Solar Physics

, Volume 278, Issue 2, pp 347–366 | Cite as

Comparing Values of the Relative Magnetic Helicity in Finite Volumes



Relative magnetic helicity, as a conserved quantity of ideal magnetohydrodynamics, has been highlighted as an important quantity to study in plasma physics. Due to its nonlocal nature, its estimation is not straightforward in both observational and numerical data. In this study we derive expressions for the practical computation of the gauge-independent relative magnetic helicity in three-dimensional finite domains. The derived expressions are easy to implement and rapid to compute. They are derived in Cartesian coordinates, but can be easily written in other coordinate systems. We apply our method to a numerical model of a force-free equilibrium containing a flux rope, and compare the results with those obtained employing known half-space equations. We find that our method requires a much smaller volume than half-space expressions to derive the full helicity content. We also prove that values of relative magnetic helicity of different magnetic fields can be compared with each other in the same sense as free-energy values can. Therefore, relative magnetic helicity can be meaningfully and directly compared between different datasets, such as those from different active regions, but also within the same dataset at different times. Typical applications of our formulae include the helicity computation in three-dimensional models of the solar atmosphere, e.g., coronal-field reconstructions by force-free extrapolation and discretized magnetic fields of numerical simulations.


Active regions, magnetic fields Magnetic field, photosphere, corona 


  1. Berger, M.A.: 1988, Astron. Astrophys. 201, 355. ADSMATHGoogle Scholar
  2. Berger, M.A.: 2003, In: Ferris-Mas, A., Nunez, M. (eds.) Adv. Nonlinear Dynamics, Taylor & Francis, London, 345. Google Scholar
  3. Berger, M.A., Field, G.B.: 1984, J. Fluid Mech. 147, 133. doi:10.1017/S0022112084002019. MathSciNetADSCrossRefGoogle Scholar
  4. Démoulin, P.: 2007, Adv. Space Res. 39, 1674. doi:10.1016/j.asr.2006.12.037. ADSCrossRefGoogle Scholar
  5. Démoulin, P., Pariat, E.: 2009, Adv. Space Res. 43, 1013. doi:10.1016/j.asr.2008.12.004. ADSCrossRefGoogle Scholar
  6. DeVore, C.R.: 2000, Astrophys. J. 539, 944. doi:10.1086/309274. ADSCrossRefGoogle Scholar
  7. Finn, J.H., Antonsen, T.M.J.: 1985, Comm. Plasma Phys. Control. Fusion 9, 111. Google Scholar
  8. Kliem, B., Török, T.: 2006, Phys. Rev. Lett. 96(25), 255002. doi:10.1103/PhysRevLett.96.255002. ADSCrossRefGoogle Scholar
  9. Kliem, B., Rust, S., Seehafer, N.: 2011, In: Bonanno, A., de Gouveia Dal Pino, E., Kosovichev, A.G. (eds.) Adv. Plasma Astrophys, IAU Symp. 274, Cambridge University Press, Cambridge, 125. doi:10.1017/S1743921311006715. Google Scholar
  10. Nakwacki, M.S., Dasso, S., Démoulin, P., Mandrini, C.H., Gulisano, A.M.: 2011, Astron. Astrophys. 535, A52. doi:10.1051/0004-6361/201015853. ADSCrossRefGoogle Scholar
  11. 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, Cambridge. MATHGoogle Scholar
  12. Roussev, I.I., Forbes, T.G., Gombosi, T.I., Sokolov, I.V., DeZeeuw, D.L., Birn, J.: 2003, Astrophys. J. Lett. 588, L45. doi:10.1086/375442. ADSCrossRefGoogle Scholar
  13. Rudenko, G.V., Myshyakov, I.I.: 2011, Solar Phys. 270, 165. doi:10.1007/s11207-011-9743-4. ADSCrossRefGoogle Scholar
  14. Seehafer, N., Kliem, B.: 2012, Solar Phys., in preparation. Google Scholar
  15. Schrijver, C.J., DeRosa, M.L., Metcalf, T., Barnes, G., Lites, B., Tarbell, T., McTiernan, J., Valori, G., Wiegelmann, T., Wheatland, M.S., Amari, T., Aulanier, G., Démoulin, P., Fuhrmann, M., Kusano, K., Régnier, S., Thalmann, J.K.: 2008, Astrophys. J. 675, 1637. doi:10.1086/527413. ADSCrossRefGoogle Scholar
  16. Thalmann, J.K., Inhester, B., Wiegelmann, T.: 2011, Solar Phys. 272, 243. doi:10.1007/s11207-011-9826-2. ADSCrossRefGoogle Scholar
  17. Titov, V.S.: 2007, Astrophys. J. 660, 863. doi:10.1086/512671. ADSCrossRefGoogle Scholar
  18. Titov, V.S., Démoulin, P.: 1999, Astron. Astrophys. 351, 707. ADSGoogle Scholar
  19. Török, T., Kliem, B.: 2005, Astrophys. J. Lett. 630, L97. doi:10.1086/462412. ADSCrossRefGoogle Scholar
  20. Török, T., Panasenco, O., Titov, V.S., Mikić, Z., Reeves, K.K., Velli, M., Linker, J.A., De Toma, G.: 2011, Astrophys. J. Lett. 739, L63. doi:10.1088/2041-8205/739/2/L63. ADSCrossRefGoogle Scholar
  21. Valori, G., Kliem, B., Török, T., Titov, V.S.: 2010, Astron. Astrophys. 519, A44. doi:10.1051/0004-6361/201014416. ADSCrossRefGoogle Scholar
  22. Valori, G., Green, L.M., Démoulin, P., Vargas Domínguez, S., van Driel-Gesztelyi, L., Wallace, A., Baker, D., Fuhrmann, M.: 2011, Solar Phys. doi:10.1007/s11207-011-9865-8. Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.LESIA, Observatoire de Paris, CNRS, UPMCUniversité Paris-DiderotMeudonFrance
  2. 2.Institute of Physics and AstronomyUniversity of PotsdamPotsdamGermany

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