Science China Earth Sciences

, Volume 60, Issue 8, pp 1466–1494 | Cite as

The Grad-Shafranov reconstruction in twenty years: 1996–2016

  • Qiang HuEmail author


We review and summarize the applications of the Grad-Shafranov (GS) reconstruction technique to space plasma structures in the Earth’s magnetosphere and in the interplanetary space. We organize our presentations following the branches of the “academic family tree” rooted on Prof. Bengt U. Ö. Sonnerup, the inventor of the GS method. Special attentions are paid to validations of the GS reconstruction results via (1) the direct validation by co-spatial in-situ measurements among multiple spacecraft, and (2) indirect validation by implications and interpretations of the physical connection between the structures reconstructed and other related processes. For the latter, the inter-comparison and interconnection between the large-scale magnetic flux ropes (i.e., Magnetic Clouds) in the solar wind and their solar source properties are presented. In addition, we also summarize various GS-type (or -like) reconstruction and an extension of the GS technique to toroidal geometry. In particular, we point to a possible advancement with added complexity of “helical symmetry” and mixed helicity, in the hope of stimulating interest in future development. We close by offering some thoughts on appreciating the scientific merit of GS reconstruction in general.


Grad-Shafranov equation Magnetohydrodynamics Magnetic clouds Magnetic flux ropes Magnetopause Current sheets Flux transfer events Plasmoids Solar flare Coronal mass ejections 


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We acknowledge the invitation by Drs. Feng XueShang and Wan WeiXing for this review. The author’s first scientific publication appeared in this journal nearly twenty years ago. As a point of reflection on the author’s own professional career, I am grateful for the mentorship, friendship, and companionship, offered by numerous mentors, colleagues, and families. In particular, the towering figures of Prof. Wei FengSi, Prof. Bengt Sonnerup, Drs. Charles Smith and Norman Ness, Prof. Gary Zank, and Prof. Wu Shi-Tsan, have provided invaluable guidance along the way during my past years as a graduate student, a postdoc, and a professional researcher, up to the present time. I am most indebted to Prof. Sonnerup and Prof. Hau Lin-Ni who laid down the foundation for and led me down the path of the GS reconstruction. I thank numerous colleagues and co-authors in shaping and enriching my professional development and growth. In particular, I benefited tremendously from the collaboration with Prof. Qiu Jiong, which has spanned one decade, and two sub-disciplines in space physics, i.e., solar physics and interplanetary physics. In addition, I have benefited greatly from collaborations with many other Chinese colleagues who have been so prolific that all their works could not possibly be included. I have been inspired by their work ethics and formidable spirit. I am particularly grateful to Prof. Chen Yao and colleagues at Shandong University, Weihai, for their hospitality and support. I wish to thank the reviewers for their enthusiastic review and knowledgeable comments that help improve the manuscript. Upon the completion of this manuscript, we were all saddened by the sudden passing of Dr. Shi-Tsan Wu, with whom the author has been working closely in the past decade or so, till the very last day. Together with many colleagues in China and around the world, we mourn the loss of such an inspirational figure in the field of solar-terrestrial physics. We will miss ST’s wisdom, his mentorship, and his passion for science. The best way to carry on ST’s legacy is to continue to encourage and inspire future generations, as ST always did. I hope this piece of writing will contribute to that noble cause. The author’s work summarized herein has been supported by National Aeronautics and Space Administration (NASA) and National Science Foundation (NSF) (Grants Nos. AGS-1062050, NNG04GF47G, NNG06GD41G, NNX12AF97G, NNX12AH50G, NNH13ZDA001N, and NNX14AF41G).


  1. Al-Haddad N, Nieves-Chinchilla T, Savani N P, Möstl C, Marubashi K, Hidalgo M A, Roussev I I, Poedts S, Farrugia C J. 2013. Magnetic field configuration models and reconstruction methods for interplanetary coronal mass ejections. Solar Phys, 284: 129CrossRefGoogle Scholar
  2. Al-Haddad N, Roussev I I, Möstl C, Jacobs C, Lugaz N, Poedts S, Farrugia C J. 2011. On the internal structure of the magnetic field in magnetic clouds and interplanetary coronal mass ejections: Writhe versus twist. Astrophys J, 738: L18CrossRefGoogle Scholar
  3. Antiochos S K, DeVore C R, Klimchuk J A. 1999. A model for solar coronal mass ejections. Astrophys J, 510: 485–493CrossRefGoogle Scholar
