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Solar Physics

, Volume 290, Issue 2, pp 437–466 | Cite as

A Minimum Energy Fit Method to Reconstruct Photospheric Velocity and Magnetic Diffusivity in Active Regions from Observed Magnetograms and Dopplergrams

  • Benoit Tremblay
  • Alain VincentEmail author
Article

Abstract

We introduce MEF-R, a generalization of the minimum energy fit (MEF; Longcope, Astrophys. J. 612, 1181, 2004) to a non-ideal (resistive) gas. The new technique requires both vector magnetograms and Doppler velocities as input fields. However, in the case of active regions observed only with the Michelson–Doppler Imager (MDI) onboard the Solar and Heliospheric Observatory (SOHO) such as AR 9077, we have only access to line-of-sight magnetograms. We reconstruct two-dimensional maps of the magnetic diffusivity η(x,y) together with velocity components v x (x,y), v y (x,y), and v z (x,y) under the linear force-free magnetic field approximation. Computed maps for v z (x,y) very well match the Doppler velocities v r (x,y). We find the average value 〈η〉≈108 m2 s−1 with a standard deviation of ≈ 1010 m2 s−1. Such high values of η(x,y) are to be expected at some places since our magnetic diffusivity is actually eddy-diffusivity. Inside AR 9077, the maps of η(x,y) do not resemble closely the maps from classical models of the magnetic diffusivity, but they are closer to η as a function of temperature than to η as a function of electric current density.

Keywords

Eddy-diffusivity Magnetograms Minimum energy fit Photosphere Velocity fields 

Notes

Acknowledgements

Alain Vincent is supported through NSERC Individual Research Grant. Computations have been done with a modified version of MEF (Longcope 2004). We have used the IDL graphics system and SAO Image DS9 from the Smithsonian Astrophysical Observatory. In this study, we have used SOHO/MDI data archives ( http://soi.stanford.edu/data/ ) as well as Solar Monitor ( http://www.solarmonitor.org/ ). We thank Frédérique Baron, Léonie Petitclerc, and Benoît Rolland for their initial contributions to data processing. Finally, we thank the anonymous reviewer for her/his constructive remarks.

