Solar Physics

, 293:165 | Cite as

The Magnetic Field Geometry of Small Solar Wind Flux Ropes Inferred from Their Twist Distribution

  • W. Yu
  • C. J. FarrugiaEmail author
  • N. Lugaz
  • A. B. Galvin
  • C. Möstl
  • K. Paulson
  • P. Vemareddy


This work extends recent efforts on the force-free modeling of large flux rope-type structures (magnetic clouds, MCs) to much smaller spatial scales. We first select small flux ropes (SFRs) by eye whose duration is unambiguous and which were observed by the Solar Terrestrial Relations Observatory (STEREO) or Wind spacecraft during solar maximum years. We inquire into which analytical technique is physically most appropriate, augmenting the numerical modeling with considerations of magnetic twist. The observational fact that these SFRs typically do not expand significantly into the solar wind makes static models appropriate for this study. SFRs can be modeled with force-free methods during the maximum phase of solar activity. We consider three models: (i) a linear force-free field (\(\nabla\times \mathbf{B} = \alpha \mathbf{B}\)) with a specific, prescribed constant \(\alpha\) (Lundquist solution), and (ii) with \(\alpha\) as a free constant parameter (“Lundquist-alpha” solution), and (iii) a uniform twist field (Gold–Hoyle solution). We retain only those cases where the impact parameter is less than one-half the flux rope (FR) radius, \(R\), so the results should be robust (29 cases). The SFR radii lie in the range \([{\approx}\,0.003, 0.059]~\mbox{AU}\). Comparing results, we find that the Lundquist-alpha and uniform twist solutions yielded comparable and small normalized \(\chi^{2}\) values in most cases. This means that analytical modeling alone cannot distinguish which of these two is better in reproducing their magnetic field geometry. We then use Grad–Shafranov (GS) reconstruction to analyze these events further in a model-independent way. The orientations derived from GS are close to those obtained from the uniform twist field model. We then considered the twist per unit length, \(\tau\), both its profile through the FR and its absolute value, applying a graphic approach to obtain \(\tau\) from the GS solution. The results are in better agreement with the uniform twist model. We find \(\tau\) to lie in the range \([5.6, 34]~\mbox{turns}/\mbox{AU}\), i.e. much higher than typical values for MCs. The GH model-derived \(\tau\) values are comparable to those obtained from GS reconstruction. We find that the twist per unit length, \(L\), is inversely proportional to \(R\), as \(\tau\approx0.17/R\). We combine MC and SFR results on \(\tau(R)\) and give a relation that is approximately valid for both sets. Using this, we find that the axial and azimuthal fluxes, \(F_{z}\) and \(F_{\phi}\), vary as \({\approx}\,2.1 B_{0} R^{2}\) (\({\times}10^{21}~\mbox{Mx}\)) and \(F_{\phi}/L \approx0.36 B_{0} R\) (\({\times}10^{21}~\mbox{Mx}/\mbox{AU}\)). The relative helicity per unit length is \(H/L \approx0.75 B_{0}^{2} R^{3}\) (\({\times}10^{42}~\mbox{Mx}^{2}/\mbox{AU}\)).


Small solar wind flux ropes Analytical models Magnetic field line twist 



Support for this work came from the following grants: NASA STEREO Quadrature grant, NSF AGS-1435785, AGS-1433086 and AGS-1433213, and NASA NNX16AO04G, NNX15AB87G, and NNX15AU01G. C.M. thanks the Austrian Science Fund (FWF): [P26174-N27]. P.V. is supported by an INSPIRE grant of AORC scheme under Department of Science and Technology.

Disclosure of Potential Conflicts of Interest

The authors declare that they have no conflicts of interest.