  4. Berger M A, Field G B. 1984. The topological properties of magnetic helicity. J Fluid Mech, 147: 133–148CrossRefGoogle Scholar
  5. Cartwright M L, Moldwin M B. 2010. Heliospheric evolution of solar wind small-scale magnetic flux ropes. J Geophys Res, 115: A08102Google Scholar
  6. Chen Y. 2013. A review of recent studies on coronal dynamics: Streamers, coronal mass ejections, and their interactions. Chin Sci Bull, 58: 1599–1624CrossRefGoogle Scholar
  7. Cheng J X, Qiu J. 2016. The nature of CME-flare-associated coronal dimming. Astrophys J, 825: 37CrossRefGoogle Scholar
  8. Chian A C L, Feng H Q, Hu Q, Loew M H, Miranda R A, Muñoz P R, Sibeck D G, Wu D J. 2016. Genesis of interplanetary intermittent turbulence: A case study of rope-rope magnetic reconnection. Astrophys J, 832: 179CrossRefGoogle Scholar
  9. Chou Y C, Hau L N. 2012. A statistical study of magnetopause structures: Tangential versus rotational discontinuities. J Geophys Res, 117: A08232CrossRefGoogle Scholar
  10. De Keyser J, Dunlop M W, Owen C J, Sonnerup B U Ö, Haaland S E, Vaivads A, Paschmann G, Lundin R, Rezeau L. 2005. Magnetopause and boundary layer. Space Sci Rev, 118: 231–320CrossRefGoogle Scholar
  11. Du D, Wang C, Hu Q. 2007. Propagation and evolution of a magnetic cloud from ACE to Ulysses. J Geophys Res, 112: A09101CrossRefGoogle Scholar
  12. Eriksson S, Hasegawa H, Teh W L, Sonnerup B U Ö, McFadden J P, Glassmeier K H, Le Contel O, Angelopoulos V, Cully C M, Larson D E, Ergun R E, Roux A, Carlson C W. 2009. Magnetic island formation between large-scale flow vortices at an undulating postnoon magnetopause for northward interplanetary magnetic field. J Geophys Res, 114: A00C17Google Scholar
  13. Eriksson S, Newman D L, Lapenta G, Angelopoulos V. 2014. On the signatures of magnetic islands and multiple X-lines in the solar wind as observed by ARTEMIS and WIND. Plasma Phys Control Fusion, 56: 064008CrossRefGoogle Scholar
  14. Farrugia C J, Berdichevsky D B, Möstl C, Galvin A B, Leitner M, Popecki M A, Simunac K D C, Opitz A, Lavraud B, Ogilvie K W, Veronig A M, Temmer M, Luhmann J G, Sauvaud J A. 2011. Multiple, distant (40°) in situ observations of a magnetic cloud and a corotating interaction region complex. J Atmos Sol-Terr Phys, 73: 1254–1269CrossRefGoogle Scholar
  15. Feng H Q, Wang J M. 2015. Observations of several unusual plasma compositional signatures within small interplanetary magnetic flux ropes. Astrophys J, 809: 112CrossRefGoogle Scholar
  16. Feng H Q, Wu D J, Lin C C, Chao J K, Lee L C, Lyu L H. 2008. Interplanetary small-and intermediate-sized magnetic flux ropes during 1995–2005. J Geophys Res, 113: A12105CrossRefGoogle Scholar
  17. Feng H Q, Zhao G Q, Wang J M. 2015. Counterstreaming electrons in small interplanetary magnetic flux ropes. J Geophys Res-Space Phys, 120: 10175–10184CrossRefGoogle Scholar
  18. Forbes T G, Linker J A, Chen J, Cid C, Kóta J, Lee M A, Mann G, Mikić Z, Potgieter M S, Schmidt J M, Siscoe G L, Vainio R, Antiochos S K, Riley P. 2006. CME theory and models. Space Sci Rev, 123: 251–302CrossRefGoogle Scholar
  19. Freidberg J P. 2014. Ideal MHD. Cambridge: Cambridge University PressCrossRefGoogle Scholar
  20. Gold T, Hoyle F. 1960. On the origin of solar flares. Mon Not Roy Astron Soc, 120: 89–105CrossRefGoogle Scholar
  21. González, A O, Domingues M O, Mendes O, Kaibara M K, Prestes A. 2015. Grad-Shafranov Reconstruction: Overview and improvement of the numerical solution used in space physics. Braz J Phys, 45: 493–509CrossRefGoogle Scholar
  22. González A O, Prestes A, Laurindo Sousa A N. 2016. Discussion about the magnetic field dimensionality, invariant axis condition, and coulomb gauge to solve the Grad-Shafranov equation. Braz J Phys, 46: 408–414CrossRefGoogle Scholar
  23. Gopalswamy N, Yashiro S, Akiyama S, Xie H. 2017. Estimation of reconnection flux using post-eruption arcades and its relevance to magnetic clouds at 1 AU. Sol Phys, 292: 65–82CrossRefGoogle Scholar
  24. Grad H, Rubin H. 1958. Hydromagnetic equilibria and force-free fields. In: Proceedings of the Second United Nations International Conference on the Peaceful Uses of Atomic Energy. Vol 31: 190–197. Geneva, United NationsGoogle Scholar
  25. Hara T, Luhmann J G, Halekas J S, Espley J R, Seki K, Brain D A, Hasegawa H, McFadden J P, Mitchell D L, Mazelle C, Harada Y, Livi R, DiBraccio G A, Connerney J E P, Andersson L, Jakosky B M. 2016. MAVEN observations of magnetic flux ropes with a strong field amplitude in the Martian magnetosheath during the ICME passage on 8 March 2015. Geophys Res Lett, 43: 4816–4824CrossRefGoogle Scholar
