation time.

 магнитосферная турбулентность диффузионный транспорт коэффициенты вихревой диффузии magnetospheric turbulence turbulent transport coefficients of eddy diffusion Исследование выполнено за счет гранта Российского научного фонда № 23-22-00076 [https://rscf.ru/project/23-22-00076/]. Найко Д.Ю. является стипендиатом Фонда развития теоретической физики и математики «БАЗИС» The work was financially supported by the Russian Science Foundation (Grant No. 23-22-00076) [https://rscf.ru/project/23-22-00076/]. Naiko D.Y. is a scholarship holder of the Theoretical Physics and Mathematics Advancement Foundation «BASIS»

Антонова Е.Е. О неадиабатической диффузии, выравнивании концентрации и температуры в плазменном слое магнитосферы Земли. Геомагнетизм и аэрономия. 1985. Т. 25, № 4. С. 623–627. Alexandrova O., Chen C.H.K., Sorriso-Valvo L., Horbury T.S., Bale S.D. Solar wind turbulence and the role of ion instabilities. Space Sci. Rev. 2013, vol. 178, pp. 101–139. DOI: 10.1007/s11214-013-0004-8. Антонова Е.Е., Овчинников И.Л. Равновесие турбулентного токового слоя и токовый слой хвоста магнитосферы Земли. Геомагнетизм и аэрономия. 1996. Т. 36, № 5. С. 7–14. Angelopoulos V., Kennel C.F., Coroniti F.V., Pellat R., Spence H.E., Kivelson M.G., Walker R.J., Baumjohann W., Feldman W.C., Gosling J.T. Characteristics of ion flow in the quiet state of the inner plasma sheet. Geophys. Res. Lett. 1993, vol. 20, no 16, pp. 1711–1714. DOI: 10.1029/93GL00847. Ермолаев Ю.И., Петрукович А.А., Зеленый Л.М. и др. Исследования структуры и динамики плазменного слоя в эксперименте Коралл проекта Интербол. Космические исследования. 2000. Т. 38, № 1. С. 16–22. Angelopoulos V., Mukai T., Kokubun S. Evidence for intermittency in Earth's plasma sheet and implications for selforganized criticality. Phys. Plasmas. 1999, vol. 6, no 11, pp. 4161–4168. DOI: 10.1063/1.873681. Найко Д.Ю., Овчинников И.Л., Антонова Е.Е. Пространственное распределение коэффициента вихревой диффузии в плазменном слое хвоста магнитосферы Земли и его зависимость от межпланетного магнитного поля и геомагнитной активности по данным MMS. Геомагнетизм и аэрономия. 2024. Т. 64, № 2. С. 172–179. DOI: 10.1134/S0016793223600996. Antonova E.E. On nonadiabatic diffusion and adjustment of concentration and temperature in the plasma sheet of the Earth’s magnetosphere. Geomagnetism and Aeronomy. 1985, vol. 2, no. 4, pp. 623–627, (In Russian). Овчинников И.Л., Антонова Е.Е. Турбулентный транспорт магнитосферы Земли: обзор результатов наблюдений и моделирования. Геомагнетизм и аэрономия. 2017. Т. 57, № 6. С. 706–714. DOI: 10.7868/S0016794017060086. Antonova E.E. Quasiturbulent transport and LLBL properties. Adv. Space Res. 2006, vol. 37, pp. 532–536. DOI: 10.1016/j.asr.2006.01.019. Овчинников И.Л., Антонова Е.Е., Ермолаев Ю.И. Определение коэффициента турбулентной диффузии в плазменном слое по данным проекта ИНТЕРБОЛ. Космические исследования. 2000. Т. 38, № 6. С. 596–601. Antonova E.E., Ovchinnikov I.L. T Turbulent current sheets and magnetospheric substorms. Proc. International Conference on Substorms. Versailles, 12–17 May 1996a. ESA SP-389. Paris: European Space Agency, 1996a. P. 255. Овчинников И.