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Multi-scale dynamics of magnetic flux tubes and inverse magnetic energy transfer

Part of: Fundamental Problems of Plasma Astrophysics Focus on Plasma Astrophysics

Published online by Cambridge University Press:  08 July 2020

Muni Zhou Open the ORCID record for Muni Zhou [Opens in a new window] ,
Nuno F. Loureiro Open the ORCID record for Nuno F. Loureiro [Opens in a new window]  and
Dmitri A. Uzdensky Open the ORCID record for Dmitri A. Uzdensky [Opens in a new window]
Muni Zhou*
Affiliation:
Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, MA02139, USA
Nuno F. Loureiro
Affiliation:
Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, MA02139, USA
Dmitri A. Uzdensky
Affiliation:
Center for Integrated Plasma Studies, Physics Department, UCB-390, University of Colorado, Boulder, CO80309, USA
*
†Email address for correspondence: munizhou@mit.edu
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Abstract

We report on an analytical and numerical study of the dynamics of a three-dimensional array of identical magnetic flux tubes in the reduced-magnetohydrodynamic description of the plasma. We propose that the long-time evolution of this system is dictated by flux-tube mergers, and that such mergers are dynamically constrained by the conservation of the pertinent (ideal) invariants, viz. the magnetic potential and axial fluxes of each tube. We also propose that in the direction perpendicular to the merging plane, flux tubes evolve in a critically balanced fashion. These notions allow us to construct an analytical model for how quantities such as the magnetic energy and the energy-containing scale evolve as functions of time. Of particular importance is the conclusion that, like its two-dimensional counterpart, this system exhibits an inverse transfer of magnetic energy that terminates only at the system scale. We perform direct numerical simulations that confirm these predictions and reveal other interesting aspects of the evolution of the system. We find, for example, that the early time evolution is characterized by a sharp decay of the initial magnetic energy, which we attribute to the ubiquitous formation of current sheets. We also show that a quantitatively similar inverse transfer of magnetic energy is observed when the initial condition is a random, small-scale magnetic seed field.

Keywords

astrophysical plasmas
Type
Research Article
Information
Journal of Plasma Physics , Volume 86 , Issue 4 , August 2020 , 535860401
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Alexakis, A., Mininni, P. D. & Pouquet, A. 2005 Shell-to-shell energy transfer in magnetohydrodynamics. I. Steady state turbulence. Phys. Rev. E 72 (4), 046301.Google ScholarPubMed
Alves, E. P., Zrake, J. & Fiuza, F. 2018 Efficient nonthermal particle acceleration by the kink instability in relativistic jets. Phys. Rev. Lett. 121 (24), 245101.CrossRefGoogle ScholarPubMed
Bhat, P., Zhou, M. & Loureiro, N. F.2020 Inverse transfer in decaying three dimensional nonhelical magnetic turbulence due to magnetic reconnection (in preparation).Google Scholar
Bhattacharjee, A., Huang, Y., Yang, H. & Rogers, B. 2009 Fast reconnection in high-Lundquist-number plasmas due to the plasmoid instability. Phys. Plasmas 16 (11), 112102.CrossRefGoogle Scholar
Biskamp, D. 1986 Magnetic reconnection via current sheets. Phys. Fluids 29, 1520–1531.CrossRefGoogle Scholar
Boldyrev, S. 2006 Spectrum of magnetohydrodynamic turbulence. Phys. Rev. Lett. 96 (11), 115002.CrossRefGoogle ScholarPubMed