  26. Hara T, Mitchell D L, McFadden J P, Seki K, Brain D A, Halekas J S, Harada Y, Espley J R, DiBraccio G A, Connerney J E P, Andersson L, Mazelle C, Jakosky B M. 2015. Estimation of the spatial structure of a detached magnetic flux rope at Mars based on simultaneous MAVEN plasma and magnetic field observations. Geophys Res Lett, 42: 8933–8941CrossRefGoogle Scholar
  27. Hara T, Seki K, Hasegawa H, Brain D A, Matsunaga K, Saito M H. 2014a. The spatial structure of Martian magnetic flux ropes recovered by the Grad-Shafranov reconstruction technique. J Geophys Res-Space Phys, 119: 1262–1271CrossRefGoogle Scholar
  28. Hara T, Seki K, Hasegawa H, Brain D A, Matsunaga K, Saito M H, Shiota D. 2014b. Formation processes of flux ropes downstream from Martian crustal magnetic fields inferred from Grad-Shafranov reconstruction. J Geophys Res-Space Phys, 119: 7947–7962CrossRefGoogle Scholar
  29. Hasegawa H. 2012. Structure and dynamics of the magnetopause and its boundary layers. Monogr Environ Earth Planets, 1: 71–119CrossRefGoogle Scholar
  30. Hasegawa H, Nakamura R, Fujimoto M, Sergeev V A, Lucek E A, Rème H, Khotyaintsev Y. 2007a. Reconstruction of a bipolar magnetic signature in an earthward jet in the tail: Flux rope or 3D guide-field reconnection? J Geophys Res, 112: A11206CrossRefGoogle Scholar
  31. Hasegawa H, Sonnerup B U Ö, Dunlop M W, Balogh A, Haaland S E, Klecker B, Paschmann G, Lavraud B, Dandouras I, Rème H. 2004. Reconstruction of two-dimensional magnetopause structures from Cluster observations: Verification of method. Ann Geophys, 22: 1251–1266CrossRefGoogle Scholar
  32. Hasegawa H, Sonnerup B U Ö, Fujimoto M, Saito Y, Mukai T. 2007b. Recovery of streamlines in the flank low-latitude boundary layer. J Geophys Res, 112: A04213Google Scholar
  33. Hasegawa H, Sonnerup B U Ö, Hu Q, Nakamura T. 2014. Reconstruction of an evolving magnetic flux rope in the solar wind: Decomposing spatial and temporal variations from single-spacecraft data. J Geophys Res-Space Phys, 119: 97–114CrossRefGoogle Scholar
  34. Hasegawa H, Sonnerup B U Ö, Klecker B, Paschmann G, Dunlop M W, Rème H. 2005. Optimal reconstruction of magnetopause structures from Cluster data. Ann Geophys, 23: 973–982CrossRefGoogle Scholar
  35. Hasegawa H, Sonnerup B U Ö, Nakamura T K M. 2010. Recovery of time evolution of Grad-Shafranov equilibria from single-spacecraft data: Benchmarking and application to a flux transfer event. J Geophys Res, 115: A11219Google Scholar
  36. Hasegawa H, Sonnerup B U Ö, Owen C J, Klecker B, Paschmann G, Balogh A, Rème H. 2006. The structure of flux transfer events recovered from Cluster data. Ann Geophys, 24: 603–618CrossRefGoogle Scholar
  37. Hasegawa H, Retinò A, Vaivads A, Khotyaintsev Y, André M, Nakamura T K M, Teh W L, Sonnerup B U Ö, Schwartz S J, Seki Y, Fujimoto M, Saito Y, Rème H, Canu P. 2009. Kelvin-Helmholtz waves at the Earth’s magnetopause: Multiscale development and associated reconnection. J Geophys Res, 114: A12207CrossRefGoogle Scholar
  38. Hasegawa H, Zhang H, Lin Y, Sonnerup B U Ö, Schwartz S J, Lavraud B, Zong Q G. 2012. Magnetic flux rope formation within a magnetosheath hot flow anomaly. J Geophys Res, 117: A09214CrossRefGoogle Scholar
  39. Hau L N, Sonnerup B U Ö. 1999. Two-dimensional coherent structures in the magnetopause: Recovery of static equilibria from single-spacecraft data. J Geophys Res, 104: 6899–6917CrossRefGoogle Scholar
  40. Hidalgo M A. 2016. A global magnetic topology model for magnetic clouds. Iv. Astrophys J, 823: 3CrossRefGoogle Scholar
  41. Hietala H, Eastwood J P, Isavnin A. 2014. Sequentially released tilted flux ropes in the Earth’s magnetotail. Plasma Phys Control Fusion, 56: 064011CrossRefGoogle Scholar
  42. Hood A W, Priest E R. 1979. Kink instability of solar coronal loops as the cause of solar flares. Sol Phys, 64: 303–321CrossRefGoogle Scholar
  43. Hu H, Liu Y D, Wang R, Möstl C, Yang Z. 2016. Sun-to-Earth characteristics of the 2012 July 12 coronal mass ejection and associated geo-effectiveness. Astrophys J, 829: 97CrossRefGoogle Scholar
  44. Hu Q. 2001. Reconstruction of two-dimensional coherent structures in space plasmas from spacecraft data. Doctoral Dissertation. Hanover: Dartmouth CollegeGoogle Scholar
  45. Hu Q. 2016. On the Grad-Shafranov (GS) reconstruction of toroidal magnetic flux ropes. In: Wang L, Bruno R, Möbius E, Vourlidas A, Zank G, eds. In: International Solar Wind 14 Conference. Volume 1720 of AIP Conf. Series. 040005Google Scholar
  46. Hu Q. 2017. The Grad-Shafranov reconstruction of toroidal magnetic flux ropes: Method development and benchmark studies. Sol Phys, doi: 10.1007/11207-017-1134-zGoogle Scholar
  47. Hu Q, Dasgupta B. 2005. Calculation of magnetic helicity of cylindrical flux rope. Geophys Res Lett, 32: L12109Google Scholar