Л., Антонова Е.Е., Ермолаев Ю.И. Турбулентность в плазменном слое во время суббурь (Исследование ряда случаев на базе наблюдений хвостового зонда проекта ИНТЕРБОЛ). Космические исследования. 2002. Т. 40, № 6. С. 563–570. Antonova E.E., Ovchinnikov I.L. Equilibrium of a turbulent current sheet and the current sheet of geomagnetic tail. Geomagnetism and Aeronomy. 1996b, vol. 36, no. 5, pp. 597–601. Овчинников И.Л., Антонова Е.Е., Найко Д.Ю. Флуктуации электрического и магнитного полей в плазменном слое хвоста магнитосферы Земли по данным MMS. Космические исследования. 2024. Т. 62, № 1. С. 13–35. DOI: 10.31857/S0023420624010021. Antonova E.E., Ovchinnikov I.L. Magnetostatically equilibrated plasma sheet with developed medium-scale turbulence: structure and implications for substorm dynamics. J. Geophys. Res. 1999, vol. 104, no. A8, pp. 17289–17297. DOI: 10.1029/1999JA900141. Рахманова Л.С., Хохлачев А.А., Рязанцева М.О. и др. Развитие турбулентности за околоземной ударной волной в периоды спокойного и возмущенного солнечного ветра. Солнечно-земная физика. 2024. Т. 10, no 2. С. 15–28. DOI: 10.12737/szf-102202402 // Rakhmanova L.S., Khokhlachev A.A., Riazantseva M.O., et al. Development of turbulence behind a near-Earth shock wave during periods of calm and disturbed solar wind. Solar-Terrestrial Physics. 2024. Vol. 10, no. 2. P. 15–28. DOI: 10.12737/stp-102202402. Antonova E.E., Stepanova M.V. The impact of turbulence on physics of the geomagnetic tail. Front. Astron. Space Sci. 2021, vol. 8, 622570. DOI: 10.3389/fspas.2021.622570. Alexandrova O., Chen C.H.K., Sorriso-Valvo L., et al. Solar wind turbulence and the role of ion instabilities. Space Sci. Rev. 2013. Vol. 178. P. 101–139. DOI: 10.1007/s11214-013-0004-8. Borovsky J.E. Plasma and magnetic-field structure of the solar wind at inertial-range scale. Sizes discerned from statistical examinations of the time-series measurements. Front. Astron. Space Sci. 2020, vol. 7, 20. DOI: 10.3389/fspas.2020.00020. Angelopoulos V., Kennel C.F., Coroniti F.V., et al. Characteristics of ion flow in the quiet state of the inner plasma sheet. Geophys. Res. Lett. 1993. Vol. 20, no 16. P. 1711–1714. DOI: 10.1029/93GL00847. Borovsky J.E., Funsten H.E. MHD turbulence in the Earth’s plasma sheet: Dynamics, dissipation and driving. J. Geophys. Res. 2003, vol. 107, no. A7. DOI: 10.1029/2002JA009625. Angelopoulos V., Mukai T., Kokubun S. Evidence for intermittency in Earth’s plasma sheet and implications for selforganized criticality. Phys. Plasmas. 1999. Vol. 6, no 11. P. 4161–4168. DOI: 10.1063/1.873681. Borovsky J.E., Elphic R.C., Funsten H.O., Thomsen M.F. The Earth’s plasma sheet as a laboratory for turbulence in high-β MHD. J. Plasma Phys. 1997, vol. 57, no. 1, pp. 1–34. DOI: 10.1017/S0022377896005259. Antonova E.E. Quasiturbulent transport and LLBL properties. Adv. Space Res. 2006. Vol. 37. P. 532–536. DOI: 10.1016/j.asr.2006.01.019. Borovsky J.E., Thomsen M.F., Elphic R.C. The driving of the plasma sheet by the solar wind. J. Geophys. Res. 