  1. Al-Haddad, N., Nieves-Chinchilla, T., Savani, N.P., Möstl, C., Marubashi, K., Hidalgo, M.A., et al.: 2013, Magnetic field configuration models and reconstruction methods for interplanetary coronal mass ejections. Solar Phys. 284, 129. DOI. ADSCrossRefGoogle Scholar
  2. Al-Haddad, N., Nieves-Chinchilla, T., Savani, N.P., Lugaz, N., Roussev, I.: 2018, Fitting and reconstruction of thirteen simple coronal mass ejections. Solar Phys. 293, 73. DOI. ADSCrossRefGoogle Scholar
  3. Aulanier, G., Janvier, M., Schmieder, B.: 2012, The standard flare model in three dimensions – I. Strong-to-weak shear transition in post-flare loops. Astron. Astrophys. 543, A110. DOI. ADSCrossRefGoogle Scholar
  4. Burlaga, L.F.: 1988, Magnetic clouds: Constant alpha force-free configurations. J. Geophys. Res. 93, 7217. DOI. ADSCrossRefGoogle Scholar
  5. Burlaga, L., Sittler, E., Mariani, F., Schwenn, R.: 1981, Magnetic loop behind an interplanetary shock: Voyager, Helios, and IMP 8 observations. J. Geophys. Res. 86, 6673. DOI. ADSCrossRefGoogle Scholar
  6. Dasso, S., Mandrini, C.H., Démoulin, P., Luoni, M.L.: 2006, A new model-independent method to compute magnetic helicity in magnetic clouds. Astron. Astrophys. 455, 349. DOI. ADSCrossRefzbMATHGoogle Scholar
  7. Démoulin, P.: 2010, Interaction of ICMEs with the solar wind. AIP Conf. Proc. 1216, 229. DOI. ADSCrossRefGoogle Scholar
  8. Farrugia, C.J., Burlaga, L.F., Osherovich, V.A., Richardson, I.G., Freeman, M.P., Lepping, R.P., et al.: 1993, A study of an expanding interplanetary magnetic cloud and its interaction with the Earth’s magnetosphere: The interplanetary aspect. J. Geophys. Res. 98, 7621. DOI. ADSCrossRefGoogle Scholar
  9. Farrugia, C.J., Janoo, L.A., Torbert, R.B., Quinn, J.M., Ogilvie, K.W., Lepping, R.P., et al.: 1999, A uniform-twist magnetic flux rope in the solar wind. AIP Conf. Proc. 471, 745. DOI. ADSCrossRefGoogle Scholar
  10. Galvin, A.B., Kistler, L.M., Popecki, M.A., Farrugia, C.J., Simunac, K.D.C., Ellis, L., et al.: 2008, The plasma and suprathermal ion composition (PLASTIC) investigation on the STEREO observatories. Space Sci. Rev. 136, 437. DOI. ADSCrossRefGoogle Scholar
  11. Gold, T., Hoyle, F.: 1960, On the origin of solar flares. Mon. Not. Roy. Astron. Soc. 120, 89. ADS. ADSCrossRefGoogle Scholar
  12. 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. DOI. ADSCrossRefGoogle Scholar
  13. Hu, Q., Qiu, J., Krucker, S.: 2015, Magnetic field line lengths inside interplanetary magnetic flux ropes. J. Geophys. Res. 120, 5266. DOI. CrossRefGoogle Scholar
  14. Hu, Q., Sonnerup, B.U.Ö: 2001, Reconstruction of magnetic flux ropes in the solar wind. Geophys. Res. Lett. 28(3), 467. DOI. ADSCrossRefGoogle Scholar
  15. Hu, Q., Sonnerup, B.U.Ö: 2002, Reconstruction of magnetic clouds in the solar wind: Orientations and configurations. J. Geophys. Res. 107, 1142. DOI. CrossRefGoogle Scholar
  16. Hu, Q., Qiu, J., Dasgupta, B., Khare, A., Webb, G.M.: 2014, Structures of interplanetary magnetic flux ropes and comparison with their solar sources. Astrophys. J. 793, 53. DOI. ADSCrossRefGoogle Scholar
  17. Janvier, M., Démoulin, P., Dasso, S.: 2013, Global axis shape of magnetic clouds deduced from the distribution of their local axis orientation. Astron. Astrophys. 556, A50. DOI. ADSCrossRefGoogle Scholar