  48. Hu Q, Farrugia C J, Osherovich V A, Möstl C, Szabo A, Ogilvie K W, Lepping R P. 2013. Effect of electron pressure on the grad-shafranov reconstruction of interplanetary coronal mass ejections. Sol Phys, 284: 275–291CrossRefGoogle Scholar
  49. Hu Q, Qiu J, Dasgupta B, Khare A, Webb G M. 2014a. Structures of interplanetary magnetic flux ropes and comparison with their solar sources. Astrophys J, 793: 53CrossRefGoogle Scholar
  50. Hu Q, Qiu J, Krucker S. 2015. Magnetic field line lengths inside interplanetary magnetic flux ropes. J Geophys Res-Space Phys, 120: 5266–5283CrossRefGoogle Scholar
  51. Hu Q, Qiu J, Zheng J. 2014b. Characteristics of magnetic flux ropes from the sun to the heliosphere. In: Hu Q, Zank G P, eds. Outstanding Problems in Heliophysics: From Coronal Heating to the Edge of the Heliosphere. Volume 484 of Astronomical Society of the Pacific Conference Series. 78–83Google Scholar
  52. Hu Q, Smith C W, Ness N F, Skoug R M. 2003. Double flux-rope magnetic cloud in the solar wind at 1 AU. Geophys Res Lett, 30: 1385Google Scholar
  53. Hu Q, Smith C W, Ness N F, Skoug R M. 2004. Multiple flux rope magnetic ejecta in the solar wind. J Geophys Res, 109: A03102Google Scholar
  54. Hu Q, Smith C W, Ness N F, Skoug R M. 2005. On the magnetic topology of October/November 2003 events. J Geophys Res, 110: A09S03CrossRefGoogle Scholar
  55. Hu Q, Sonnerup B U Ö. 2000. Magnetopause transects from two spacecraft: A comparison. Geophys Res Lett, 27: 1443–1446CrossRefGoogle Scholar
  56. Hu Q, Sonnerup B U Ö. 2001. Reconstruction of magnetic flux ropes in the solar wind. Geophys Res Lett, 28: 467–470CrossRefGoogle Scholar
  57. Hu Q, Sonnerup B U Ö. 2002. Reconstruction of magnetic clouds in the solar wind: Orientations and configurations. J Geophys Res, 107: 1142CrossRefGoogle Scholar
  58. Hu Q, Sonnerup B U Ö. 2003. Reconstruction of two-dimensional structures in the magnetopause: Method improvements. J Geophys Res, 108: 1011CrossRefGoogle Scholar
  59. Isavnin A, Kilpua E K J, Koskinen H E J. 2011. Grad-Shafranov reconstruction of magnetic clouds: Overview and improvements. Sol Phys, 273: 205–219CrossRefGoogle Scholar
  60. Isavnin A, Vourlidas A, Kilpua E K J. 2013. Three-Dimensional evolution of erupted flux ropes from the Sun (2–20 R) to 1 AU. Sol Phys, 284: 203–215CrossRefGoogle Scholar
  61. Isavnin A, Vourlidas A, Kilpua E K J. 2014. Three-dimensional evolution of flux-rope cmes and its relation to the local orientation of the heliospheric current sheet. Sol Phys, 289: 2141–2156CrossRefGoogle Scholar
  62. Jiang C, Wu S T, Feng X, Hu Q. 2016a. Data-driven magnetohydrodynamic modelling of a flux-emerging active region leading to solar eruption. Nat Commun, 7: 11522CrossRefGoogle Scholar
  63. Jiang C, Wu S T, Yurchyshyn V, Wang H, Feng X, Hu Q. 2016b. How did a major confined flare occur in super solar active region 12192? Astrophys J, 828: 62CrossRefGoogle Scholar
  64. Kahler S W, Krucker S, Szabo A. 2011. Solar energetic electron probes of magnetic cloud field line lengths. J Geophys Res, 116: A01104CrossRefGoogle Scholar
  65. Kazachenko M D, Canfield R C, Longcope D W, Qiu J. 2012. Predictions of energy and helicity in four major eruptive solar flares. Sol Phys, 277: 165–183CrossRefGoogle Scholar
  66. Khrabrov A V, Sonnerup B U Ö. 1998. DeHoffmann-Teller analysis, in analysis methods for multi-spacecraft data. In: Paschmann G, Daly P W. Chap. 8. 221–248. Int Space Sci Inst, BernGoogle Scholar
  67. Kilpua E K J, Jian L K, Li Y, Luhmann J G, Russell C T. 2011. Multipoint ICME encounters: Pre-STEREO and STEREO observations. J Atmos Sol-Terr Phys, 73: 1228–1241CrossRefGoogle Scholar
  68. Larson D E, Lin R P, McTiernan J M, McFadden J P, Ergun R E, McCarthy M, Rème H, Sanderson T R, Kaiser M, Lepping R P, Mazur J. 1997. Tracing the topology of the October 18–20, 1995, magnetic cloud with~0.1–102 keV electrons. Geophys Res Lett, 24: 1911–1914CrossRefGoogle Scholar
  69. Li H J, Feng X S, Xiang J, Zuo P B. 2013. New approach for solving the inverse boundary value problem of Laplace’s equation on a circle: Technique renovation of the Grad-Shafranov (GS) reconstruction. J Geophys Res-Space Phys, 118: 2876–2881CrossRefGoogle Scholar
  70. Li H J, Feng X S, Zuo P B, Xie Y Q. 2009a. Inferring interplanetary flux rope orientation with the minimum residue method. J Geophys Res, 114: A03102Google Scholar
  71. Li H J, Feng X S, Zuo P B, Xie Y Q. 2009b. Observations of the field-aligned residual flow inside magnetic cloud structure. Sci China Ser E-Tech Sci, 52: 2555–2566CrossRefGoogle Scholar
  72. Li H J, Li C Y, Feng X S, Xiang J, Huang Y Y, Zhou S D. 2017. Data completion with Hilbert transform over plane rectangle: Technique renovation for the Grad-Shafranov reconstruction. J Geophys Res-Space Phys, 122: 3949–3960CrossRefGoogle Scholar