1998, vol. 103, no. A8, pp. 17617–17639. DOI: 10.1029/97JA02986. Antonova E.E., Ovchinnikov I.L. Turbulent current sheets and magnetospheric substorms. Proc. International Conference on Substorms. Versailles, 12–17 May 1996. ESA SP-389. Paris: European Space Agency, 1996. P. 255. Bruno R., Carbone, V. The Solar Wind as a Turbulence Laboratory. Living Rev. Solar Phys. 2013, vol. 10, pp. 1–208. DOI: 10.12942/lrsp-2013-2. Antonova E.E., Ovchinnikov I.L. Magnetostatically equilibrated plasma sheet with developed medium-scale turbulence: structure and implications for substorm dynamics. J. Geophys. Res. 1999. Vol. 104, no. A8. P. 17289–17297. DOI: 10.1029/1999JA900141. Budaev V.P., Zeleny L.M., Savin S.P. Generalized self-similarity of intermittent plasma turbulence in space and laboratory plasmas. J. Plasma Phys. 2015, vol. 81, 395810602. DOI: 10.1017/S0022377815001099. Antonova E.E., Stepanova M.V. The impact of turbulence on physics of the geomagnetic tail. Front. Astron. Space Sci. 2021. Vol. 8. 622570. DOI: 10.3389/fspas.2021.622570. Burch J.L., Moore T.E., Torbert R.B., Giles B.L. Magnetospheric Multiscale overview and science objectives. Space Sci. Rev. 2016, vol. 199, no. 1-4, pp. 5–21. DOI: 10.1007/s11214-015-0164-9. Borovsky J.E. Plasma and magnetic-field structure of the solar wind at inertial-range scale. Sizes discerned from statistical examinations of the time-series measurements. Front. Astron. Space Sci. 2020. Vol. 7. 20. DOI: 10.3389/fspas.2020.00020. Eyelade A.V., Espinoza C.M., Stepanova M., Antonova E.E., Ovchinnikov I.L., Kirpichev I.P. Influence of MHD turbulence on ion kappa distributions in the Earth’s plasma sheet as a function of plasma β parameter. Front. Astron. Space Sci. 2021, vol. 8, 647121. DOI: 10.3389/fspas. 2021.647121. Borovsky J.E., Funsten H.E. MHD turbulence in the Earth’s plasma sheet: Dynamics, dissipation and driving. J. Geophys. Res. 2003. Vol. 107, no. A7. DOI: 10.1029/2002JA009625. Eyelade A.V., Stepanova M., Espinoza C.M., Antonova E.E., Kirpichev I.P. The response of the Earth magnetosphere to changes in the solar wind dynamic pressure: 1. Plasma and magnetic pressures. J. Geophys. Res.: Space Phys. 2024a, vol. 129, e2023JA031948. DOI: 10.1029/2023JA031948. Borovsky J.E., Elphic R.C., Funsten H.O., Thomsen M.F. The Earth’s plasma sheet as a laboratory for turbulence in high-β MHD. J. Plasma Phys. 1997. Vol. 57, no. 1. P. 1–34. DOI: 10.1017/S0022377896005259. Eyelade A.V., Stepanova M., Espinoza C.M., Antonova E.E., Kirpichev I.P. The response of the magnetosphere to changes in the solar wind dynamic pressure: 2. Ion and electron kappa distribution functions. J. Geophys. Res.: Space Phys. 2024b, vol. 129, e2023JA031949. DOI: 10.1029/2023JA031949. Borovsky J.E., Thomsen M.F., Elphic R.C. The driving of the plasma sheet by the solar wind. J. Geophys. Res. 1998. Vol. 103, no. A8. P. 17617–17639. DOI: 10.1029/97JA02986. Iijima T., Potemra T.A. The amplitude distribution of field-aligned currents at northern high latitudes observed by Triad. J. Geophys. Res. 1976, vol. 81, no.13, pp. 2165–2174. DOI: 