  18. Kahler, S.W., Haggerty, D.K., Richardson, I.G.: 2011, Magnetic field-line lengths in interplanetary coronal mass ejections inferred from energetic electron events. Astrophys. J. 736, 106. DOI. ADSCrossRefGoogle Scholar
  19. Khrabrov, A.V., Sonnerup, B.U.Ö.: 1998, deHoffmann–Teller analysis. In: Analysis Methods for Multi-Spacecraft Data, International Space Science Institute, Bern, 221. Google Scholar
  20. Larson, D.E., Lin, R.P., McTieman, J.M., McFadden, J.P., Ergun, R.E., McCarthy, M., et al.: 1997, Tracing the topology of the October 18 – 20, 1995, magnetic cloud with \({\sim}\,0.1\,\mbox{--}\,10^{2}~\mbox{keV}\) electrons. Geophys. Res. Lett. 24(15), 1911. DOI. ADSCrossRefGoogle Scholar
  21. Lepping, R.P., Jones, J.A., Burlaga, L.F.: 1990, Magnetic field structure of interplanetary magnetic clouds at 1 AU. J. Geophys. Res. 95, 11957. DOI. ADSCrossRefGoogle Scholar
  22. Lepping, R.P., Acũna, M.H., Brulaga, L.F., Farrell, W.M., Slavin, J.A., Schatten, K.H., et al.: 1995, The Wind Magnetic Field Investigation. Space Sci. Rev. 71, 207. DOI. ADSCrossRefGoogle Scholar
  23. Lepping, R.P., Berdichevsky, D.B., Wu, C.-C., Szabo, A., Narock, T., Mariani, F., et al.: 2006, A summary of WIND magnetic clouds for years 1995 – 2003: Model-fitted parameters, associated errors and classifications. Ann. Geophys. 24, 215. DOI. ADSCrossRefGoogle Scholar
  24. Liu, Y., Luhmann, J.G., Huttunen, K.E.J., Lin, R.P., Bale, S.D., Russell, C.T., et al.: 2008, Reconstruction of the 2007 May 22 magnetic cloud: How much can we trust the flux-rope geometry of CMEs? Astrophys. J. 677, L133. DOI. ADSCrossRefGoogle Scholar
  25. Luhmann, J.G., Curtis, D.W., Schroeder, P., McCauley, J., Lin, R.P., Larson, D.E., et al.: 2008, STEREO IMPACT investigation goals, measurements, and data products overview. Space Sci. Rev. 136, 117. DOI. ADSCrossRefGoogle Scholar
  26. Lundquist, S.: 1950, Magnetohydrostatic fields. Ark. Fys. 2, 361. MathSciNetzbMATHGoogle Scholar
  27. Mandrini, C.H., Pohjolainen, S., Dasso, S., Green, L.M., Démoulin, P., van Driel-Gesztelyi, L., et al.: 2005, Interplanetary flux rope ejected from an X-ray bright point – The smallest magnetic cloud source-region ever observed. Astron. Astrophys. 434(2), 725. DOI. ADSCrossRefGoogle Scholar
  28. Markwardt, C.B.: 2009, Non-linear least squares fitting in IDL with MPFIT. In: Bohlender, D., Dowler, P., Durand, D. (eds.) Proc. ADASS XVIII. ASP Conf. Ser., 411, Astronomical Society of the Pacific, San Francisco, 251. Google Scholar
  29. Marubashi, K., Lepping, R.P.: 2007, Long-duration magnetic clouds: A comparison of analyses using torus- and cylinder-shaped flux rope models. Ann. Geophys. 25, 2453. DOI. ADSCrossRefGoogle Scholar
  30. 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(1), 57. DOI. ADSCrossRefGoogle Scholar
  31. Möstl, C., Miklenic, C., Farrugia, C.J., Temmer, M., Veronig, A., Galvin, A.B., et al.: 2008, Two-spacecraft reconstruction of a magnetic cloud and comparison to its solar source. Ann. Geophys. 26, 3139. DOI. ADSCrossRefGoogle Scholar
  32. Möstl, C., Farrugia, C.J., Biernat, H.K., Leitner, M., Kilpua, E.K.J., Galvin, A.B., et al.: 2009, Optimized Grad–Shafranov reconstruction of a magnetic cloud using STEREO-Wind observations. Solar Phys. 256, 427. DOI. ADSCrossRefGoogle Scholar