  73. Li Z Y, Chen T, Yan G Q. 2016. New method for determining central axial orientation of flux rope embedded within current sheet using multipoint measurements. Sci China Earth Sci, 59: 2037–2052CrossRefGoogle Scholar
  74. Linton M G, Moldwin M B. 2009. A comparison of the formation and evolution of magnetic flux ropes in solar coronal mass ejections and magnetotail plasmoids. J Geophys Res, 114: A00B09CrossRefGoogle Scholar
  75. Liu Y, Luhmann J G, Huttunen K E J, Lin R P, Bale S D, Russell C T, Galvin A B. 2008a. Reconstruction of the 2007 May 22 magnetic cloud: How much can we trust the flux-rope geometry of CMEs? Astrophys J, 677: L133–L136CrossRefGoogle Scholar
  76. Liu Y, Luhmann J G, Müller-Mellin R, Schroeder P C, Wang L, Lin R P, Bale S D, Li Y, Acuña M H, Sauvaud J A. 2008b. A comprehensive view of the 2006 December 13 CME: From the sun to interplanetary space. Astrophys J, 689: 563–571CrossRefGoogle Scholar
  77. Liu Y, Richardson J D, Belcher J W, Wang C, Hu Q, Kasper J C. 2006. Constraints on the global structure of magnetic clouds: Transverse size and curvature. J Geophys Res, 111: A12S03CrossRefGoogle Scholar
  78. Liu Y, Thernisien A, Luhmann J G, Vourlidas A, Davies J A, Lin R P, Bale S D. 2010. Reconstructing coronal mass ejections with coordinated imaging and in situ observations: Global structure, kinematics, and implications for space weather forecasting. Astrophys J, 722: 1762–1777CrossRefGoogle Scholar
  79. Liu Y D, Hu H, Wang C, Luhmann J G, Richardson J D, Yang Z, Wang R. 2016. On sun-to-earth propagation of coronal mass ejections: II. Slow events and comparison with others. Astrophys J Suppl Ser, 222: 23CrossRefGoogle Scholar
  80. Liu Y D, Hu H, Wang R, Yang Z, Zhu B, Liu Y A, Luhmann J G, Richardson J D. 2015. Plasma and magnetic field characteristics of solar coronal mass ejections in relation to geomagnetic storm intensity and variability. Astrophys J, 809: L34CrossRefGoogle Scholar
  81. Liu Y D, Luhmann J G, Kajdic P, Kilpua E K J, Lugaz N, Nitta N V, Möstl C, Lavraud B, Bale S D, Farrugia C J, Galvin A B. 2014. Observations of an extreme storm in interplanetary space caused by successive coronal mass ejections. Nat Commun, 5: 3481Google Scholar
  82. Longcope D W, Beveridge C. 2007. A quantitative, topological model of reconnection and flux rope formation in a two-ribbon flare. Astrophys J, 669: 621–635CrossRefGoogle Scholar
  83. Longcope D, Beveridge C, Qiu J, Ravindra B, Barnes G, Dasso S. 2007. Modeling and measuring the flux reconnected and ejected by the tworibbon Flare/CME event on 7 November 2004. Sol Phys, 244: 45–73CrossRefGoogle Scholar
  84. Lu S W, Zong Q G, Vogiatzis I, Wang Y F, Tian A M. 2015. Reconstruction of plasmoid and traveling compression region in the near-Earth magnetotail. Sci China Tech Sci, 58: 330–337CrossRefGoogle Scholar
  85. Lui A T Y. 2011. Grad-Shafranov reconstruction of magnetic flux ropes in the near-earth space. Space Sci Rev, 158: 43–68CrossRefGoogle Scholar
  86. Lui A T Y, Sibeck D G, Phan T, Angelopoulos V, McFadden J, Carlson C, Larson D, Bonnell J, Glassmeier K H, Frey S. 2008a. Reconstruction of a magnetic flux rope from THEMIS observations. Geophys Res Lett, 35: L17S05CrossRefGoogle Scholar
  87. Lui A T Y, Sibeck D G, Phan T, McFadden J P, Angelopoulos V, Glassmeier K H. 2008b. Reconstruction of a flux transfer event based on observations from five THEMIS satellites. J Geophys Res, 113: A00C01Google Scholar
  88. Marubashi K, Akiyama S, Yashiro S, Gopalswamy N, Cho K S, Park Y D. 2015. Geometrical relationship between interplanetary flux ropes and their solar sources. Sol Phys, 290: 1371–1397CrossRefGoogle Scholar
  89. Moldwin M B, Ford S, Lepping R, Slavin J, Szabo A. 2000. Small-scale magnetic flux ropes in the solar wind. Geophys Res Lett, 27: 57–60CrossRefGoogle Scholar
  90. Moore T E, Burch J L, Daughton W S, Fuselier S A, Hasegawa H, Petrinec S M, Pu Z. 2013. Multiscale studies of the three-dimensional dayside X-line. J Atmos Sol-Terr Phys, 99: 32–40CrossRefGoogle Scholar
  91. Möstl C, Farrugia C J, Biernat H K, Leitner M, Kilpua E K J, Galvin A B, Luhmann J G. 2009a. Optimized Grad - Shafranov reconstruction of a magnetic cloud using STEREO-wind observations. Sol Phys, 256: 427–441CrossRefGoogle Scholar
  92. Möstl C, Farrugia C J, Kilpua E K J, Jian L K, Liu Y, Eastwood J P, Harrison R A, Webb D F, Temmer M, Odstrcil D, Davies J A, Rollett T, Luhmann J G, Nitta N, Mulligan T, Jensen E A, Forsyth R, Lavraud B, de Koning C A, Veronig A M, Galvin A B, Zhang T L, Anderson B J. 2012. Multipoint shock and flux rope analysis of multiple interplanetary coronal mass ejections around 2010 August 1 in the inner heliosphere. Astrophys J, 758: 10CrossRefGoogle Scholar