10.1029/JA081i013p02165. Bruno R., Carbone, V. The Solar Wind as a Turbulence Laboratory. Living Rev. Solar Phys. 2013. Vol. 10. P. 1–208. DOI: 10.12942/lrsp-2013-2. Montgomery D. Remarks on the MHD problem of generic magnetospheres and magnetotails. Magnetotail Phys. Johns Hopkins University Press, Baltimore. Md. 1987, pp. 203–204. Budaev V.P., Zeleny L.M., Savin S.P. Generalized self-similarity of intermittent plasma turbulence in space and laboratory plasmas. J. Plasma Phys. 2015. Vol. 81. 395810602. DOI: 10.1017/S0022377815001099. Nagata D., Machida S., Ohtani S., Saito Y., Mukai T. Solar wind control of plasma number density in the near Earth plasma sheet: three-dimensional structure. Ann. Geophys. 2008, vol. 26, no. 12, pp. 4031–4049. DOI: 10.5194/angeo-26-4031-2008. Burch J.L., Moore T.E., Torbert R.B., Giles B.L. Magnetospheric Multiscale overview and science objectives. Space Sci. Rev. 2016. Vol. 199, no. 1-4. P. 5–21. DOI: 10.1007/s11214-015-0164-9. Naiko D.Yu., Ovchinnikov I.L, Antonova E.E. Spatial distribution of the eddy diffusion coefficient in the plasma sheet of Earth’s magnetotail and its dependence on the interplanetary magnetic field and geomagnetic activity based on MMS satellite data. Geomagnetism and Aeronomy. 2024, vol. 64, no. 2, pp. 172–179. DOI: 10.1134/S0016793223600996. Eyelade A.V., Espinoza C.M., Stepanova M., et al. Influence of MHD turbulence on ion kappa distributions in the Earth’s plasma sheet as a function of plasma β parameter. Front. Astron. Space Sci. 2021. Vol. 8. 647121. DOI: 10.3389/fspas.2021.647121. Newell P.T., Gjerloev J.W. Evaluation of SuperMAG auroral electrojet indices as indicators of substorms and auroral power. J. Geophys. Res. 2011, vol. 116, A12211. DOI: 10.1029/2011JA016779. Eyelade A.V., Stepanova M., Espinoza C.M., et al. The response of the Earth magnetosphere to changes in the solar wind dynamic pressure: 1. Plasma and magnetic pressures. J. Geophys. Res.: Space Phys. 2024a. Vol. 129. e2023JA031948. DOI: 10.1029/2023JA031948. Ovchinnikov I.L., Antonova E.E. Turbulent transport of the Earth magnetosphere: Review of the results of observations and modeling. Geomagnetism and Aeronomy. 2017, vol. 57, no. 6, pp. 655–663. DOI: 10.1134/S0016793217060081. Eyelade A.V., Stepanova M., Espinoza C.M., et al. The response of the magnetosphere to changes in the solar wind dynamic pressure: 2. Ion and electron kappa distribution functions. J. Geophys. Res.: Space Phys. 2024b. Vol. 129. e2023JA031949. DOI: 10.1029/2023JA031949. Ovchinnikov I.L., Antonova E.E., Yermolaev Yu.I. Determination of the turbulent diffusion coefficient in the plasma sheet using the project INTERBALL data. Cosmic Res. 2000, vol. 38, no. 6, pp. 557–561. (In Russian). Iijima T., Potemra T.A. The amplitude distribution of field-aligned currents at northern high latitudes observed by Triad. J. Geophys. Res. 1976. Vol. 81, no.13. P. 2165–2174. DOI: 10.1029/JA081i013p02165. Ovchinnikov I.L., Antonova E.E., Yermolaev Yu.I. Turbulence in the plasma sheet during substorms: A case study for three events observed by the INTERBALL Tail probe. Cosmic Res. 