  33. Newbury, J.A., Russell, C.T., Phillips, J.L., Gary, S.P.: 1998, Electron temperature in the ambient solar wind: Typical properties and a lower bound at 1 AU. J. Geophys. Res. 103, 9553. DOI. ADSCrossRefGoogle Scholar
  34. Ogilvie, K.W., Chornay, D.J., Fritzenreiter, R.J., Hunsaker, F., Keller, J., Lobell, J., et al.: 1995, SWE, A comprehensive plasma instrument for the Wind spacecraft. Space Sci. Rev. 71, 55. DOI. ADSCrossRefGoogle Scholar
  35. Riley, P., Linker, J.A., Lionello, R., Mikić, Z., Odstrcil, D., Hidalgo, M.A., et al.: 2004, Fitting flux ropes to a global MHD solution: A comparison of techniques. J. Atmos. Solar-Terr. Phys. 66, 1321. DOI. ADSCrossRefGoogle Scholar
  36. Sonnerup, B.U.Ö., Scheible, M.: 1998, Minimum and maximum variance analysis. In: Analysis Methods for Multi-Spacecraft Data, 185. ADS. Google Scholar
  37. Sonnerup, B.U.Ö., Hasegawa, H., Teh, W.-L., Hau, L.-N.: 2006, Grad–Shafranov reconstruction: An overview. J. Geophys. Res. 111, A09204. DOI. ADSCrossRefGoogle Scholar
  38. Sturrock, P.A.: 1994, Plasma Physics: An Introduction to the Theory of Astrophysical, Geophysical and Laboratory Plasmas, Cambridge University Press, New York, 209. ADS. CrossRefGoogle Scholar
  39. van Ballegooijen, A.A., Mackay, D.H.: 2007, Model for the coupled evolution of subsurface and coronal magnetic fields in solar active regions. Astrophys. J. 659, 1713. DOI. ADSCrossRefGoogle Scholar
  40. Vemareddy, P., Möstl, C., Amerstorfer, T., Mishra, W., Farrugia, C.J., Leitner, M.: 2016, Comparison of magnetic properties in a magnetic cloud and its solar source on 2013 April 11 – 14. Astrophys. J. 828(12), 3139. DOI. CrossRefGoogle Scholar
  41. Wang, Y., Zhou, Z., Shen, C., Liu, R., Wang, S.: 2015, Investigating plasma motion of magnetic clouds at 1 AU through a velocity-modified cylindrical force-free flux rope model. J. Geophys. Res. 120, 1543. DOI. CrossRefGoogle Scholar
  42. Wang, Y., Zhuang, B., Hu, Q., Liu, R., Shen, C., Chi, Y.: 2016, On the twists of interplanetary magnetic flux ropes observed at 1 AU. J. Geophys. Res. 121, 9316. DOI. CrossRefGoogle Scholar
  43. Yu, W., Farrugia, C.J., Lugaz, N., Galvin, A.B., Kilpua, E.K.J., Kucharek, H., et al.: 2014, A statistical analysis of properties of small transients in the solar wind 2007 – 2009: STEREO and Wind observations. J. Geophys. Res. 119, 689. DOI. CrossRefGoogle Scholar
  44. Yu, W., Farrugia, C.J., Galvin, A.B., Lugaz, N., Luhmann, J.G., Simunac, K.D.C., et al.: 2016, Small solar wind transients at 1 AU: STEREO observations (2007 – 2014) and comparison with near-Earth wind results (1995 – 2014). J. Geophys. Res. 121, 5005. DOI. CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  • W. Yu
    • 1
  • C. J. Farrugia
    • 1
    Email author
  • N. Lugaz
    • 1
  • A. B. Galvin
    • 1
  • C. Möstl
    • 2
  • K. Paulson
    • 1
  • P. Vemareddy
    • 3
  1. 1.Space Science Center and Department of PhysicsUniversity of New HampshireDurhamUSA
  2. 2.Space Research InstituteAustrian Academy of SciencesGrazAustria
  3. 3.Indian Institute of AstrophysicsBengaluruIndia

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