Solar Physics

, Volume 289, Issue 12, pp 4453–4480

Validation and Benchmarking of a Practical Free Magnetic Energy and Relative Magnetic Helicity Budget Calculation in Solar Magnetic Structures

  • K. Moraitis
  • K. Tziotziou
  • M. K. Georgoulis
  • V. Archontis
Article

Abstract

In earlier works we introduced and tested a nonlinear force-free (NLFF) method designed to self-consistently calculate the coronal free magnetic energy and the relative magnetic helicity budgets of observed solar magnetic structures. In principle, the method requires only a single, photospheric or low-chromospheric, vector magnetogram of a quiet-Sun patch or an active region and performs calculations without three-dimensional magnetic and velocity-field information. In this work we strictly validate this method using three-dimensional coronal magnetic fields. Benchmarking employs both synthetic, three-dimensional magnetohydrodynamic simulations and nonlinear force-free field extrapolations of the active-region solar corona. Our time-efficient NLFF method provides budgets that differ from those of more demanding semi-analytical methods by a factor of approximately three, at most. This difference is expected to come from the physical concept and the construction of the method. Temporal correlations show more discrepancies that are, however, soundly improved for more complex, massive active regions, reaching correlation coefficients on the order of, or exceeding, 0.9. In conclusion, we argue that our NLFF method can be reliably used for a routine and fast calculation of the free magnetic energy and relative magnetic helicity budgets in targeted parts of the solar magnetized corona. As explained in this article and in previous works, this is an asset that can lead to valuable insight into the physics and triggering of solar eruptions.

Keywords

Helicity, magnetic Magnetic fields, corona Active regions, magnetic fields Magnetohydrodynamics 