2002a, vol. 40, no. 6, pp. 521–528. (In Russian). Montgomery D. Remarks on the MHD problem of generic magnetospheres and magnetotails. Magnetotail Phys. Johns Hopkins University Press, Baltimore. Md. 1987. P. 203–204. Ovchinnikov I.L., Antonova E.E., Yermolaev Yu.I. Plasma sheet heating during substorm and the values of the plasma sheet diffusion coefficient obtained on the base of Interball/Tail probe observations. Adv. Space Res. 2002b, vol. 30, no. 7, pp. 1821–1824. DOI: 10.1016/S0273-1177(02)00456-8. Nagata D., Machida S., Ohtani S., et al. Solar wind control of plasma number density in the near Earth plasma sheet: three-dimensional structure. Ann. Geophys. 2008. Vol. 26, no. 12. P. 4031–4049. DOI: 10.5194/angeo-26-4031-2008. Ovchinnikov I.L, Naiko D.Yu., Antonova E.E. Fluctuations of the electric and magnetic fields in the plasma sheet of the Earth’s magnetotail according to MMS data. Cosmic Res. 2024, vol. 62, no. 1, pp. 10–33. DOI: 10.1134/S0010952523700788. Newell P.T., Gjerloev J.W. Evaluation of SuperMAG auroral electrojet indices as indicators of substorms and auroral power. J. Geophys. Res. 2011. Vol. 116. A12211. DOI: 10.1029/2011JA016779. Pinto V., Stepanova M., Antonova E.E., Valdivia J.A. Estimation of the eddy-diffusion coefficients in the plasma sheet using THEMIS satellite data. J. Atmos. Solar-Terr. Phys. 2011, vol. 73, no. 7, pp. 1472–1477. DOI: 10.1016/j.jastp.2011.05.007. Ovchinnikov I.L., Antonova E.E., Yermolaev Yu.I. Plasma sheet heating during substorm and the values of the plasma sheet diffusion coefficient obtained on the base of Interball/Tail probe observations. Adv. Space Res. 2002. Vol. 30, no. 7. P. 1821–1824. DOI: 10.1016/S0273-1177(02)00456-8. Podesta J.J., Borovsky J.E. Scale invariance of normalized cross-helicity throughout the inertial range of solar wind turbulence. Phys. Plasmas. 2010, vol. 17, no 11, 112905. DOI: 10.1063/1.3505092. Pinto V., Stepanova M., Antonova E.E., Valdivia J.A. Estimation of the eddy-diffusion coefficients in the plasma sheet using THEMIS satellite data. J. Atmos. Solar-Terr. Phys. 2011. Vol. 73, no. 7. P. 1472–1477. DOI: 10.1016/j.jastp.2011.05.007. Pollock C., Moore T., Jacques A., Burch J., Gliese U., Saito Y., et al. Fast plasma investigation for magnetospheric multiscale. Space Sci. Rev. 2016, vol. 199, no. 1-4, pp. 331–406. DOI: 10.1007/s11214-016-0245-4. Podesta J.J., Borovsky J.E. Scale invariance of normalized cross-helicity throughout the inertial range of solar wind turbulence. Phys. Plasmas. 2010. Vol. 17, no 11. 112905. DOI: 10.1063/1.3505092. Rakhmanova L., Riazantseva M., Zastenker G., Verigin M. Kinetic-scale ion flux fluctuations behind the quasi-parallel and quasi-perpendicular bow shock. J. Geophys. Res.: Space Phys. 2018, vol. 123, no. 7, pp. 5300–5314. DOI: 10.1029/2018JA025179. Pollock C., Moore T., Jacques A., et al. Fast plasma investigation for magnetospheric multiscale. Space Sci. Rev. 2016. Vol. 199, no. 1-4. P. 331–406. DOI: 10.1007/s11214-016-0245-4. Rakhmanova L., Riazantseva M., Zastenker G., Yermolaev Y., Lodkina I. Dynamics of plasma turbulence at Earth’s