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The topological properties of magnetic helicity

Published online by Cambridge University Press:  20 April 2006

Mitchell A. Berger  and
George B. Field
Mitchell A. Berger
Affiliation:
Astronomy Department, Harvard University
George B. Field
Affiliation:
Astronomy Department, Harvard University
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Abstract

The relation of magnetic helicity to the topological structure of field lines is discussed. If space is divided into a collection of flux tubes, magnetic helicity arises from internal structure within a flux tube, such as twist and kinking, and external relations between flux tubes, i.e. linking and knotting. The concepts of twist number and writhing number are introduced from the mathematical-biology literature to describe the contributions to helicity from twist about the axis of a flux tube, and from the structure of the axes themselves.

There exists no absolute measure of the helicity within a subvolume of space if that subvolume is not bounded by a magnetic surface. However, a topologically meaningful and gauge-invariant relative measure of helicity for such volumes is presented here. The time derivative of this relative measure is calculated, which leads to an expression for the flow of topological structure across boundaries.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 147 , October 1984 , pp. 133 - 148
Copyright
© 1984 Cambridge University Press

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References

Arnol'd, V. I. 1974 The asymptotic Hopf invariant and its applications (in Russian). In Proc. Summer School in Differential Equations, Erevan, 1974. Armenian SSR Acad. Sci.
Berger, M. A. 1982 Rapid reconnection and the conservation of magnetic helicity (abstract). Bull. Am. Astron. Soc. 14, 978.Google Scholar
Berger, M. A. 1984 Rigorous new limits on magnetic helicity dissipation in the solar corona. Geophys. Astrophys. Fluid Dyn. (to appear).Google Scholar
CĂlugĂreanu, G. 1959 L'intégrale de Gauss et l'analyse des noeuds tridimensionelles. Rev. Math. Pures Appl. 4, 5.Google Scholar
Carter, B. 1979 Perfect fluid and magnetic field conservation laws in the theory of black hole accretion disks. In Active Galactic Nuclei (ed. C. Hazard & S. Milton), p. 273. Cambridge University Press.
Crick, F. H. C. 1976 Linking numbers and nucleosomes. Proc. Natl Acad. Sci. USA 73, 2639.Google Scholar
Elsasser, W. M. 1956 Hydromagnetic dynamo theory. Rev. Mod. Phys. 28, 135.Google Scholar
Frank-Kamenetskii, M. D., Lukashin, A. V. & Vologodski, A. V. 1975 Statistical mechanics and topology of polymer chains. Nature 258, 298.Google Scholar
Fuller, F. B. 1971 The writhing number of a space curve. Proc. Natl Acad. Sci. USA 68, 815.Google Scholar
Fuller, F. B. 1978 Decomposition of the linking number of a closed ribbon: a problem from molecular biology. Proc. Natl Acad. Sci. USA 75, 3557.Google Scholar
Kruskal, M. D. & Kulsrud, R. M. 1958 Equilibrium of a magnetically confined plasma in a toroid. Phys. Fluids 1, 265.Google Scholar
Matthaeus, W. H. & Smith, C. 1981 Structure of correlation tensors in homogeneous anisotropic turbulence. Phys. Rev. A 24, 2135.Google Scholar
Moffatt, H. K. 1969 The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35, 117.Google Scholar
Moffatt, H. K. 1978 Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press.
Moffatt, H. K. 1981 Some developments in the theory of turbulence. J. Fluid Mech. 106, 27.Google Scholar
Montgomery, D. & Turner, L. 1981 Anisotropic magnetohydrodynamic turbulence in a strong external magnetic field. Phys. Fluids 24, 825.Google Scholar
Moreau, J. J. 1961 Constantes d'un îlot tourbillonnaire en fluid parfait barotrope. C. R. Acad. Sci. Paris 252, 2810.Google Scholar
Newcomb, W. A. 1958 Motion of magnetic lines of force. Ann. Phys. 3, 347.Google Scholar
Pohl, W. F. 1980 DNA and differential geometry. Math. Intelligencer 3, 20.Google Scholar
Rolfsen, D. 1976 Knots and Links. Publish or Perish.
Taylor, J. B. 1974 Relaxation of toroidal plasma and generation of reverse magnetic fields. Phys. Rev. Lett. 33, 1139.Google Scholar
Taylor, J. B. 1981 Relaxation revisited. In Proc. Reversed Field Pinch Theory Workshop, Los Alamos 1980, p. 239.
White, J. H. 1969 Self-linking and the Gauss integral in higher dimensions. Am. J. Maths 91, 693.Google Scholar
Winfree, A. T. & Strogatz, S. 1984 Singular filaments organize chemical waves in three dimensions: III. Knotted waves; IV. Wave taxonomy. Physica D (to appear).
Woltjer, L. 1958 A theorem on force-free magnetic fields. Proc. Natl Acad. Sci. USA 44, 489.Google Scholar
Worcel, A., Strogatz, S. & Riley, D. 1981 Structure of chromatin and the linking number of DNA. Proc. Natl Acad. Sci. USA 78, 1461.Google Scholar