Brandenburg, A., Kahniashvili, T. & Tevzadze, A. G. 2015 Nonhelical inverse transfer of a decaying turbulent magnetic field. Phys. Rev. Lett. 114 (7), 075001.CrossRefGoogle ScholarPubMed
Burgers, J. M. 1948 A mathematical model illustrating the theory of turbulence. In Adv. Appl. Mech., vol. 1, pp. 171–199. Elsevier.Google Scholar
Cho, J. & Vishniac, E. T. 2000 The anisotropy of magnetohydrodynamic Alfvénic turbulence. Astrophys. J. 539 (1), 273–282.CrossRefGoogle Scholar
Christensson, M., Hindmarsh, M. & Brandenburg, A. 2001 Inverse cascade in decaying three-dimensional magnetohydrodynamic turbulence. Phys. Rev. E 64 (5), 056405.Google ScholarPubMed
Dmitruk, P. & Gómez, D. O. 1999 Scaling law for the heating of solar coronal loops. Astrophys. J. Lett. 527, L63–L66.CrossRefGoogle ScholarPubMed
Drake, J. F., Opher, M., Swisdak, M. & Chamoun, J. N. 2010 A magnetic reconnection mechanism for the generation of anomalous cosmic rays. Astrophys. J. 709, 963–974.CrossRefGoogle Scholar
East, W. E., Zrake, J., Yuan, Y. & Blandford, R. D. 2015 Spontaneous decay of periodic magnetostatic equilibria. Phys. Rev. Lett. 115 (9), 095002.CrossRefGoogle ScholarPubMed
Einaudi, G. & Velli, M. 1999 The distribution of flares, statistics of magnetohydrodynamic turbulence and coronal heating. Phys. Plasmas 6, 4146–4153.CrossRefGoogle Scholar
Fermo, R. L., Drake, J. F. & Swisdak, M. 2010 A statistical model of magnetic islands in a current layer. Phys. Plasmas 17 (1), 010702.CrossRefGoogle Scholar
Finn, J. M. & Kaw, P. K. 1977 Coalescence instability of magnetic islands. Phys. Fluids 20 (1), 72–78.CrossRefGoogle Scholar
Fried, B. D. 1959 Mechanism for instability of transverse plasma waves. Phys. Fluids 2 (3), 337–337.CrossRefGoogle Scholar
Furno, I., Intrator, T. P., Hemsing, E. W., Hsu, S. C., Abbate, S., Ricci, P. & Lapenta, G. 2005 Coalescence of two magnetic flux ropes via collisional magnetic reconnection. Phys. Plasmas 12 (5), 055702.CrossRefGoogle Scholar
Gekelman, W., De Haas, T., Daughton, W., Van Compernolle, B., Intrator, T. & Vincena, S. 2016 Pulsating magnetic reconnection driven by three-dimensional flux-rope interactions. Phys. Rev. Lett. 116 (23), 235101.CrossRefGoogle ScholarPubMed
Gekelman, W., Lawrence, E. & Van Compernolle, B. 2012 Three-dimensional reconnection involving magnetic flux ropes. Astrophys. J. 753 (2), 131.CrossRefGoogle Scholar
Goldreich, P. & Lynden-Bell, D. 1965 II. Spiral arms as sheared gravitational instabilities. Mon. Not. R. Astron. Soc. 130, 125.CrossRefGoogle Scholar
Goldreich, P. & Sridhar, S. 1995 Toward a theory of interstellar turbulence. 2. Strong alfvenic turbulence. Astrophys. J. 438, 763–775.CrossRefGoogle Scholar
Gruzinov, A. 2001 Gamma-ray burst phenomenology, shock dynamo, and the first magnetic fields. Astrophys. J. Lett. 563 (1), L15.CrossRefGoogle Scholar
Hazeltine, R. D. & Meiss, J. D. 2003 Plasma Confinement. Courier Corporation.Google Scholar
Hu, Q., Chen, Y. & le Roux, J. 2019 Radial evolution of the properties of small-scale magnetic flux ropes in the solar wind. J. Phys.: Conf. Ser. 1332, 012005.Google Scholar
Huang, Y. & Bhattacharjee, A. 2010 Scaling laws of resistive magnetohydrodynamic reconnection in the high-Lundquist-number, plasmoid-unstable regime. Phys. Plasmas 17 (6), 062104.CrossRefGoogle Scholar
Huntington, C. M., Fiuza, F., Ross, J. S., Zylstra, A. B., Drake, R. P., Froula, D. H., Gregori, G., Kugland, N. L., Kuranz, C. C., Levy, M. C. et al. 2015 Observation of magnetic field generation via the Weibel instability in interpenetrating plasma flows. Nat. Phys. 11 (2), 173.CrossRefGoogle Scholar
