Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-30T00:38:15.983Z Has data issue: false hasContentIssue false
  • English
  • Français

Relative field-line helicity in bounded domains

Part of: PLA Topological Methods in MHD, Fluids, and Plasmas

Published online by Cambridge University Press:  26 November 2018

Anthony R. Yeates Open the ORCID record for Anthony R. Yeates [Opens in a new window]  and
Marcus H. Page
Anthony R. Yeates*
Affiliation:
Department of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, UK
Marcus H. Page
Affiliation:
Department of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, UK
*
†Email address for correspondence: anthony.yeates@durham.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Models for astrophysical plasmas often have magnetic field lines that leave the boundary rather than closing within the computational domain. Thus, the relative magnetic helicity is frequently used in place of the usual magnetic helicity, so as to restore gauge invariance. We show how to decompose the relative helicity into a relative field-line helicity that is an ideal-magnetohydrodynamic invariant for each individual magnetic field line, and vanishes along any field line where the original field matches the reference field. Physically, this relative field-line helicity is a magnetic flux, whose specific definition depends on the gauge of the reference vector potential on the boundary. We propose a particular ‘minimal’ gauge that depends only on the reference field and minimises this boundary contribution, so as to reveal topological information about the original magnetic field. We illustrate the effect of different gauge choices using the Low–Lou and Titov–Démoulin models of solar active regions. Our numerical code to compute appropriate vector potentials and relative field-line helicity in Cartesian domains is open source and freely available.