References

  1. Abbett, W.P.: 2007, The magnetic connection between the convection zone and corona in the quiet Sun. Astrophys. J. 665, 1469.ADSCrossRefGoogle Scholar
  2. Abbett, W.P., Fisher, G.H.: 2012, Radiative cooling in MHD models of the quiet Sun convection zone and corona. Solar Phys. 277, 3.ADSCrossRefGoogle Scholar
  3. Abramenko, V.I., Carbone, V., Yurchyshyn, V., Goode, P.R., Stein, R.F., Lepreti, F., Capparelli, V., Vecchio, A.: 2011, Turbulent diffusion in the photosphere as derived from photospheric bright point motion. Astrophys. J. 743, 133.ADSCrossRefGoogle Scholar
  4. Alexander, D.: 2006, An introduction to the pre-CME corona. Space Sci. Rev. 123, 81.ADSCrossRefGoogle Scholar
  5. Alfvén, H.: 1942, On the existence of electromagnetic-hydromagnetic waves. Ark. Mat. Astron. Fys. 29B, 1.ADSGoogle Scholar
  6. Archontis, V., Hood, A.W.: 2012, Magnetic flux emergence: a precursor of solar plasma expulsion. Astron. Astrophys. 537, A62.ADSCrossRefGoogle Scholar
  7. Aschwanden, M.J.: 2008, Solar flare physics enlivened by TRACE and RHESSI. Solar Phys. 29, 115.Google Scholar
  8. Babin, A.N., Koval, A.N.: 2007, Ni i 6768 Å line profile variations during a solar flare and their effect on the SOHO/MDI magnetic field measurement. Bull. Crimean Astrophys. Obs. 103, 63.ADSCrossRefGoogle Scholar
  9. Bellan, P.M.: 2000, Spheromaks. A Practical Application of Magnetohydrodynamic Dynamos and Plasma Self-Organization, Imperial College Press, London, 71.CrossRefGoogle Scholar
  10. Bellot Rubio, L.R., Rodriguez Hidalgo, I., Collados, M., Khomenko, E., Ruiz Cobo, B.: 2001, Observation of convective collapse and upward-moving shocks in the quiet Sun. Astrophys. J. 560, 1010.ADSCrossRefGoogle Scholar
  11. Bommier, V., Degl’Innocenti, E.L., Schmieder, B., Gelly, B.: 2011, Vector magnetic field and vector current density in and around the δ-spot NOAA 10808. In: Choudhary, D.P., Strassmeier, K.G. (eds.) The Physics of Sun and Star Spots, IAU Symp. 273, 338.Google Scholar
  12. Brownstein, K.R.: 1994, Nonexistence of spatially bounded force-free magnetic fields: A scaling point of view. IEEE Trans. Plasma Sci. 22, 275.ADSCrossRefGoogle Scholar
  13. Büchner, J., Nikutowski, B., Otto, A.: 2004, Magnetic coupling of photosphere and corona: MHD simulation for multi-wavelength observations. In: Stepanov, A.V., Benevolenskaya, E.E., Kosovichev, A.G. (eds.) Multi-Wavelength Investigations of Solar Activity, IAU Symp. 223, 353.Google Scholar
  14. Cameron, R., Vögler, A., Schüssler, M.: 2011, Decay of a simulated mixed-polarity magnetic field in the solar surface layers. Astron. Astrophys. 533, A86.CrossRefGoogle Scholar
  15. Cao, W., Goode, P.R., Ahn, K., Gorceix, N., Schmidt, W., Lin, H.: 2012, NIRIS – The second generation near-infrared imaging spectro-polarimeter for the 1.6 meter New Solar Telescope. In: Rimmele, T., Tritschler, A., Wöger, F., Collados, V., Socas-Navarro, H., Schlichenmaier, R., Carlsson, M., Berger, T., Cadavid, A., Gilbert, P., Goode, P., Knölker, M. (eds.) 2nd ATST-EAST Workshop in Solar Physics: Magnetic Fields from the Photosphere to the Corona, ASP Conf. Ser. 463, 291.Google Scholar
  16. Chae, J., Litvinenko, Y.E., Sakurai, T.: 2008, Determination of magnetic diffusivity from high-resolution solar magnetograms. Astrophys. J. 683, 1153.ADSCrossRefGoogle Scholar
  17. Chae, J., Sakurai, T.: 2008, A test of three optical flow techniques-LCT, DAVE, and NAVE. Astrophys. J. 689, 593.ADSCrossRefGoogle Scholar
  18. Chen, P.F., Shibata, K.: 2000, An emerging flux trigger mechanism for coronal mass ejections. Astrophys. J. 545, 524.ADSCrossRefGoogle Scholar
  19. Chen, F., Peter, H., Bingert, S., Cheung, M.C.M.: 2014, A model for the formation of the active region corona driven by magnetic flux emergence. arXiv.