Intrator, T. P., Sun, X., Dorf, L., Sears, J. A., Feng, Y., Weber, T. E. & Swan, H. O. 2013 Flux ropes and 3D dynamics in the relaxation scaling experiment. Plasma Phys. Control. Fusion 55 (12), 124005.CrossRefGoogle Scholar
Jara-Almonte, J., Ji, H., Yamada, M., Yoo, J. & Fox, W. 2016 Laboratory observation of resistive electron tearing in a two-fluid reconnecting current sheet. Phys. Rev. Lett. 117 (9), 095001.CrossRefGoogle Scholar
Kadomtsev, B. B. & Pogutse, O. P. 1973 Nonlinear helical perturbations of a plasma in the tokamak. Sov. Phys. JETP 5, 575–590.Google Scholar
Kato, T. N. 2005 Saturation mechanism of the Weibel instability in weakly magnetized plasmas. Phys. Plasmas 12 (8), 080705.CrossRefGoogle Scholar
Kato, T. N. & Takabe, H. 2008 Nonrelativistic collisionless shocks in unmagnetized electron-ion plasmas. Astrophys. J. 681 (2), L93.CrossRefGoogle Scholar
Katz, B., Keshet, U. & Waxman, E. 2007 Self-similar collisionless shocks. Astrophys. J. 655 (1), 375.CrossRefGoogle Scholar
Khabarova, O., Zank, G. P., Li, G., le Roux, J. A., Webb, G. M., Dosch, A. & Malandraki, O. E. 2015 Small-scale magnetic islands in the solar wind and their role in particle acceleration. I. Dynamics of magnetic islands near the heliospheric current sheet. Astrophys. J. 808 (2), 181.CrossRefGoogle Scholar
Klimchuk, J. A., Patsourakos, S. & Cargill, P. J. 2008 Highly efficient modeling of dynamic coronal loops. Astrophys. J. 682 (2), 1351–1362.CrossRefGoogle Scholar
Kulsrud, R. M. & Zweibel, E. G. 2008 On the origin of cosmic magnetic fields. Rep. Prog. Phys. 71 (4), 046901.Google Scholar
Lapenta, G. 2008 Self-feeding turbulent magnetic reconnection on macroscopic scales. Phys. Rev. Lett. 100 (23), 235001.CrossRefGoogle ScholarPubMed
Lazarian, A. & Opher, M. 2009 A model of acceleration of anomalous cosmic rays by reconnection in the heliosheath. Astrophys. J. 703, 8–21.CrossRefGoogle Scholar
Le Roux, J. A., Zank, G. P., Webb, G. M. & Khabarova, O. 2015 A kinetic transport theory for particle acceleration and transport in regions of multiple contracting and reconnecting inertial-scale flux ropes. Astrophys. J. 801 (2), 112.CrossRefGoogle Scholar
Loureiro, N. F., Dorland, W., Fazendeiro, L., Kanekar, A., Mallet, A., Vilelas, M. S. & Zocco, A. 2016 Viriato: a Fourier–Hermite spectral code for strongly magnetized fluid–kinetic plasma dynamics. Comput. Phys. Commun. 206, 45–63.CrossRefGoogle Scholar
Loureiro, N. F., Samtaney, R., Schekochihin, A. A. & Uzdensky, D. A. 2012 Magnetic reconnection and stochastic plasmoid chains in high-lundquist-number plasmas. Phys. Plasmas 19 (4), 042303.CrossRefGoogle Scholar
Loureiro, N. F., Schekochihin, A. A. & Cowley, S. C. 2007 Instability of current sheets and formation of plasmoid chains. Phys. Plasmas 14 (10), 100703–100703.CrossRefGoogle Scholar
Loureiro, N. F., Schekochihin, A. A. & Uzdensky, D. A. 2013 Plasmoid and Kelvin-Helmholtz instabilities in sweet-parker current sheets. Phys. Rev. E 87 (1), 013102.Google ScholarPubMed
Loureiro, N. F. & Uzdensky, D. A. 2016 Magnetic reconnection: from the Sweet-Parker model to stochastic plasmoid chains. Plasma Phys. Control. Fusion 58 (1), 014021.CrossRefGoogle Scholar
Lyutikov, M., Sironi, L., Komissarov, S. & Porth, O. 2017a Explosive X-point collapse in relativistic magnetically dominated plasma. J. Plasma Phys. 83 (6), 635830601.CrossRefGoogle Scholar
Lyutikov, M., Sironi, L., Komissarov, S. S. & Porth, O. 2017b Particle acceleration in relativistic magnetic flux-merging events. J. Plasma Phys. 83 (6), 635830602.CrossRefGoogle Scholar
