Abstract—
The mirror asymmetry of hydrodynamics flows and magnetic fields plays a crucial role in magnetic field evolution in various celestial bodies including the Earth, Sun and Milky Way galaxy. A natural measure of the mirror asymmetry is associated with the Gauss invariant, i.e. the linkage number of vortex and magnetic lines. We consider the relation between magnetic field generation and the knotting of vortex and magnetic lines in terms of hydrodynamic and magnetic helicities. Higher helicity invariants known in topology generalize in various ways the Gauss invariant. We argue that these higher helicities are less important for magnetic field evolution than the classical Gauss invariant.
Similar content being viewed by others
REFERENCES
Akhmet’ev, P.M., Quadratic magnetic helicity and magnetic energy, Proc. Steklov Math. Inst., 2012, vol. 278, pp. 16–28.
Anufriev, A.P., Reshetnyak, M.Yu., and Sokoloff, D.D., Estimating the dynamo number for a model of the turbulent-effect in the Earth’s liquid core, Geomagn. Aeron. (Engl. Transl.), 1997, vol. 37, no. 5, pp. 628–631.
Arnold, V.I. and Khesin, B.A., Topological Methods in Hydrodynamics, New York: Springer, 1998.
Babcock, H.W., The topology of the Sun’s magnetic field and the 22-year cycle, Astrophys. J., 1961, vol. 133, pp. 572–587.
Baer, K.E., v.: Über ein allgemeines Gesetz in der Gestaltung der Flußbetten. Kaspische Stud., 1860, vol. 8, pp. 1–6.
Beck, R., Brandenburg, A., Moss, D., Shukurov, A., and Sokoloff, D., Galactic magnetism: recent developments and perspectives, Ann. Rev. Astron. Astrophys., 1996, vol. 34, pp. 155–206.
Brandenburg, A., Why coronal mass ejections are necessary for the dynamo, Highlights Astron., 2007, vol. 14, pp. 291–292.
Brandenburg, A., Sokoloff, D., and Subramanian, K., Current status of turbulent dynamo theory. From large-scale to small-scale dynamos, Space Sci. Rev., 2012, vol. 169, pp. 123–157.
Choudhuri, A.R., Schüssler, M., and Dikpati, M., The solar dynamo with meridional circulation, Astron. Astrophys., 1995, vol. 303, pp. L29–L32.
Dikpati, M. and Gilman, P.A., Flux-transport dynamos with alpha-effect from global instability of tachocline differential rotation: A solution for magnetic parity selection in the Sun, Astrophys. J., 2001, vol. 559, pp. 428–442.
Einstein, A., Die Ursache der Mäanderbildung der flußläufe und des sogenannten Baerschen Gesetzes, Naturwissenschaften, 1926, vol. 14, no. 11, pp. 223–224.
Enciso, A. and Peralta-Salas, D., and Torres de Lizaur, F., Helicity is the only integral invariant of volume-preserving transformations, Proc. Natl. Acad. Sci. U.S.A., 2016, vol. 113, pp. 2035–2040.
Frisch, U., Turbulence: The Legacy of A.N. Kolmogorov, Cambridge Univ., 1995.
Gailitis, A., Lielausis, O., Dement’ev, S., et al., Detection of a flow induced magnetic field eigenmode in the Riga dynamo facility, Phys. Rev. Lett., 2000, vol. 84, pp. 4365–4368.
Gruzinov, A.V. and Diamond, P.H., Self-consistent theory of mean-field electrodynamics, Phys. Rev. Lett., 1994, vol. 72, pp. 1651–1653.
Kolmogorov, A., The local structure of turbulence in incompressible viscous fluid for very large Reynolds’ numbers, Dokl. Akad. Nauk SSSR, 1941, vol. 30, pp. 301–305.
Komm, R., Gosain, S., and Pevtsov, A.A., Hemispheric distribution of subsurface kinetic helicity and its variation with magnetic activity, Sol. Phys., 2014, vol. 289, pp. 2399–2418.
Krause, F., Eine Lösung des Dynamoproblems auf der Grundlage einer linearen Theorie der magnetohydrodynamischen Turbulenz, Doctoral Dissertation, University of Jena, 1967.
Krause, F. and Rädler, K.-H., Mean-Field Magnetohydrodynamics and Dynamo Theory, Oxford: Pergamon, 1980.
Kulsrud, R.M. and Anderson, S.W., The spectrum of random magnetic fields in the mean field dynamo theory of the galactic magnetic field, Astrophys. J., 1992, vol. 396, pp. 606–630.
Lee, T.D. and Yang, C.N., Question of parity conservation in weak interactions, Phys. Rev., 1956, vol. 104, pp. 254–258.
Leighton, R.B., Transport of magnetic fields on the sun, Astrophys. J., 1964, vol. 140, pp. 1547–1562.
