Abstract
We derive the magnetic helicity for configurations formed by flux tubes contained fully or only partially in the spatial domain considered (called closed and open configurations, respectively). In both cases, magnetic helicity is computed as the sum of mutual helicity over all possible pairs of magnetic flux tubes weighted by their magnetic fluxes. We emphasize that these mutual helicities have properties which are not those of mutual inductances in classical circuit theory. For closed configurations, the mutual helicity of two closed flux tubes is their relative winding around each other (known as the Gauss linkage number). For open configurations, the magnetic helicity is derived directly from the geometry of the interlaced flux tubes so it can be computed without reference to a ground state (such as a potential field). We derive the explicit expression in the case of a planar and spherical boundary. The magnetic helicity has two parts. The first one is given only by the relative positions of the flux tubes on the boundary. It is the only part if all flux tubes are arch-shaped. The second part counts the integer number of turns each pair of flux tubes wind about each other. This provides a general method to compute the magnetic helicity with discrete or continuous distributions of magnetic field. The method sets closed and open configurations on an equal level within the same theoretical framework.
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References
Berger, M. A.: 1984, Geophys. Astrophys. Fluid Dyn. 30, 79
Berger, M. A.: 1986, Geophys. Astrophys. Fluid Dyn. 34, 265
Berger, M. A.: 2003, in A. Ferriz-Mas and M. Núñez (eds.), Advances in Nonlinear Dynamos, Gordon and Breach, London, p. 345.
Berger, M. A. and Field, G. B.: 1984, J. Fluid. Mech. 147, 133
Berger, M. A.: 1999, Magnetic Helicity in Space and Laboratory Plasmas, Geophys. Monograph, Vol. 111, American Geophysical Union, p. 1.
Brandenburg, A.: 2001, Astrophys. J. 550, 824
Brown, M. R., Canfield, R. C., and Pevtsov, A. A. (eds.): 1999, Magnetic Helicity in Space and Laboratory Plasmas, Geophys. Monograph, Vol. 111, American Geophysical Union.
Chae, J., Moon, Y., and Park, Y.: 2004, Solar Phys. 223, 39
Conway, J. B.: 1975, Functions of One Complex Variable, Springer-Verlag, New York.
Dasso, S., Mandrini, C. H., Démoulin, P., and Farrugia, C. J.: 2003, J. Geophys. Res. 108, 1362 (SSH 3,1 doi: 10.1029/2003JA009942).
Démoulin, P., Mandrini, C. H., van Driel-Gesztelyi, L., Thompson, B. J., Plunkett, S., Kovári, Zs., Aulanier, G., and Young, A.: 2002, Astron. Astrophys. 382, 650
Elsasser, W. M.: 1956, Rev. Mod. Phys. 28, 135.
Finn, J. H. and Antonsen, T. M.: 1985, Comments Plasma Phys. Contr. Fusion 9, 111.
Green, L. M., López Fuentes, M. C., Mandrini, C. H., Démoulin, P., van Driel-Gesztelyi, L., and Culhane, J. L.: 2002, Solar Phys. 208, 43
Heyvaerts, J. and Priest, E. R.: 1984, Astron. Astrophys. 137, 63
Kimura, Y. and Okamoto, H.: 1987, J. Phys. Soc. Japan 56, 2024.
Kusano, K., Maeshiro, T., Yokoyama, T., and Sakurai, T.: 2002, Astrophys. J. 577, 501
Kusano, K., Maeshiro, T., Yokoyama, T., and Sakurai, T.: 2004, Astrophys. J. 610, 537
Low, B. C.: 1997, in N. Crooker, J. A. Joselyn, and J. Feynman (eds.), Coronal Mass Ejections, Geophys. Monograph, Vol. 99, American Geophysical Union, p. 39.
Mandrini, C. H., Pohjolainen, S., Dasso, S., Green, L. M, Démoulin, P., van Driel-Gesztelyi, L., Copperwheat, C., and Foley, C.: 2005, Astron. Astrophys. 434, 725
Melrose, D.: 2004, Solar Phys. 221, 121
Moffatt, H. K.: 1969, J. Fluid Mech. 35, 117.
Nindos, A., Zhang, J., and Zhang, H.: 2003, Astrophys. J. 594, 1033
Pariat, E., Démoulin, P., and Berger, M. A.: 2005, Astron. Astrophys. 439, 1191
Rust, D. M.: 1994, Geophys. Res. Lett. 21, 241
Wright, A. N. and Berger, M. A.: 1988, J. Geophys. Res. 94A2, 1295
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Demoulin, P., Pariat, E. & Berger, M.A. Basic Properties of Mutual Magnetic Helicity. Sol Phys 233, 3–27 (2006). https://doi.org/10.1007/s11207-006-0010-z
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DOI: https://doi.org/10.1007/s11207-006-0010-z