  20. Chertok, I.M., Grechnev, V.V.: 2005, Large-scale activity in the Bastille day 2000 solar event. Solar Phys. 229, 95.ADSCrossRefGoogle Scholar
  21. Contopoulos, I., Kalapotharakos, C., Georgoulis, M.K.: 2011, Nonlinear force-free reconstruction of the global solar magnetic field: methodology. Solar Phys. 269, 351.ADSCrossRefGoogle Scholar
  22. Deloach, A.C., Hagyard, M.J., Rabin, D., Moore, R.L., Smith, B.J. Jr., West, E.A.: 1984, Photospheric electric current and transition region brightness within an active region. Solar Phys. 91, 235.ADSCrossRefGoogle Scholar
  23. DeRosa, M.L., Schrijver, C.J., Barnes, G., Leka, K.D., Lites, B.W., Aschwanden, M.J., Amari, T., Canou, A., McTiernan, J.M., Régnier, S., Thalmann, J.K., Valori, G., Wheatland, M.S., Wiegelmann, T., Cheung, M.C.M., Conlon, P.A., Fuhrmann, M., Inhester, B., Tadess, T.: 2009, A critical assessment of nonlinear force-free field modeling of the solar corona for active region 10953. Astrophys. J. 696, 1780.ADSCrossRefGoogle Scholar
  24. Falchi, A., Mauas, P.J.D.: 2002, Chromospheric models of a solar flare including velocity fields. Astron. Astrophys. 387, 678.ADSCrossRefGoogle Scholar
  25. Fan, Y., Zweibel, E.G., Linton, M.G., Fisher, G.H.: 1999, The rise of kink-unstable magnetic flux tubes and the origin of δ-configuration sunspots. Astrophys. J. 521, 460.ADSCrossRefGoogle Scholar
  26. Fan, Y.L., Wang, H.N., He, H., Zhu, X.S.: 2011, Study of the Poynting flux in active region 10930 using data-driven magnetohydrodynamic simulation. Astrophys. J. 737, 39 (9 pp.).ADSCrossRefGoogle Scholar
  27. Fisher, G.H., Welsch, B.T., Abbet, W.P.: 2012b, Can we determine electric fields and Poynting fluxes from vector magnetograms and Doppler measurements? Solar Phys. 277, 153.ADSCrossRefGoogle Scholar
  28. Fisher, G.H., Welsch, B.T., Abbett, W.P., Bercik, D.J.: 2010, Estimating electric fields from vector magnetogram sequences. Astrophys. J. 715, 242.ADSCrossRefGoogle Scholar
  29. Fisher, G.H., Bercik, D.J., Welsch, B.T., Hudson, H.S.: 2012a, Global forces in eruptive solar flares: the Lorentz force acting on the solar atmosphere and the solar interior. Solar Phys. 277, 56.ADSGoogle Scholar
  30. Fuhrmann, M., Seehafer, N., Valori, G., Wiegelmann, T.: 2011, A comparison of preprocessing methods for solar force-free magnetic field extrapolation. Astron. Astrophys. 526, A70.ADSCrossRefGoogle Scholar
  31. Garcia-Martinez, P.L.: 2012, Dynamics of magnetic relaxation in Spheromaks. In: Zheng, L. (ed.) Topics in Magnetohydrodynamics, InTech, Rijeka, 85.Google Scholar
  32. Georgoulis, M.K., LaBonte, B.J.: 2005, Reconstruction of an inductive velocity field vector from Doppler motions and a pair of solar vector magnetograms. Astrophys. J. 636, 475.ADSCrossRefGoogle Scholar
  33. Griffiths, D.J.: 2007 Introduction to Electrodynamics, Harlow, Essex, 345, 555.Google Scholar
  34. Hao, J., Zhang, M.: 2011, Hemispheric helicity trend for solar cycle 24. Astrophys. J. Lett. 733, L27.ADSCrossRefGoogle Scholar
  35. Harra, L.K., Archontis, V., Pedram, E., Hood, A.W., Shelton, D.L., van Driel-Gesztelyi, L.: 2012, The creation of outflowing plasma in the corona at emerging flux regions: Comparing observations and simulations. Solar Phys. 278, 47.ADSCrossRefGoogle Scholar
  36. Hathaway, D.H., Beck, J.G., Bogart, R.S., Bachmann, K.T., Khatri, G., Petitto, J.M., Han, S., Raymond, J.: 2000, The photospheric convection spectrum. Solar Phys. 193, 299.ADSCrossRefGoogle Scholar
  37. Heggland, L., De Pontieu, B., Hansteen, V.H.: 2009, Observational signatures of simulated reconnection events in the solar chromosphere and transition region. Astrophys. J. 702, 1.ADSCrossRefGoogle Scholar
  38. Ilonidis, S., Zhao, J., Kosovichev, A.: 2011, Detection of emerging sunspot regions in the solar interior. Science 333, 993.ADSCrossRefGoogle Scholar