bow shock and through the magnetosheath. Astrophys. J. 2020, vol. 901, no. 30. DOI: 10.3847/1538-4357/abae00. Rakhmanova L., Riazantseva M., Zastenker G., Verigin M. Kinetic-scale ion flux fluctuations behind the quasi-parallel and quasi-perpendicular bow shock. J. Geophys. Res.: Space Phys. 2018. Vol. 123, no. 7. P. 5300–5314. DOI: 10.1029/2018JA025179. Rakhmanova L.S., Khokhlachev A.A., Riazantseva M.O., Yermolaev Yu.I., Zastenker G.N. Development of turbulence behind a near-Earth shock wave during periods of calm and disturbed solar wind. Solar-Terrestrial Physics. 2024, vol. 10, no. 2, pp. 15–28. DOI: 10.12737/stp-102202402. Rakhmanova L., Riazantseva M., Zastenker G., et al. Dynamics of plasma turbulence at Earth’s bow shock and through the magnetosheath. Astrophys. J. 2020. Vol. 901, no. 30. DOI: 10.3847/1538-4357/abae00. Riazantseva M., Budaev V., Rakhmanova L., Zastenker G., Yermolaev Y., Lodkina I., et al. Variety of shapes of solar wind ion flux spectra: Spektr-R measurements. J. Plasma Phys. 2017, vol. 83, no. 04, 705830401. DOI: 10.1017/S0022377817000502. Riazantseva M., Budaev V., Rakhmanova L., et al. Variety of shapes of solar wind ion flux spectra: Spektr-R measurements. J. Plasma Phys. 2017. Vol. 83, no. 04. 705830401. DOI: 10.1017/S0022377817000502. Sahraoui F., Hadid L., Huang S. Magnetohydrodynamic and kinetic scale turbulence in the near Earth space plasmas: a (short) biased review. Rev. Modern Plasma Phys. 2020, vol. 4, no. 4. DOI: 10.1007/s41614-020-0040-2. Sahraoui F., Hadid L., Huang S. Magnetohydrodynamic and kinetic scale turbulence in the near Earth space plasmas: a (short) biased review. Rev. Modern Plasma Phys. 2020. Vol. 4, no. 4. DOI: 10.1007/s41614-020-0040-2. Stepanova M., Antonova E.E. Modeling of the turbulent plasma sheet during quiet geomagnetic conditions. J. Atmos. Solar-Terr. Phys. 2011. Vol. 73, no. 8. P. 1636–1642. DOI: 10.1016/j.jastp.2011.02.009. Stepanova M., Antonova E.E. Modeling of the turbulent plasma sheet during quiet geomagnetic conditions. J. Atmos. Solar-Terr. Phys. 2011. Vol. 73, no. 8. P. 1636–1642. DOI: 10.1016/j.jastp.2011.02.009. Stepanova M.V., Vucina-Parga T., Antonova E.E., Ovchinnikov I.L., Yermolaev Yu.I. Variation of the plasma turbulence in the central plasma sheet during substorm phases observed by the Interball/tail satellite. J. Atmos. Solar-Terr. Phys. 2005, vol. 67, no. 11, pp. 1815–1820. DOI: 10.1016/j.jastp.2005.01.013. Stepanova M.V., Vucina-Parga T., Antonova E.E., et al. Variation of the plasma turbulence in the central plasma sheet during substorm phases observed by the Interball/tail satellite. J. Atmos. Solar-Terr. Phys. 2005. Vol. 67, no. 11. P. 1815–1820. DOI: 10.1016/j.jastp.2005.01.013. Stepanova M., Antonova E.E., Paredes-Davis D., Ovchinnikov I.L., Yermolaev Y. Spatial variation of eddy-diffusion coefficients in the turbulent plasma sheet during substorms. Ann. Geophys. 2009, vol. 27, no. 4, pp. 1407–1411. DOI: 10.5194/angeo-27-1407-2009. Stepanova M., Antonova E.E., Paredes-Davis D., et al. Spatial variation of eddy-diffusion coefficients in the turbulent plasma sheet during substorms. Ann. Geophys. 