Mallet, A., Schekochihin, A. A., Chandran, B. D. G., Chen, C. H. K., Horbury, T. S., Wicks, R. T. & Greenan, C. C. 2016 Measures of three-dimensional anisotropy and intermittency in strong Alfvénic turbulence. Mon. Not. R. Astron. Soc. 459 (2), 2130–2139.CrossRefGoogle Scholar
Matthaeus, W. H. & Lamkin, S. L. 1986 Turbulent magnetic reconnection. Phys. Fluids 29, 2513–2534.CrossRefGoogle Scholar
McComas, D. J., Allegrini, F., Bochsler, P., Bzowski, M., Christian, E. R., Crew, G. B., DeMajistre, R., Fahr, H., Fichtner, H., Frisch, P. C. et al. 2009 Global observations of the interstellar interaction from the Interstellar Boundary Explorer (IBEX). Science 326, 959.CrossRefGoogle Scholar
Medvedev, M. V., Fiore, M., Fonseca, R. A., Silva, L. O. & Mori, W. B. 2005 Long-time evolution of magnetic fields in relativistic gamma-ray burst shocks. Astrophys. J. Lett. 618 (2), L75.CrossRefGoogle Scholar
Medvedev, M. V. & Loeb, A. 1999 Generation of magnetic fields in the relativistic shock of gamma-ray burst sources. Astrophys. J. 526 (2), 697.CrossRefGoogle Scholar
Olesen, P. 1997 Inverse cascades and primordial magnetic fields. Phys. Lett. B 398 (3-4), 321–325.CrossRefGoogle Scholar
O’Neill, S. M., Beckwith, K. & Begelman, M. C. 2012 Local simulations of instabilities in relativistic jets – I. Morphology and energetics of the current-driven instability. Mon. Not. R. Astron. Soc. 422 (2), 1436–1452.CrossRefGoogle Scholar
Opher, M., Drake, J. F., Swisdak, M., Schoeffler, K. M., Richardson, J. D., Decker, R. B. & Toth, G. 2011 Is the magnetic field in the heliosheath laminar or a turbulent sea of bubbles? Astrophys. J. 734, 71.CrossRefGoogle Scholar
Parker, E. N. 1955 Hydromagnetic dynamo models. Astrophys. J. 122, 293.CrossRefGoogle Scholar
Parker, E. N. 1957 Sweet’s mechanism for merging magnetic fields in conducting fluids. J. Geophys. Res. 62, 509–520.CrossRefGoogle Scholar
Parker, E. N. 1983 Magnetic neutral sheets in evolving fields. I – General theory. Astrophys. J. 264, 635–647.CrossRefGoogle Scholar
Polomarov, O., Kaganovich, I. & Shvets, G. 2008 Merging of super-alfvénic current filaments during collisionless weibel instability of relativistic electron beams. Phys. Rev. Lett. 101, 175001.CrossRefGoogle ScholarPubMed
Pouquet, A. 1978 On two-dimensional magnetohydrodynamic turbulence. J. Fluid Mech. 88 (1), 1–16.CrossRefGoogle Scholar
Reppin, J. & Banerjee, R. 2017 Nonhelical turbulence and the inverse transfer of energy: a parameter study. Phys. Rev. E 96 (5), 053105.Google ScholarPubMed
Ruyer, C. & Fiuza, F. 2018 Disruption of current filaments and isotropization of the magnetic field in counterstreaming plasmas. Phys. Rev. Lett. 120, 245002.CrossRefGoogle ScholarPubMed
Samtaney, R., Loureiro, N. F., Uzdensky, D. A., Schekochihin, A. A. & Cowley, S. C. 2009 Formation of plasmoid chains in magnetic reconnection. Phys. Rev. Lett. 103 (10), 105004.CrossRefGoogle ScholarPubMed
Schekochihin, A. A., Cowley, S. C., Dorland, W., Hammett, G. W., Howes, G. G., Quataert, E. & Tatsuno, T. 2009 Astrophysical gyrokinetics: kinetic and fluid turbulent cascades in magnetized weakly collisional plasmas. Astrophys. J. Suppl. 182 (1), 310.CrossRefGoogle Scholar
Sears, J., Feng, Y., Intrator, T. P., Weber, T. E. & Swan, H. O. 2014 Flux rope dynamics in three dimensions. Plasma Phys. Control. Fusion 56 (9), 095022.CrossRefGoogle Scholar
Servidio, S., Matthaeus, W. H., Shay, M. A., Cassak, P. A. & Dmitruk, P. 2009 Magnetic reconnection in two-dimensional magnetohydrodynamic turbulence. Phys. Rev. Lett. 102 (11), 115003.CrossRefGoogle ScholarPubMed