  93. Möstl C, Farrugia C J, Miklenic C, Temmer M, Galvin A B, Luhmann J G, Kilpua E K J, Leitner M, Nieves-Chinchilla T, Veronig A, Biernat H K. 2009b. Multispacecraft recovery of a magnetic cloud and its origin from magnetic reconnection on the Sun. J Geophys Res, 114: A04102CrossRefGoogle Scholar
  94. Möstl C, Temmer M, Rollett T, Farrugia C J, Liu Y, Veronig A M, Leitner M, Galvin A B, Biernat H K. 2010. STEREO and Wind observations of a fast ICME flank triggering a prolonged geomagnetic storm on 5–7 April 2010. Geophys Res Lett, 37: L24103CrossRefGoogle Scholar
  95. Osherovich V A, Fainberg J, Stone R G. 1999. Multi-tube model for interplanetary magnetic clouds. Geophys Res Lett, 26: 401–404CrossRefGoogle Scholar
  96. Paschmann G, Sonnerup B U Ö. 2008. Proper frame determination and Walen test. ISSI Scientific Reports Ser, 8: 65–74Google Scholar
  97. Press W H, Teukolsky S A, Vetterling W T, Flannery B P. 2007. Numerical Recipes: The Art of Scientific Computing. New York: Cambridge University PressGoogle Scholar
  98. Priest E R, Longcope D W. 2017. Flux-rope twist in eruptive flares and CMEs: Due to zipper and main-phase reconnection. Sol Phys, 292, 25Google Scholar
  99. Priest E R, Longcope D W, Janvier M. 2016. Evolution of magnetic helicity during eruptive flares and coronal mass ejections. Sol Phys, 291: 2017–2036CrossRefGoogle Scholar
  100. Qiu J. 2009. Observational analysis of magnetic reconnection sequence. Astrophys J, 692: 1110–1124CrossRefGoogle Scholar
  101. Qiu J, Hu Q, Howard T A, Yurchyshyn V B. 2007. On the magnetic flux budget in low-corona magnetic reconnection and interplanetary coronal mass ejections. Astrophys J, 659: 758–772CrossRefGoogle Scholar
  102. Qiu J, Longcope D W, Cassak P A, Priest E R. 2017. Elongation of flare ribbons. Astrophys J, 838: 17CrossRefGoogle Scholar
  103. Riley P, Linker J A, Lionello R, Mikic Z, Odstrcil D, Hidalgo M A, Cid C, Hu Q, Lepping R P, Lynch B J, Rees A. 2004. Fitting flux ropes to a global MHD solution: A comparison of techniques. J Atmos Sol-Terr Phys, 66: 1321–1331CrossRefGoogle Scholar
  104. Rong Z J, Wan W X, Shen C, Zhang T L, Lui A T Y, Wang Y, Dunlop M W, Zhang Y C, Zong Q G. 2013. Method for inferring the axis orientation of cylindrical magnetic flux rope based on single-point measurement. J Geophys Res-Space Phys, 118: 271–283CrossRefGoogle Scholar
  105. Ruzmaikin A, Martin S, Hu Q. 2003. Signs of magnetic helicity in interplanetary coronal mass ejections and associated prominences: Case study. J Geophys Res, 108: 1096CrossRefGoogle Scholar
  106. Shafranov V D. 1958. Magnetohydrodynamical equilibrium configurations. Sov Phys JETP, 6: 545–554Google Scholar
  107. Sharma R, Srivastava N. 2012. Presence of solar filament plasma detected in interplanetary coronal mass ejections by in situ spacecraft. J Space Weather Space Clim, 2: A10CrossRefGoogle Scholar
  108. Sharma R, Srivastava N, Chakrabarty D, Möstl C, Hu Q. 2013. Interplanetary and geomagnetic consequences of 5 January 2005 CMEs associated with eruptive filaments. J Geophys Res-Space Phys, 118: 3954–3967CrossRefGoogle Scholar
  109. Song H Q, Zhong Z, Chen Y, Zhang J, Cheng X, Zhao L, Hu Q, Li G. 2016. A statistical study of the average iron charge state distributions inside magnetic clouds for solar cycle 23. Astrophys J Suppl Ser, 224: 27CrossRefGoogle Scholar
  110. Sonnerup B U Ö, Denton R E, Hasegawa H, Swisdak M. 2013. Axis and velocity determination for quasi two-dimensional plasma/field structures from Faraday’s law: A second look. J Geophys Res-Space Phys, 118: 2073–2086CrossRefGoogle Scholar
  111. Sonnerup B U Ö, Guo M. 1996. Magnetopause transects. Geophys Res Lett, 23: 3679–3682CrossRefGoogle Scholar
  112. Sonnerup B U Ö, Hasegawa H. 2010. On slowly evolving Grad-Shafranov equilibria. J Geophys Res, 115: A11218CrossRefGoogle Scholar
  113. Sonnerup B U Ö, Hasegawa H. 2011. Reconstruction of steady, three-dimensional, magnetohydrostatic field and plasma structures in space: Theory and benchmarking. J Geophys Res, 116: A09230CrossRefGoogle Scholar
  114. Sonnerup B U Ö, Hasegawa H, Denton R E, Nakamura T K M. 2016. Reconstruction of the electron diffusion region. J Geophys Res-Space Phys, 121: 4279–4290CrossRefGoogle Scholar
  115. Sonnerup B U Ö, Hasegawa H, Paschmann G. 2004. Anatomy of a flux transfer event seen by Cluster. Geophys Res Lett, 31: L11803CrossRefGoogle Scholar
  116. Sonnerup B U Ö, Hasegawa H, Teh W L, Hau L N. 2006. Grad-Shafranov reconstruction: An overview. J Geophys Res, 111: A09204Google Scholar
  117. Sonnerup B U Ö, Scheible M. 1998. Minimum and maximum variance analysis. In: Paschmann G, Daly P W, eds. Analysis Methods for Multi-Spacecraft Data. Bern: Int Space Sci Inst. 185–220Google Scholar