References

  1. Arber, T.D., Haynes, M., Leake, J.E.: 2007, Emergence of a flux tube through a partially ionized solar atmosphere. Astrophys. J. 666, 541. DOI. ADS. ADSCrossRefGoogle Scholar
  2. Arber, T.D., Longbottom, A.W., Gerrard, C.L., Milne, A.M.: 2001, A staggered grid, Lagrangian–Eulerian remap code for 3D MHD simulations. J. Comput. Phys. 171, 151. ADS. ADSCrossRefMATHMathSciNetGoogle Scholar
  3. Archontis, V., Hood, A.W.: 2012, Magnetic flux emergence: a precursor of solar plasma expulsion. Astron. Astrophys. 537, A62. DOI. ADS. ADSCrossRefGoogle Scholar
  4. Archontis, V., Hood, A.W., Tsinganos, K.: 2013, The emergence of weakly twisted magnetic fields in the Sun. Astrophys. J. 778, 42. DOI. ADS. ADSCrossRefGoogle Scholar
  5. Archontis, V., Hood, A.W., Tsinganos, K.: 2014, Recurrent explosive eruptions and the “sigmoid-to-arcade” transformation in the Sun driven by dynamical magnetic flux emergence. Astrophys. J. Lett. 786, L21. DOI. ADS. ADSCrossRefGoogle Scholar
  6. Archontis, V., Moreno-Insertis, F., Galsgaard, K., Hood, A., O’Shea, E.: 2004, Emergence of magnetic flux from the convection zone into the corona. Astron. Astrophys. 426, 1047. DOI. ADS. ADSCrossRefGoogle Scholar
  7. Berger, M.A.: 1999, Introduction to magnetic helicity. Plasma Phys. Control. Fusion 41, 167. DOI. ADS. ADSCrossRefGoogle Scholar
  8. Berger, M.A., Field, G.B.: 1984, The topological properties of magnetic helicity. J. Fluid Mech. 147, 133. ADSCrossRefMathSciNetGoogle Scholar
  9. Borrero, J.M., Tomczyk, S., Kubo, M., Socas-Navarro, H., Schou, J., Couvidat, S., Bogart, R.: 2011, VFISV: very fast inversion of the Stokes vector for the Helioseismic and Magnetic Imager. Solar Phys. 273, 267. DOI. ADS. ADSCrossRefGoogle Scholar
  10. Chae, J.: 2001, Observational determination of the rate of magnetic helicity transport through the solar surface via the horizontal motion of field line footpoints. Astrophys. J. Lett. 560, L95. DOI. ADS. ADSCrossRefGoogle Scholar
  11. De Rosa, 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., Tadesse, T.: 2009, A critical assessment of nonlinear force-free field modeling of the solar corona for Active Region 10953. Astrophys. J. 696, 1780. DOI. ADS. ADSCrossRefGoogle Scholar
  12. Démoulin, P., Pariat, E., Berger, M.A.: 2006, Basic properties of mutual magnetic helicity. Solar Phys. 233, 3. DOI. ADS. ADSCrossRefGoogle Scholar
  13. DeVore, C.R.: 2000, Magnetic helicity generation by solar differential rotation. Astrophys. J. 539, 944. DOI. ADS. ADSCrossRefGoogle Scholar
  14. Finn, J.M., Antonsen, T.M.: 1985, Magnetic helicity: what is it and what is it good for? Comments Plasma Phys. Control. Fusion 9, 111. Google Scholar
  15. Freedman, M.H., Berger, M.A.: 1993, Combinatorial relaxation of magnetic fields. Geophys. Astrophys. Fluid Dyn. 73, 91. DOI. ADS. ADSCrossRefMathSciNetGoogle Scholar
  16. Gary, G.A., Hagyard, M.J.: 1990, Transformation of vector magnetograms and the problems associated with the effects of perspective and the azimuthal ambiguity. Solar Phys. 126, 21. DOI. ADS. ADSCrossRefGoogle Scholar
  17. Georgoulis, M.K.: 2005, A new technique for a routine azimuth disambiguation of solar vector magnetograms. Astrophys. J. Lett. 629, L69. DOI. ADS. ADSCrossRefGoogle Scholar
  18. Georgoulis, M.K., LaBonte, B.J.: 2007, Magnetic energy and helicity budgets in the active region solar corona. I. Linear force-free approximation. Astrophys. J. 671, 1034. DOI. ADS. ADSCrossRefGoogle Scholar
  19. Georgoulis, M.K., Rust, D.M.: 2007, Quantitative forecasting of major solar flares. Astrophys. J. Lett. 661, L109. DOI. ADS. ADSCrossRefGoogle Scholar
  20. Georgoulis, M.K., Tziotziou, K., Raouafi, N.-E.: 2012, Magnetic energy and helicity budgets in the active-region solar corona. II. Nonlinear force-free approximation. Astrophys. J. 759, 1. DOI. ADS. ADSCrossRefGoogle Scholar
  21. Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P.: 1983, Optimization by simulated annealing. Science 220, 671. DOI. ADS. ADSCrossRefMATHMathSciNetGoogle Scholar
  22. Kusano, K., Maeshiro, T., Yokoyama, T., Sakurai, T.: 2002, Measurement of magnetic helicity injection and free energy loading into the solar corona. Astrophys. J. 577, 501. DOI. ADS. ADSCrossRefGoogle Scholar
  23. Low, B.C.: 1994, Magnetohydrodynamic processes in the solar corona: flares, coronal mass ejections, and magnetic helicity. Phys. Plasmas 1, 1684. DOI. ADS. ADSCrossRefGoogle Scholar
  24. Low, B.C., Lou, Y.Q.: 1990, Modeling solar force-free magnetic fields. Astrophys. J. 352, 343. DOI. ADS. ADSCrossRefGoogle Scholar