  39. Keller, C.U., Solis Team: 2001, The SOLIS Vector-Spectromagnetograph (VSM). In: Sigwarth, M. (ed.) Advanced Solar Polarimetry – Theory, Observation, and Instrumentation, ASP Conf. Ser. 236, 16.Google Scholar
  40. Klimas, A.J., Uritsky, V.M., Vassiliadis, D., Baker, D.N.: 2004, Reconnection and scale-free avalanching in a driven current-sheet model. J. Geophys. Res. 109, 2218.CrossRefGoogle Scholar
  41. Kosugi, T., Matsuzaki, K., Sakao, T., Shimizu, T., Sone, Y., Tachikawa, S., Hashimoto, T., Minesugi, K., Ohnishi, A., Yamada, T., Tsuneta, S., Hara, H., Ichimoto, K., Suematsu, Y., Shimojo, M., Watanabe, T., Shimada, S., Davis, J.M., Hill, L.D., Owens, J.K., Title, A.M., Culhane, J.L., Harra, L.K., Doschek, G.A., Golub, L.: 2007, The Hinode (Solar-B) mission: An overview. Solar Phys. 243, 3.ADSCrossRefGoogle Scholar
  42. Krall, K.R., Smith, J.B. Jr., Hagyard, M.J., West, E.A., Cummings, N.P.: 1982, Vector magnetic field evolution, energy storage, and associated photospheric velocity shear within a flare-productive active region. Solar Phys. 79, 59.ADSCrossRefGoogle Scholar
  43. Kumar, P., Kumar, N., Uddin, W.: 2011, Reconnection in photospheric-chromospheric current sheet and coronal heating. Plasma Phys. Rep. 37, 161.ADSCrossRefGoogle Scholar
  44. Lantz, S.R., Fan, Y.: 1999, Anelastic magnetohydrodynamic equations for modeling solar and stellar convection zones. Astron. Astrophys. Suppl. 121, 247.ADSGoogle Scholar
  45. Liu, Y., Schuck, P.W.: 2012, Magnetic energy and helicity in two emerging active regions in the Sun. Astrophys. J. 761, 105.ADSCrossRefGoogle Scholar
  46. Liu, Y., Zhang, H.: 2001, Relationship between magnetic field evolution and major flare event on July 14, 2000. Astron. Astrophys. 372, 1019.ADSCrossRefGoogle Scholar
  47. Liu, Y., Zhao, J., Schuck, P.W.: 2013, Horizontal flows in the photosphere and subphotosphere of two active regions. Solar Phys. 287, 279.ADSCrossRefGoogle Scholar
  48. Longcope, D.W.: 2004, Inferring a photospheric velocity field from a sequence of vector magnetograms: The minimum energy fit. Astrophys. J. 612, 1181.ADSCrossRefGoogle Scholar
  49. Lu, E.T.: 1995, Avalanches in continuum driven dissipative systems. Phys. Rev. Lett. 74, 2511.ADSCrossRefGoogle Scholar
  50. Metcalf, T.R., Jiao, L., McClymont, A.N., Caneld, R.C., Uitenbroek, H.: 1995, Is the solar chromospheric magnetic field force-free? Astrophys. J. 439, 474.ADSCrossRefGoogle Scholar
  51. Metcalf, T.R., Leka, K.D., Barnes, G., Lites, B.W., Georgoulis, M.K., Pevtsov, A.A., Balasubramaniam, K.S., Gary, G.A., Jing, J., Li, J., Liu, Y., Wang, H.N., Abramenko, V., Yurchyshyn, V., Moon, Y.-J.: 2006, An overview of existing algorithms for resolving the 180 ambiguity in vector magnetic fields: Quantitative tests with synthetic data. Solar Phys. 237, 267.ADSCrossRefGoogle Scholar
  52. Miyagoshi, T., Yokoyama, T.: 2004, Magnetohydrodynamic simulation of solar coronal chromospheric evaporation jets caused by magnetic reconnection associated with magnetic flux emergence. Astrophys. J. 614, 1042.ADSCrossRefGoogle Scholar
  53. Moon, Y.-J., Choe, G.S., Yun, H.S., Park, Y.D., Mickey, D.L.: 2002, Force-freeness of solar magnetic fields in the photosphere. Astrophys. J. 568, 422.ADSCrossRefGoogle Scholar
  54. Nakagawa, Y., Raadu, M.A.: 1972, On practical representation of the magnetic field. Solar Phys. 25, 127.ADSCrossRefGoogle Scholar
  55. Otto, A.: 2001, Geospace Environment Modeling (GEM) magnetic reconnection challenge: MHD and Hall MHD – Constant and current dependent resistivity models. J. Geophys. Res. 106, 3751.ADSCrossRefGoogle Scholar
  56. Pandey, B.P., Wardle, M.: 2012, Hall instability of solar flux tubes in the presence of shear flows. Mon. Not. Roy. Astron. Soc. 426, 1436.ADSCrossRefGoogle Scholar
  57. Pandey, B.P., Wardle, M.: 2013, Magnetic diffusion driven shear instability of solar flux tubes. Mon. Not. Roy. Astron. Soc. 431, 570.ADSCrossRefGoogle Scholar