  118. Sonnerup B U Ö, Teh W L. 2008. Reconstruction of two-dimensional coherent MHD structures in a space plasma: The theory. J Geophys Res, 113: A05202Google Scholar
  119. Sonnerup B U Ö, Teh W L. 2009. Reconstruction of two-dimensional coherent structures in ideal and resistive Hall MHD: The theory. J Geophys Res, 114: A04206CrossRefGoogle Scholar
  120. Teh W L, Hau L N. 2004. Evidence for pearl-like magnetic island structures at dawn and dusk side magnetopause. Earth Planet Space, 56: 681–686CrossRefGoogle Scholar
  121. Teh W L, Hau L N. 2007. Triple crossings of a string of magnetic islands at duskside magnetopause encountered by AMPTE/IRM satellite on 8 August 1985. J Geophys Res, 112: A08207CrossRefGoogle Scholar
  122. Teh W L, Eriksson S, Sonnerup B U Ö, Ergun R, Angelopoulos V, Glassmeier K H, McFadden J P, Bonnell J W. 2010a. THEMIS observations of a secondary magnetic island within the Hall electromagnetic field region at the magnetopause. Geophys Res Lett, 37: L21102CrossRefGoogle Scholar
  123. Teh W L, Sonnerup B U Ö. 2008. First results from ideal 2-D MHD reconstruction: magnetopause reconnection event seen by Cluster. Ann Geophys, 26: 2673–2684CrossRefGoogle Scholar
  124. Teh W L, Sonnerup B U Ö, Birn J, Denton R E. 2010b. Resistive MHD reconstruction of two-dimensional coherent structures in space. Ann Geophys, 28: 2113–2125CrossRefGoogle Scholar
  125. Teh W L, Sonnerup B U Ö, Hau L N. 2007. Grad-Shafranov reconstruction with field-aligned flow: First results. Geophys Res Lett, 34: L05109CrossRefGoogle Scholar
  126. Teh W L, Sonnerup B U Ö, Hu Q, Farrugia C J. 2009. Reconstruction of a large-scale reconnection exhaust structure in the solar wind. Ann Geophys, 27: 807–822CrossRefGoogle Scholar
  127. Teh W L, Sonnerup B U Ö, Paschmann G, Haaland S E. 2011a. Local structure of directional discontinuities in the solar wind. J Geophys Res, 116: A04105Google Scholar
  128. Teh W L, Nakamura R, Baumjohann W. 2013. Magnetic field topology of the plasma sheet boundary layer. J Geophys Res-Space Phys, 118: 4059–4065CrossRefGoogle Scholar
  129. Teh W L, Nakamura R, Karimabadi H, Baumjohann W, Zhang T L. 2014. Correlation of core field polarity of magnetotail flux ropes with the IMFBy: Reconnection guide field dependency. J Geophys Res-Space Phys, 119: 2933–2944CrossRefGoogle Scholar
  130. Teh W L, Nakamura R, Sonnerup B U Ö, Eastwood J P, Volwerk M, Fazakerley A N, Baumjohann W. 2011b. Evidence of the origin of the Hall magnetic field for reconnection: Hall MHD reconstruction results from Cluster observations. J Geophys Res, 116: A11218Google Scholar
  131. Tian A M, Shi Q Q, Zong Q G, Du J, Fu S Y, Dai Y N. 2014. Analysis of magnetotail flux rope events by ARTEMIS observations. Sci China Tech Sci, 57: 1010–1019CrossRefGoogle Scholar
  132. Tian H, Yao S, Zong Q, He J, Qi Y. 2010. Signatures of magnetic reconnection at boundaries of interplanetary small-scale magnetic flux ropes. Astrophys J, 720: 454–464CrossRefGoogle Scholar
  133. Tian A M, Zong Q G. 2009. Study of magnetotail plasma sheet vortices with GS velocity field reconstruction method. Chin J Geophys, 52: 743–753CrossRefGoogle Scholar
  134. Tian A M, Zong Q G, Shi Q Q. 2012. Reconstruction of morningside plasma sheet compressional ULF Pc5 wave. Sci China Tech Sci, 55: 1092–1100CrossRefGoogle Scholar
  135. Tian A M, Zong Q G, Wang Y F, Shi Q Q, Fu S Y, Pu Z Y. 2010. A series of plasma flow vortices in the tail plasma sheet associated with solar wind pressure enhancement. J Geophys Res, 115: A09204CrossRefGoogle Scholar
  136. Trenchi L, Bruno R, Telloni D, D’amicis R, Marcucci M F, Zurbuchen T H, Weberg M. 2013. Solar energetic particle modulations associated with coherent magnetic structures. Astrophys J, 770: 11CrossRefGoogle Scholar
  137. Trenchi L, Fear R C, Trattner K J, Mihaljcic B, Fazakerley A N. 2016. A sequence of flux transfer events potentially generated by different generation mechanisms. J Geophys Res-Space Phys, 121: 8624–8639CrossRefGoogle Scholar
  138. van Ballegooijen A A, Martens P C H. 1989. Formation and eruption of solar prominences. Astrophys J, 343: 971–984CrossRefGoogle Scholar
  139. Vemareddy P, Möstl C, Amerstorfer T, Mishra W, Farrugia C, Leitner M. 2016. Comparison of magnetic properties in a magnetic cloud and its solar source on 2013 April 11–14. Astrophys J, 828: 12CrossRefGoogle Scholar
  140. Vogiatzis I I, Isavnin A, Zong Q G, Sarris E T, Lu S W, Tian A M. 2015. Dipolarization fronts in the near-Earth space and substorm dynamics. Ann Geophys, 33: 63–74CrossRefGoogle Scholar
  141. Walthour D W, Sonnerup B U Ö. 1995. Remote sensing of 2D magnetopause structures. Washington DC: American Geophysical Union. Geophys Monograph Ser, 90: 247Google Scholar