Keywords

astrophysical plasmasspace plasma physics
Type
Research Article
Information
Journal of Plasma Physics , Volume 84 , Issue 6 , December 2018 , 775840602
Copyright
© Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arnold, V. I. 1986 The asymptotic Hopf invariant and its applications. Sel. Math. Sov. 5, 327–345.Google Scholar
Barnes, D. C. 1988 Mechanical injection of magnetic helicity. Phys. Fluids 31, 2214–2220.Google Scholar
Berger, M. A. 1984 Rigorous new limits on magnetic helicity dissipation in the solar corona. Geophys. Astrophys. Fluid Dyn. 30, 79–104.Google Scholar
Berger, M. A. 1988 An energy formula for nonlinear force-free magnetic fields. Astron. Astrophys. 201, 355–361.Google Scholar
Berger, M. A. & Field, G. B. 1984 The topological properties of magnetic helicity. J. Fluid Mech. 147, 133–148.Google Scholar
Berger, M. A. & Ruzmaikin, A. 2000 Rate of helicity production by solar rotation. J. Geophys. Res. 105, 10481–10490.Google Scholar
Browning, P. K. 1988 Helicity injection and relaxation in a solar-coronal magnetic loop with a free surface. J. Plasma Phys. 40, 263–280.Google Scholar
Cheung, M., Schüssler, M. & Moreno-Insertis, F. 2005 3D magneto-convection and flux emergence in the photosphere. In Chromospheric and Coronal Magnetic Fields (ed. Innes, D. E., Lagg, A. & Solanki, S. K.), Special Publication, vol. 596, p. 54.1. ESA.Google Scholar
Del Sordo, F., Candelaresi, S. & Brandenburg, A. 2010 Magnetic-field decay of three interlocked flux rings with zero linking number. Phys. Rev. E 81 (3), 036401.Google Scholar
Démoulin, P. & Pariat, E. 2009 Modelling and observations of photospheric magnetic helicity. Adv. Space Res. 43, 1013–1031.Google Scholar
DeVore, C. R. 2000 Magnetic helicity generation by solar differential rotation. Astrophys. J. 539, 944–953.Google Scholar
Dixon, A. M., Berger, M. A., Priest, E. R. & Browning, P. K. 1989 A generalization of the Woltjer minimum-energy principle. Astron. Astrophys. 225, 156–166.Google Scholar
Finn, J. M. & Antonsen, T. M. Jr 1985 Magnetic helicity: what is it and what is it good for? Comments Plasma Phys. Control. Fusion 9 (3), 111–126.Google Scholar
Hornig, G.2006 A universal magnetic helicity integral. ArXiv Astrophysics e-prints, arXiv:astro-ph/0606694.Google Scholar
Longcope, D. W. & Malanushenko, A. 2008 Defining and calculating self-helicity in coronal magnetic fields. Astrophys. J. 674, 1130–1143.Google Scholar
Low, B. C. & Lou, Y. Q. 1990 Modeling solar force-free magnetic fields. Astrophys. J. 352, 343–352.Google Scholar
Lowder, C. & Yeates, A. 2017 Magnetic flux rope identification and characterization from observationally driven solar coronal models. Astrophys. J. 846, 106.Google Scholar
Mackay, D. H., Green, L. M. & van Ballegooijen, A. 2011 Modeling the dispersal of an active region: quantifying energy input into the corona. Astrophys. J. 729, 97.Google Scholar
Magara, T. 2008 Emergence of a partially split flux tube into the solar atmosphere. Publ. Astron. Soc. Japan 60, 809–826.Google Scholar
Malanushenko, A., Longcope, D. W., Fan, Y. & Gibson, S. E. 2009 Additive self-helicity as a kink mode threshold. Astrophys. J. 702, 580–592.Google Scholar
Moffatt, H. K. 1969 The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35, 117–129.Google Scholar
Moraitis, K., Tziotziou, K., Georgoulis, M. K. & Archontis, V. 2014 Validation and benchmarking of a practical free magnetic energy and relative magnetic helicity budget calculation in solar magnetic structures. Solar Phys. 289, 4453–4480.Google Scholar
Pariat, E., Leake, J. E., Valori, G., Linton, M. G., Zuccarello, F. P. & Dalmasse, K. 2017 Relative magnetic helicity as a diagnostic of solar eruptivity. Astron. Astrophys. 601, A125.Google Scholar
Pariat, E., Valori, G., Démoulin, P. & Dalmasse, K. 2015 Testing magnetic helicity conservation in a solar-like active event. Astron. Astrophys. 580, A128.Google Scholar
Pontin, D. I., Wilmot-Smith, A. L., Hornig, G. & Galsgaard, K. 2011 Dynamics of braided coronal loops. II. Cascade to multiple small-scale reconnection events. Astron. Astrophys. 525, A57.Google Scholar
Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flannery, B. P. 1992 Numerical Recipes in FORTRAN. The Art of Scientific Computing. Cambridge University Press.Google Scholar
Prior, C. & Yeates, A. R. 2014 On the helicity of open magnetic fields. Astrophys. J. 787, 100.Google Scholar
Russell, A. J. B., Yeates, A. R., Hornig, G. & Wilmot-Smith, A. L. 2015 Evolution of field line helicity during magnetic reconnection. Phys. Plasmas 22 (3), 032106.Google Scholar
Rust, D. M. 1997 Helicity conservation. In Coronal Mass Ejections (ed. Crooker, N., Joselyn, J. A. & Feynman, J.), Geophysical Monograph Series, vol. 99, pp. 119–125. American Geophysical Union.Google Scholar
Sturrock, Z., Hood, A. W., Archontis, V. & McNeill, C. M. 2015 Sunspot rotation. I. A consequence of flux emergence. Astron. Astrophys. 582, A76.Google Scholar
Titov, V. S. & Démoulin, P. 1999 Basic topology of twisted magnetic configurations in solar flares. Astron. Astrophys. 351, 707–720.Google Scholar
Valori, G., Pariat, E., Anfinogentov, S., Chen, F., Georgoulis, M. K., Guo, Y., Liu, Y., Moraitis, K., Thalmann, J. K. & Yang, S. 2016 Magnetic helicity estimations in models and observations of the solar magnetic field. Part I: finite volume methods. Space Sci. Rev. 201, 147–200.Google Scholar
Woltjer, L. 1958 A theorem on force-free magnetic fields. Proc. Natl Acad. Sci. 44, 489–491.Google Scholar
Yang, S., Büchner, J., Santos, J. C. & Zhang, H. 2013 Evolution of relative magnetic helicity: method of computation and its application to a simulated solar corona above an active region. Solar Phys. 283, 369–382.Google Scholar
Yang, S. & Zhang, H. 2012 Large-scale magnetic helicity fluxes estimated from MDI magnetic synoptic charts over the solar cycle 23. Astrophys. J. 758, 61.Google Scholar
Yardley, S. L., Mackay, D. H. & Green, L. M. 2018 Simulating the coronal evolution of AR 11437 using SDO/HMI magnetograms. Astrophys. J. 852, 82.Google Scholar
Yeates, A. R. & Hornig, G. 2011 A generalized flux function for three-dimensional magnetic reconnection. Phys. Plasmas 18 (10), 102118.Google Scholar
Yeates, A. R. & Hornig, G. 2013 Unique topological characterization of braided magnetic fields. Phys. Plasmas 20 (1), 012102.Google Scholar
Yeates, A. R. & Hornig, G. 2014 A complete topological invariant for braided magnetic fields. J. Phys.: Conf. Ser. 544, 012002.Google Scholar
Yeates, A. R. & Hornig, G. 2016 The global distribution of magnetic helicity in the solar corona. Astron. Astrophys. 594, A98.Google Scholar
Yee, K. 1966 Numerical solution of inital boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans. Antennas Propag. 14, 302–307.Google Scholar