  58. Petrie, G.J.D., Sudol, J.J.: 2010, Abrupt longitudinal magnetic field changes in flaring active regions. Astrophys. J. 724, 1218.ADSCrossRefGoogle Scholar
  59. Petrovay, K.: 1994, Theory of passive magnetic field transport. In: Rutten, R.J., Schrijver, C.J. (eds.) Solar Surface Magnetism, Kluwer Academic, Dordrecht, 415.CrossRefGoogle Scholar
  60. Potts, H.E., Diver, D.A.: 2009, A repository of precision flat fields for high-resolution MDI continuum data. Solar Phys. 258, 343.ADSCrossRefGoogle Scholar
  61. Ravindra, B., Longcope, D.W., Abbett, W.P.: 2008, Inferring photospheric velocity fields using combination of minimum energy fit, local correlation tracking and Doppler velocity. Astrophys. J. 677, 751.ADSCrossRefGoogle Scholar
  62. Régnier, S., Priest, E.R.: 2007, Free magnetic energy in solar active regions above the minimum-energy relaxed state. Astrophys. J. Lett. 669, L53.ADSCrossRefGoogle Scholar
  63. Sakurai, T., Hagino, M.: 2003, Magnetic helicity of solar active regions and its implications. J. Korean Astron. Soc. 36, 7.CrossRefGoogle Scholar
  64. Santos, J.C., Büchner, J., Zhang, H.: 2008, Inferring plasma flow velocities from photospheric vector magnetic field observations for the investigation of flare onsets. Adv. Space Res. 42, 812.ADSCrossRefGoogle Scholar
  65. Scherrer, P.H., Bogart, R.S., Bush, R.I., Hoeksema, J.T., Kosovichev, A.G., Schou, J., Rosenberg, W., Springer, L., Tarbell, T.D., Title, A., Wolfson, C.J., Zayer, I., MDI Engineering Team: 1995, The solar oscillations investigation Michelson Doppler imager. Solar Phys. 162, 129.ADSCrossRefGoogle Scholar
  66. Schou, J., Scherrer, P.H., Bush, R.I., Wachter, R., Couvidat, S., Rabello-Soares, M.C., Bogart, R.S., Hoeksema, J.T., Liu, Y., Duvall, T.L. Jr., Akin, D.J., Allard, B.A., Miles, J.W., Rairden, R., Shine, R.A., Tarbell, T.D., Title, A.M., Wolfson, C.J., Elmore, D.F., Norton, A.A., Tomczyk, S.: 2012, Design and ground calibration of the helioseismic and magnetic imager (HMI) instrument on the Solar Dynamics Observatory (SDO). Solar Phys. 275, 229.ADSCrossRefGoogle Scholar
  67. Schrijver, C.J., Derosa, M.L., Metcalf, T.R., Liu, Y., Mctiernan, J., Régnier, S., Valori, G., Wheatland, M.S., Wiegelmann, T.: 2006, Nonlinear force-free modeling of coronal magnetic fields part I: A quantitative comparison of methods. Solar Phys. 235, 161.ADSCrossRefGoogle Scholar
  68. Schuck, P.W.: 2005, Local correlation tracking and the magnetic induction equation. Astrophys. J. Lett. 632, L53.ADSCrossRefGoogle Scholar
  69. Schuck, P.W.: 2008, Tracking vector magnetograms with the magnetic induction equation. Astrophys. J. 683, 1134.ADSCrossRefGoogle Scholar
  70. Schuck, P.W.: 2010, The photospheric energy and helicity budgets of the flux-injection hypothesis. Astrophys. J. 713, 1.CrossRefGoogle Scholar
  71. Seehafer, N., Fuhrmann, M., Valori, G., Kliem, B.: 2007, Force-free magnetic fields in the solar atmosphere. Astron. Nachr. 328, 1166.ADSCrossRefzbMATHGoogle Scholar
  72. Simon, G.W., Weiss, N.O.: 1997, Kinematic modeling of vortices in the solar photosphere. Astrophys. J. 489, 960.ADSCrossRefGoogle Scholar
  73. Singh, K.A.P., Shibata, K., Nishizuka, N., Isobe, H.: 2011, Chromospheric anemone jets and magnetic reconnection in partially ionized solar atmosphere. Phys. Plasmas 18, 111210.ADSCrossRefGoogle Scholar
  74. Smagorinsky, J.: 1963, General circulation experiments with the primitive equations. Mon. Weather Rev. 91, 99.ADSCrossRefGoogle Scholar
  75. Snodgrass, H.B.: 1984, Separation of large-scale photospheric Doppler patterns. Solar Phys. 94, 13.ADSCrossRefGoogle Scholar
  76. Snodgrass, H., Ulrich, R.: 1990, Rotation of Doppler features in the solar photosphere. Astrophys. J. 351, 309.ADSCrossRefGoogle Scholar