Servidio, S., Matthaeus, W. H., Shay, M. A., Dmitruk, P., Cassak, P. A. & Wan, M. 2010 Statistics of magnetic reconnection in two-dimensional magnetohydrodynamic turbulence. Phys. Plasmas 17 (3), 032315.CrossRefGoogle Scholar
Silva, L. O., Fonseca, R. A., Tonge, J. W., Dawson, J. M., Mori, W. B. & Medvedev, M. V. 2003 Interpenetrating plasma shells: near-equipartition magnetic field generation and nonthermal particle acceleration. Astrophys. J. Lett. 596 (1), L121.CrossRefGoogle Scholar
Spitkovsky, A. 2008 On the structure of relativistic collisionless shocks in electron-ion plasmas. Astrophys. J. Lett. 673 (1), L39.CrossRefGoogle Scholar
Stone, E. C., Cummings, A. C., McDonald, F. B., Heikkila, B. C., Lal, N. & Webber, W. R. 2005 Voyager 1 explores the termination shock region and the heliosheath beyond. Science 309, 2017–2020.CrossRefGoogle ScholarPubMed
Stone, E. C., Cummings, A. C., McDonald, F. B., Heikkila, B. C., Lal, N. & Webber, W. R. 2008 An asymmetric solar wind termination shock. Nature 454, 71–74.CrossRefGoogle ScholarPubMed
Strauss, H. R. 1976 Nonlinear, three-dimensional magnetohydrodynamics of noncircular tokamaks. Phys. Fluids 19 (1), 134–140.CrossRefGoogle Scholar
Sweet, P. A. 1958 The neutral point theory of solar flares. In Electromagnetic Phenomena in Cosmical Physics (ed. Lehnert, B.), IAU Symposium, vol. 6, p. 123. Cambridge University Press.Google Scholar
Told, D., Jenko, F., TenBarge, J. M., Howes, G. G. & Hammett, G. W. 2015 Multiscale nature of the dissipation range in gyrokinetic simulations of alfvénic turbulence. Phys. Rev. Lett. 115 (2), 025003.CrossRefGoogle ScholarPubMed
Towns, J., Cockerill, T., Dahan, M., Foster, I., Gaither, K., Grimshaw, A., Hazlewood, V., Lathrop, S., Lifka, D., Peterson, G. D. et al. 2014 Xsede: accelerating scientific discovery. Comput. Sci. Engng 16 (5), 62–74.CrossRefGoogle Scholar
Uzdensky, D. A. & Goodman, J. 2008 Statistical description of a magnetized corona above a turbulent accretion disk. Astrophys. J. 682, 608–629.CrossRefGoogle Scholar
Uzdensky, D. A., Loureiro, N. F. & Schekochihin, A. A. 2010 Fast magnetic reconnection in the plasmoid-dominated regime. Phys. Rev. Lett. 105 (23), 235002.CrossRefGoogle ScholarPubMed
Weibel, E. S. 1959 Spontaneously growing transverse waves in a plasma due to an anisotropic velocity distribution. Phys. Rev. Lett. 2 (3), 83.CrossRefGoogle Scholar
Yamada, M., Ji, H., Hsu, S., Carter, T., Kulsrud, R., Bretz, N., Jobes, F., Ono, Y. & Perkins, F. 1997 Study of driven magnetic reconnection in a laboratory plasma. Phys. Plasmas 4 (5), 1936–1944.CrossRefGoogle Scholar
Yuan, Y., Nalewajko, K., Zrake, J., East, W. E. & Blandford, R. D. 2016 Kinetic study of radiation-reaction-limited particle acceleration during the relaxation of unstable force-free equilibria. Astrophys. J. 828 (2), 92.CrossRefGoogle Scholar
Zank, G. P., Hunana, P., Mostafavi, P., Le Roux, J. A., Li, G., Webb, G. M., Khabarova, O., Cummings, A., Stone, E. & Decker, R. 2015 Diffusive shock acceleration and reconnection acceleration processes. Astrophys. J. 814 (2), 137.CrossRefGoogle Scholar
Zank, G. P. & Matthaeus, W. H. 1992 The equations of reduced magnetohydrodynamics. J. Plasma Phys. 48, 85–100.CrossRefGoogle Scholar
Zhou, M., Bhat, P., Loureiro, N. F. & Uzdensky, D. A. 2019 (referred to as Z19) magnetic island merger as a mechanism for inverse magnetic energy transfer. Phys. Rev. Res. 1, 012004.CrossRefGoogle Scholar
Zrake, J. 2014 Inverse cascade of nonhelical magnetic turbulence in a relativistic fluid. Astrophys. J. Lett. 794 (2), L26.CrossRefGoogle Scholar
Zrake, J. & Arons, J. 2017 Turbulent magnetic relaxation in pulsar wind nebulae. Astrophys. J. 847 (1), 57.CrossRefGoogle Scholar