  142. Walthour D W, Sonnerup B U O, Paschmann G, Luehr H, Klumpar D, Potemra T. 1993. Remote sensing of two-dimensional magnetopause structures. J Geophys Res, 98: 1489–1504CrossRefGoogle Scholar
  143. Wang J, Dunlop M W, Pu Z Y, Zhou X Z, Zhang X G, Wei Y, Fu S Y, Xiao C J, Fazakerley A, Laakso H, Taylor M G G T, Bogdanova Y, Pitout F, Davies J, Zong Q G, Shen C, Liu Z X, Carr C, Perry C, Rème H, Dandouras I, Escoubet P, Owen C J. 2007. TC1 and Cluster observation of an FTE on 4 January 2005: A close conjunction. Geophys Res Lett, 34: L03106Google Scholar
  144. Wang W S, Liu R, Wang Y M, Hu Q, Shen C L, Jiang C W, Zhu C M. 2017. Formation of a highly twisted magnetic flux rope during the course of a solar eruption. Nature Communications, submittedGoogle Scholar
  145. Wang Y M, Zhang Q H, Liu J J, Shen C L, Shen F, Yang Z C, Zic T, Vrsnak B, Webb D F, Liu R, Wang S, Zhang J, Hu Q, Zhuang B. 2016a. On the propagation of a geoeffective coronal mass ejection during 15–17 Marc. 2015. J Geophys Res-Space Phys, 121: 7423–7434CrossRefGoogle Scholar
  146. Wang Y M, Zhuang B, Hu Q, Liu R, Shen C L, Chi Y T. 2016b. On the twists of interplanetary magnetic flux ropes observed at 1 AU. J Geophys Res-Space Phys, 121: 9316–9339CrossRefGoogle Scholar
  147. Webb D F, Möstl C, Jackson B V, Bisi M M, Howard T A, Mulligan T, Jensen E A, Jian L K, Davies J A, de Koning C A, Liu Y, Temmer M, Clover J M, Farrugia C J, Harrison R A, Nitta N, Odstrcil D, Tappin S J, Yu H S. 2013. Heliospheric imaging of 3D density structures during the multiple coronal mass ejections of late July to early August 2010. Sol Phys, 285: 317–348CrossRefGoogle Scholar
  148. Webb G M, Hu Q, Dasgupta B, Zank G P. 2010. Homotopy formulas for the magnetic vector potential and magnetic helicity: The Parker spiral interplanetary magnetic field and magnetic flux ropes. J Geophys Res, 115: A10112CrossRefGoogle Scholar
  149. Wei F, Schwenn R, Hu Q. 1997. Magnetic reconnection events in the interplanetary space. Sci China Ser E-Tech Sci, 40: 463–471CrossRefGoogle Scholar
  150. Wood B E, Rouillard A P, Möstl C, Battams K, Savani N P, Marubashi K, Howard R A, Socker D G. 2012. Connecting coronal mass ejections and magnetic clouds: A case study using an event from 22 June 2009. Sol Phys, 16: 369–389Google Scholar
  151. Yang Y Y, Shen C, Zhang Y C, Rong Z J, Li X, Dunlop M, Ma Y H, Liu Z X, Carr C M, Rème H. 2014. The force-free configuration of flux ropes in geomagnetotail: Cluster observations. J Geophys Res-Space Phys, 119: 6327–6341CrossRefGoogle Scholar
  152. Yu W, Farrugia C J, Lugaz N, Galvin A B, J. Kilpua E K, Kucharek H, Möstl C, Leitner M, Torbert R B, C. Simunac K D, Luhmann J G, Szabo A, Wilson Iii L B, Ogilvie K W, Sauvaud J A. 2014. A statistical analysis of properties of small transients in the solar wind 2007–2009: STEREO and Wind observations. J Geophys Res-Space Phys, 119: 689–708CrossRefGoogle Scholar
  153. Yurchyshyn V, Hu Q, Abramenko V. 2005. Structure of magnetic fields in NOAA active regions 0486 and 0501 and in the associated interplanetary ejecta. Space Weather, 3: S08C02CrossRefGoogle Scholar
  154. Zank G P, le Roux J A, Webb G M, Dosch A, Khabarova O. 2014. Particle acceleration via reconnection processes in the supersonic solar wind. Astrophys J, 797: 28CrossRefGoogle Scholar
  155. Zhang T L, Baumjohann W, Teh W L, Nakamura R, Russell C T, Luhmann J G, Glassmeier K H, Dubinin E, Wei H Y, Du A M, Lu Q M, Wang S, Balikhin M. 2012. Giant flux ropes observed in the magnetized ionosphere at Venus. Geophys Res Lett, 39: L23103Google Scholar
  156. Zhang Y C, Shen C, Liu Z X, Rong Z J, Zhang T L, Marchaudon A, Zhang H, Duan S P, Ma Y H, Dunlop M W, Yang Y Y, Carr C M, Dandouras I. 2013. Two different types of plasmoids in the plasma sheet: Cluster multisatellite analysis application. J Geophys Res-Space Phys, 118: 5437–5444CrossRefGoogle Scholar
  157. Zhang Y C, Shen C, Liu Z X, Narita Y. 2010. Magnetic helicity of a flux rope in the magnetotail: THEMIS results. Ann Geophys, 28: 1687–1693CrossRefGoogle Scholar
  158. Zhang Y C, Liu Z X, Shen C, Fazakerley A, Dunlop M, Réme H, Lucek E, Walsh A P, Yao L. 2007. The magnetic structure of an earthwardmoving flux rope observed by Cluster in the near-tail. Ann Geophys, 25: 1471–1476CrossRefGoogle Scholar
  159. Zheng J L, Hu Q. 2016. Observations and analysis of small-scale magnetic flux ropes in the solar wind. J Phys-Conf Ser, 767: 012028CrossRefGoogle Scholar

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© Science China Press and Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Space Science and CSPARUniversity of Alabama in HuntsvilleHuntsvilleUSA

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