  77. Solanki, S.K., Walther, U., Livingston, W.: 1993, Infrared lines as probes of solar magnetic features: VI. The thermal-magnetic relation and Wilson depression of a sunspot. Astron. Astrophys. 277, 639.ADSGoogle Scholar
  78. Somov, B.V.: 2007, The Bastille Day 2000 flare. Plasma Astrophys. 341, 468.ADSCrossRefGoogle Scholar
  79. Spangler, S.R.: 2009, Joule heating and anomalous resistivity in the solar corona. Nonlinear Proc. Geophys. 16, 443.ADSCrossRefGoogle Scholar
  80. Spitzer, L. Jr.: 1962, Physics of Fully Ionized Gases, Interscience, New York, 136.Google Scholar
  81. Steffen, M.: 2009, Solar photosphere and chromosphere. In: Trümper, J.E. (ed.) The Landolt-Börnstein Database, Springer, Berlin, 39.Google Scholar
  82. Sudol, J.J., Harvey, J.W.: 2005, Longitudinal magnetic field changes accompanying solar flares. Astrophys. J. 635, 647.ADSCrossRefGoogle Scholar
  83. Taylor, J.: 1974, Relaxation of toroidal plasma and generation of reverse magnetic fields. Phys. Rev. Lett. 33, 1139.ADSCrossRefGoogle Scholar
  84. Theobald, M.L., Fox, P.A., Sofia, S.: 1994, A subgridscale resistivity for magnetohydrodynamics. Phys. Plasmas 1, 3016.ADSCrossRefGoogle Scholar
  85. Tikhonov, A.N.: 1963, Solution of incorrectly formulated problems and regularization method. Sov. Math. Dokl. 4, 1035.Google Scholar
  86. Tiwari, S.K.: 2011, Are the photospheric sunspots magnetically force-free in nature? In: Choudhary, D.P., Strassmeier, K.G. (eds.) The Physics of Sun and Star Spots, IAU Symp. 273, 1.Google Scholar
  87. Uritsky, V.M., Klimas, A.J.: 2005, Hysteresis-controlled instability waves in a scale-free driven current sheet model. Nonlinear Proc. Geophys. 12, 827.ADSCrossRefGoogle Scholar
  88. Veronig, A.M., Karlick, M., Vrsnak, B., Temmer, M., Magdalenic, J., Dennis, B.R., Otruba, W., Pötzi, W.: 2006, X-ray sources and magnetic reconnection in the X3.9 flare of 2003 November 3. Astron. Astrophys. 446, 675.ADSCrossRefGoogle Scholar
  89. Vincent, A., Charbonneau, P., Dubé, C.: 2012, Numerical simulation of a solar active region. I: Bastille Day flare. Solar Phys. 278, 367.ADSCrossRefGoogle Scholar
  90. Wang, H., Liu, Ch.: 2010, Observational evidence of back reaction on the solar surface associated with coronal magnetic restructuring in solar eruptions. Astrophys. J. Lett. 716, L195.ADSCrossRefGoogle Scholar
  91. Welsch, B.T., Fisher, G.H., Abbet, W.P., Regnier, S.: 2004, ILCT: Recovering photospheric velocities from magnetograms by combining the induction equation with local correlation tracking. Astrophys. J. 610, 1148.ADSCrossRefGoogle Scholar
  92. Welsch, B.T., Abbett, W.P., DeRosa, M.L., Fisher, G.H., Georgoulis, M.K., Kusano, K., Longcope, D.W., Ravindra, B., Schuck, P.W.: 2007, Tests and comparisons of velocity inversion techniques. Astrophys. J. 670, 1434.ADSCrossRefGoogle Scholar
  93. Welsch, B.T., Li, Y., Schuck, P.W., Fisher, G.H.: 2009, What is the relationship between photospheric flow fields and solar flares? Astrophys. J. 705, 821.ADSCrossRefGoogle Scholar
  94. Woltjer, L.: 1958, A theorem on force-free magnetic fields. Proc. Natl. Acad. Sci. USA 44, 489.ADSCrossRefzbMATHMathSciNetGoogle Scholar
  95. Wu, S.T., Wang, A.H., Plunkett, S.P., Michels, D.J.: 2000, Evolution of global-scale coronal magnetic field due to magnetic reconnection: The formation of the observed blob motion in the coronal streamer belt. Astrophys. J. 545, 1101.ADSCrossRefGoogle Scholar
  96. Zheligovsky, V.A., Podvigina, O.M., Frisch, U.: 2001, Dynamo effect in parity-invariant flow with large and moderate separation of scales. Geophys. Astrophys. Fluid Dyn. 95, 227.ADSCrossRefMathSciNetGoogle Scholar

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© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Département de PhysiqueUniversité de MontréalMontréalCanada

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