  25. 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. DOI. ADS. ADSCrossRefGoogle Scholar
  26. Metcalf, T.R., De Rosa, M.L., Schrijver, C.J., Barnes, G., van Ballegooijen, A.A., Wiegelmann, T., Wheatland, M.S., Valori, G., McTtiernan, J.M.: 2008, Nonlinear force-free modeling of coronal magnetic fields. II. Modeling a filament arcade and simulated chromospheric and photospheric vector fields. Solar Phys. 247, 269. DOI. ADS. ADSCrossRefGoogle Scholar
  27. Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: 1953, Equation of state calculations by fast computing machines. J. Chem. Phys. 21, 1087. DOI. ADS. ADSCrossRefGoogle Scholar
  28. Pariat, E., Antiochos, S.K., DeVore, C.R.: 2009, A model for solar polar jets. Astrophys. J. 691, 61. DOI. ADS. ADSCrossRefGoogle Scholar
  29. Pesnell, W.D., Thompson, B.J., Chamberlin, P.C.: 2012, The Solar Dynamics Observatory (SDO). Solar Phys. 275, 3. DOI. ADS. ADSCrossRefGoogle Scholar
  30. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: 1992, Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd edn., Cambridge University Press, New York. ISBN 0521437164. ADS. Google Scholar
  31. Raouafi, N.-E., Georgoulis, M.K., Rust, D.M., Bernasconi, P.N.: 2010, Micro-sigmoids as progenitors of coronal jets: is eruptive activity self-similarly multi-scaled? Astrophys. J. 718, 981. DOI. ADS. ADSCrossRefGoogle Scholar
  32. Régnier, S., Amari, T., Canfield, R.C.: 2005, Self and mutual magnetic helicities in coronal magnetic configurations. Astron. Astrophys. 442, 345. DOI. ADS. ADSCrossRefGoogle Scholar
  33. Scherrer, P.H., Schou, J., Bush, R.I., Kosovichev, A.G., Bogart, R.S., Hoeksema, J.T., Liu, Y., Duvall, T.L., Zhao, J., Title, A.M., Schrijver, C.J., Tarbell, T.D., Tomczyk, S.: 2012, The Helioseismic and Magnetic Imager (HMI) investigation for the Solar Dynamics Observatory (SDO). Solar Phys. 275, 207. DOI. ADS. ADSCrossRefGoogle Scholar
  34. Schmidt, H.U.: 1964, On the observable effects of magnetic energy storage and release connected with solar flares, NASA SP-50, 107. ADS.
  35. Schrijver, C.J., De Rosa, 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. DOI. ADS. ADSCrossRefGoogle Scholar
  36. Sun, X., Hoeksema, J.T., Liu, Y., Wiegelmann, T., Hayashi, K., Chen, Q., Thalmann, J.: 2012, Evolution of magnetic field and energy in a major eruptive active region based on SDO/HMI observation. Astrophys. J. 748, 77. DOI. ADS. ADSCrossRefGoogle Scholar
  37. Swartztrauber, P.N., Sweet, R.A.: 1979, Algorithm 541: efficient Fortran subprograms for the solution of separable elliptic partial differential equations [D3]. ACM Trans. Math. Softw. 5(3), 352. DOI. CrossRefGoogle Scholar
  38. Tziotziou, K., Georgoulis, M.K., Raouafi, N.-E.: 2012, The magnetic energy–helicity diagram of solar active regions. Astrophys. J. Lett. 759, L4. DOI. ADS. ADSCrossRefGoogle Scholar
  39. Tziotziou, K., Georgoulis, M.K., Liu, Y.: 2013, Interpreting eruptive behavior in NOAA AR 11158 via the region’s magnetic energy and relative-helicity budgets. Astrophys. J. 772, 115. DOI. ADS. ADSCrossRefGoogle Scholar
  40. Tziotziou, K., Tsiropoula, G., Georgoulis, M.K., Kontogiannis, I.: 2014, Energy and helicity budgets of solar quiet regions. Astron. Astrophys. 564, A86. DOI. ADS. ADSCrossRefGoogle Scholar
  41. Valori, G., Démoulin, P., Pariat, E.: 2012, Comparing values of the relative magnetic helicity in finite volumes. Solar Phys. 278, 347. DOI. ADS. ADSCrossRefGoogle Scholar
  42. Valori, G., Démoulin, P., Pariat, E., Masson, S.: 2013, Accuracy of magnetic energy computations. Astron. Astrophys. 553, A38. DOI. ADS. ADSCrossRefGoogle Scholar
  43. Welsch, B.T., Abbett, W.P., De Rosa, 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. DOI. ADS. ADSCrossRefGoogle Scholar
  44. Wheatland, M.S., Sturrock, P.A., Roumeliotis, G.: 2000, An optimization approach to reconstructing force-free fields. Astrophys. J. 540, 1150. DOI. ADS. ADSCrossRefGoogle Scholar
  45. Wiegelmann, T.: 2004, Optimization code with weighting function for the reconstruction of coronal magnetic fields. Solar Phys. 219, 87. DOI. ADS. ADSCrossRefGoogle Scholar
  46. Wiegelmann, T., Inhester, B., Sakurai, T.: 2006, Preprocessing of vector magnetograph data for a nonlinear force-free magnetic field reconstruction. Solar Phys. 233, 215. DOI. ADS. ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • K. Moraitis
    • 1
  • K. Tziotziou
    • 1
  • M. K. Georgoulis
    • 1
  • V. Archontis
    • 2
  1. 1.Research Center for Astronomy and Applied Mathematics (RCAAM)Academy of AthensAthensGreece
  2. 2.School of Mathematics and StatisticsSt. Andrews UniversitySt. AndrewsUK

Personalised recommendations