2009. Vol. 27, no. 4. P. 1407–1411. DOI: 10.5194/angeo-27-1407-2009. Stepanova M., Pinto V., Valdivia J.A., Antonova E.E. Spatial distribution of the eddy diffusion coefficients in the plasma sheet during quiet time and substorms from THEMIS satellite data. J. Geophys. Res. 2011, vol. 116, no. 1, DOI: 10.1029/2010JA015887. Stepanova M., Pinto V., Valdivia J.A., Antonova E.E. Spatial distribution of the eddy diffusion coefficients in the plasma sheet during quiet time and substorms from THEMIS satellite data. J. Geophys. Res. 2011. Vol. 116, no. 1. DOI: 10.1029/2010JA015887. Torbert R.B., Russell C.T., Magnes W., Ergun R.E., Lindqvist P.-A., LeContel O., et al. The FIELDS instrument suite on MMS: Scientific objectives, measurements, and data products. Space Sci. Rev. 2016, vol. 199, no. 1-4, pp. 105–135. DOI: 10.1007/s11214-014-0109-8. Torbert R.B., Russell C.T., Magnes W., et al. The FIELDS instrument suite on MMS: Scientific objectives, measurements, and data products. Space Sci. Rev. 2016. Vol. 199, no. 1-4. P. 105–135. DOI: 10.1007/s11214-014-0109-8. Troshichev O.A., Antonova E.E., Kamide Y. Inconsistence of magnetic field and plasma velocity variations in the distant plasma sheet: violation of the “frozen-in” criterion? Adv. Space Res. 2002, vol. 30, no 12, pp. 2683–2687. DOI: 10.1016/S0273-1177(02)80382-9. Troshichev O.A., Antonova E.E., Kamide Y. Inconsistence of magnetic field and plasma velocity variations in the distant plasma sheet: violation of the “frozen-in” criterion? Adv. Space Res. 2002. Vol. 30, no 12. P. 2683–2687. DOI: 10.1016/S0273-1177(02)80382-9. Tu C.Y., Marsch E. Magnetohydrodynamic structures waves and turbulence in the solar wind: Observations and theories. Space Sci. Rev. 1995, vol. 73, no. 1-2, pp. 1–210. DOI: 10.1007/BF00748891. Tu C.Y., Marsch E. Magnetohydrodynamic structures waves and turbulence in the solar wind: Observations and theories. Space Sci. Rev. 1995. Vol. 73, no. 1-2. P. 1–210. DOI: 10.1007/BF00748891. Volwerk M., Vörös Z., Baumjohann W., Nakamura R., Runov A., T.L. Zhang, et al. Multi-scale analysis of turbulence in the Earth’s current sheet. Ann. Geophys. 2004, vol. 22, no. 7, pp. 2525–2533. DOI: 10.5194/angeo-22-2525-2004. Volwerk M., Vörös Z., Baumjohann W., et al. Multi-scale analysis of turbulence in the Earth’s current sheet. Ann. Geophys. 2004. Vol. 22, no. 7. P. 2525–2533. DOI: 10.5194/angeo-22-2525-2004. Vörös W., Baumjohann W., Nakamura R., Runov A., Zhang T.L., Volwerk M., et al. Multi-scale magnetic field intermittence in the plasma sheet. Ann. Geophys. 2003, vol. 21, no. 9, pp. 1955–1964. DOI: 10.5194/angeo-21-1955-2003. Vörös W., Baumjohann W., Nakamura R., et al. Multi-scale magnetic field intermittence in the plasma sheet. Ann. Geophys. 