Moffatt, H.K., The degree of knottedness of tangled vortex lines, J. Fluid Mech., 1969, vol. 35, pp. 117–129.
Moffatt, H.K., Magnetic Field Generation in Electrically Conducting Fluids, Cambridge Univ., 1978.
Parker, E.N., Hydromagnetic dynamo models, Astrophys. J., 1955, vol. 122, pp. 293–314.
Parker, E.N., Cosmical Magnetic Fields: Their Origin and Their Activity, Oxford: Clarendon, 1979.
Parker, E.N., Conversations on Electric and Magnetic Fields in the Cosmos, Princeton Univ., 2007.
Redford, D.B., Akhenaten: The Heretic King, Princeton Univ., 1984.
Ricca, R.L. and Nipoti, B., Gauss’ linking number revisited, J. Knot Theory Ramifications, 2011, vol. 20, no. 10, pp. 1325–1343.
Ruzmaikin, A. and Akhmetiev, P., Topological invariants of magnetic fields, and the effect of reconnections, Phys. Plasmas, 1994, vol. 1, pp. 331–336.
Ruzmaikin, A.A., Shukurov, A.M., and Sokoloff, D.D., Magnetic Fields of Galaxies, Dordrecht: Kluwer, 1988.
Shukurov, A., Sokoloff, D., Subramanian, K., and Brandenburg, A., Galactic dynamo and helicity losses through fountain flow, Astron. Astrophys., 2006, vol. 448, L33–L36.
Sokoloff, D.D., Stepanov, R.A., and Frick, P.G., Dynamos: from an astrophysical model to laboratory experiments, Phys.-Usp., 2014, vol. 57, pp. 292–311.
Sokoloff, D.D., Illarionov, E.A., and Akhmet’ev, P.M., Higher helicity invariants and solar dynamo, Geomagn. Aeron. (Engl. Transl.), 2017, vol. 57, no. 1, pp. 113–118.
Sokoloff, D., Akhmet’ev, P., and Illarionov, E., Magnetic helicity and higher helicity invariants as constraints for dynamo action, Fluid Dyn. Res., 2018, vol. 50, id 011407.
Steenbeck, M., Krause, F., and Rädler, K.-H., Berechnung der mittleren Lorentz-Feldstärke \(vxB\) für ein elektrisch leitendes Medium in turbulenter, durch Coriolis-Kräfte beeinflußter Bewegung, Z. Naturforsch. A, 1966, vol. 21, no. 4, pp. 369–376.
Stenflo, J.O. and Kosovichev, A.G., Bipolar magnetic regions on the sun: Global analysis of the SOHO/MDI data set, Astrophys. J., 2012, vol. 745, id 129.
Stepanov, R., Volk, R., Denisov, S., et al., Induction, helicity, and alpha effect in a toroidal screw flow of liquid gallium, Phys. Rev. E, 2006, vol. 73, id 046310.
Stieglitz, R. and Müller, U., Experimental demonstration of a homogeneous two-scale dynamo, Phys. Fluids, 2001, vol. 13, pp. 561–564.
Tamm, I.E., Fundamentals of the Theory of Electricity, Moscow: Mir, 1966.
Tlatov, A., Illarionov, E., Sokoloff, D., and Pipin, V., A new dynamo pattern revealed by the tilt angle of bipolar sunspot groups, Mon. Not. R. Astron. Soc., 2013, vol. 432, pp. 2975–2984.
Vainshtein, S.I. and Cattaneo, F., Nonlinear restrictions on dynamo action, Astrophys. J., 1992, vol. 393, pp. 165–171.
Zaslavsky, G.M., Chaos in Dynamic Systems, New York: Harwood, 1985.
Zaslavsky, G.M., Sagdeev, R.Z., Usikov, D.A., and Chemikov, A.A., Weak Chaos and Quasi-Regular Patterns, Cambridge Univ., 1991.
Zeldovich, Ya.B., Ruzmaikin, A.A., and Sokoloff, D.D., Magnetic Fields in Astrophysics, New York: Gordon and Breach, 1983.
Zhang, H., Sakurai, T., Pevtsov, A., et al., A new dynamo pattern revealed by solar helical magnetic fields, Mon. Not. R. Astron. Soc., 2010, vol. 402, pp. L30–L33.
ACKNOWLEDGMENTS
Critical reading of the manuscript by Dr. D. Moss (Manchester, UK) is acknowledged.
Funding
The work was supported by the Russian Foundation for Basic Research, project no. 18-02-00085 and the Foundation for Support of Theoretical Physics Studies Basis, project no. 18-1-77-1.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sokoloff, D.D. Mirror Asymmetry and Helicity Invariants in Astrophysical Dynamos. Geomagn. Aeron. 59, 799–805 (2019). https://doi.org/10.1134/S0016793219070223
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0016793219070223