2003. Vol. 21, no. 9. P. 1955–1964. DOI: 10.5194/angeo-21-1955-2003. Vörös Z., Baumjohann W., Nakamura R., Volwerk M., et al. Magnetic turbulence in the plasma sheet. J. Geophys. Res. 2004, vol. 109, no. 11. DOI: 10.1029/2004JA010404. Vörös Z., Baumjohann W., Nakamura R., et al. Magnetic turbulence in the plasma sheet. J. Geophys. Res. 2004. Vol. 109, no. 11. DOI: 10.1029/2004JA010404. Vörös Z., Baumjohann W., Nakamura R., Volwerk M., Runov A. Bursty bulk flow driven turbulence in the Earth’s plasma sheet. Space Sci. Rev. 2006, vol. 122, no. 1-4, pp. 301–311. DOI: 10.1007/s11214-006-6987-7. Vörös Z., Baumjohann W., Nakamura R., et al. Bursty bulk flow driven turbulence in the Earth’s plasma sheet. Space Sci. Rev. 2006. Vol. 122, no. 1-4. P. 301–311. DOI: 10.1007/s11214-006-6987-7. Vörös Z., Baumjohann W., Nakamura R., Volwerk M., Volwerk M., Asano Y., Jankovičová D., Lucek E.A., Rème H. Spatial structure of plasma flow associated turbulence in the Earth’s plasma sheet. Ann. Geophys. 2007, vol. 25, no. 2, pp. 13–17. DOI: 10.5194/angeo-25-13-2007. Vörös Z., Baumjohann W., Nakamura R., et al. Spatial structure of plasma flow associated turbulence in the Earth’s plasma sheet. Ann. Geophys. 2007. Vol. 25, no. 2. P. 13–17. DOI: 10.5194/angeo-25-13-2007. Wang C.-P., Lyons L.R., Nagai T., Weygand J.M., Lui A.T.Y. Evolution of plasma sheet particle content under different interplanetary magnetic field conditions. J. Geophys. Res. 2010, vol. 115, no. 6. DOI: 10.1029/2009JA015028. Wang C.-P., Lyons L.R., Nagai T., et al. Evolution of plasma sheet particle content under different interplanetary magnetic field conditions. J. Geophys. Res. 2010. Vol. 115, no. 6. DOI: 10.1029/2009JA015028. Weygand J.M., Kivelson M.G., Khurana K.K., Schwarzl H.K., Thompson S.M., McPherron R.L., et al. Plasma sheet turbulence observed by Cluster II. J. Geophys. Res. 2005, vol. 110, no. 2. DOI: 10.1029/2004JA010581. Weygand J.M., Kivelson M.G., Khurana K.K., Schwarzl H.K., et al. Plasma sheet turbulence observed by Cluster II. J. Geophys. Res. 2005. Vol. 110, no. 2. DOI: 10.1029/2004JA010581. Yermolaev Y.I., Petrukovich A.A., Zelenyi L.M., Antonova E.E., Ovchinnikov I.L., Sergeev V.A. Investigation of the structure and dynamics of the plasma sheet: the CORALL experiment of the INTERBALL project. Cosmic Res. 1995, vol. 38, no. 1, pp. 16–22. (In Russian). Yordanova E., Vaivads A., André M., Buchert S.C., Vörös Z. Magnetosheath plasma turbulence and its spatiotemporal evolution as observed by the Cluster spacecraft. Phys. Rev. Lett. 2008. Vol. 100. 205003. DOI: 10.1103/PHYSREVLETT.100.205003. Yordanova E., Vaivads A., André M., Buchert S.C., Vörös Z. Magnetosheath plasma turbulence and its spatiotemporal evolution as observed by the Cluster spacecraft. Phys. Rev. Lett. 2008, vol. 100, 205003. DOI: 10.1103/PHYSREVLETT.100. 205003. URL: https://lasp.colorado.edu/mms/sdc/public/data/ (дата обращения 26 ноября 2024 г.). URL: https://lasp.colorado.edu/mms/sdc/public/data/ (accessed November 26, 2024). URL: https://omniweb.gsfc.nasa.gov/ (дата обращения 26 ноября 2024 г.). URL: https://omniweb.gsfc.nasa.gov/ (accessed November 26, 2024). URL: https://supermag.jhuapl.edu/info/ (дата обращения 26 ноября 2024 г.). URL: https://supermag.jhuapl.edu/info/ (accessed November 26, 2024). URL: https://rscf.ru/project/23-22-00076/ (дата обращения 26 ноября 2024 г.). URL: http://space.fmi.fi/image (accessed November 26, 2024).