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Hall MHD and Electron Inertia Effects in Current Sheet Formation at a Magnetic Neutral Line

Published online by Cambridge University Press:  28 May 2015

Yuri E. Litvinenko  and
Liam C. McMahon
Yuri E. Litvinenko
Affiliation:
Department of Mathematics, University of Waikato, Private Bag 3105, Hamilton, New Zealand
Liam C. McMahon*
Affiliation:
Department of Mathematics, University of Waikato, Private Bag 3105, Hamilton, New Zealand
*
*Corresponding author. Email address: lcm21@students.waikato.ac.nz (L. C. McMahon)
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Abstract

An exact self-similar solution is used to investigate current sheet formation at a magnetic neutral line in incompressible Hall magnetohydrodynamics. The collapse to a current sheet is modelled as a finite-time singularity in the solution for electric current density at the neutral line. We establish that a finite-time collapse to the current sheet can occur in Hall magnetohydrodynamics, and we find a criterion for the finite-time singularity in terms of the initial conditions. We derive an asymptotic solution for the singularity formation and a formula for the singularity formation time. The analytical results are illustrated by numerical solutions, and we also investigate an alternative similarity reduction. Finally, we generalise our solution to incorporate resistive, viscous and electron inertia terms.

Keywords

76W0585A30Finite-time singularitiesHall MHDmagnetic reconnectioncurrent sheet formation
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 5 , Issue 2 , May 2015 , pp. 109 - 125
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Bhattacharjee, A., Impulsive magnetic reconnection in the Earth’s magnetotail and the solar corona, Annu. Rev. Astron. Astrophys. 42, 365–384 (2004).CrossRefGoogle Scholar
[2]Birn, J., Drake, J.F., Shay, M.A., Rogers, B.N., Denton, R.E., Hesse, M., Kuznetsova, M., Ma, Z.W., Bhattacharjee, A., Otto, A. and Pritchett, P.L., Geospace Environmental Modeling (GEM) magnetic reconnection challenge, J. Geophys. Res. 106, 3715–3720 (2001).Google Scholar
[3]Birn, J., Galsgaard, K., Hesse, M., Hoshino, M., Huba, J., Lapenta, G., Pritchett, P.L., Schindler, K., Yin, L., Büchner, J., Neukirch, T. and Priest, E.R., Forced magnetic reconnection, Geophys. Res. Lett. 32, 6105 (2005).Google Scholar
[4]Bulanov, S.V. and Olshanetskii, M.A., Magnetic collapse near zero points of the magnetic field, Phys. Letts. 100A, 35–38 (1984).Google Scholar
[5]Cassak, P.A.Drake, J.F. and Shay, M.A., A model for spontaneous onset of fast magnetic reconnection, Astrophys. J. 644, L145 (2006).CrossRefGoogle Scholar
[6]Cassak, P.A.Baylor, R. N., Fermo, R. L., Beidler, M. T., Shay, M. A., Swisdak, M., Drake, J. F. and Karimabadi, H., Fast magnetic reconnection due to anisotropic electron pressure, Phys. Plasmas 22, 020705 (2015).CrossRefGoogle Scholar
[7]Chapman, S. and Kendall, P.C., Liquid instability and energy transformation near a magnetic neutral line: A soluble non-linear hydromagnetic problem, Proc. Roy. Soc. London, Ser. A 271, 435–448 (1963).Google Scholar
[8]Chapman, S. and Kendall, P.C., Comment on “Some Exact Solutions of Magnetohydrodynamics”, Phys. Fluids 9, 2306–2307 (1966).Google Scholar
[9]Craig, I.J.D. and Litvinenko, Y.E., Influence of the Hall effect on the reconnection rate at line-tied magnetic X-points, Astron. Astrophys. 484, 847–850 (2008).CrossRefGoogle Scholar
[10]Craig, I.J.D. and Watson, P.G., Exact models for Hall current reconnection with axial guide fields, Phys. Plasmas 12, 012306 (2005).Google Scholar
[11]Dorelli, J.C., Effects of Hall electric fields on the saturation of forced antiparallel magnetic field merging, Phys. Plasmas 10, 3309–3314 (2003).Google Scholar
[12]Drake, J.F., Shay, M.A. and Swisdak, M., The Hall fields and fast magnetic reconnection, Phys. Plasmas 15, 042306 (2008).Google Scholar
[13]Dungey, J., Conditions for the occurrence of electrical discharges in astrophysical systems, Phil. Mag. 44, 725–738 (1953).CrossRefGoogle Scholar
[14]Dungey, J.W., The neutral point discharge theory of solar flares. A reply to Cowling’s criticism, in Electromagnetic Phenomena in Cosmical Physics, IAU Symp., vol. 6, edited by Lehnert, B., pp. 135–140 (1958).Google Scholar
[15]Forbes, T.G., Implosion of a uniform current sheet in a low-beta plasma, J. Plasma Phys. 27, 491–505 (1982).CrossRefGoogle Scholar
[16]Grauer, R. and Marliani, C., Geometry of singular structures in magnetohydrodynamic flows, Phys. Plasmas 5, 2544 (1998).Google Scholar
[17]Harris, E.G., On a plasma sheath separating regions of oppositely directed magnetic field, Nuovo Cimento 23, 115–121 (1962).CrossRefGoogle Scholar
[18]Hosking, R.J., New instabilities due to Hall effect, Phys. Rev. Lett. 15, 344–345 (1965).CrossRefGoogle Scholar
[19]Hosking, R.J. and Marinoff, G.M., Magneto-viscous effects on the ideal and resistive gravitational instabilities in Cartesian geometry, Plasma Phys. 15, 327–341 (1973).Google Scholar
[20]Imshennik, V.S. and Syrovatskiǐ, S.I., Two-dimensional flow of an ideally conducting gas in the vicinity of the zero line of a magnetic field, Sov. Phys.–JETP 25, 656 (1967).Google Scholar
[21]Klapper, I., Constraints on finite-time current sheet formation at null points in two-dimensional ideal incompressible magnetohydrodynamics, Phys. Plasmas 5, 910–914 (1998).Google Scholar
[22]Knoll, D.A. and Chacón, L., Coalescence of magnetic islands in the low-resistivity, Hall-MHD regime, Phys. Rev. Lett. 96, 135001 (2006).Google Scholar
[23]Litvinenko, Y.E., Current sheet formation at a magnetic neutral line in Hall magnetohydrody-namics, Phys. Plasmas 14, 112303 (2007).Google Scholar
[24]Malyshkin, L.M., Model of Hall reconnection, Phys. Rev. Lett. 101, 225001 (2008).Google Scholar
[25]McLaughlin, J.A., Hood, A.W. and Moortel, I. de, MHD wave propagation near coronal null points of magnetic fields, Space Sci. Rev. 158, 205–236 (2011).CrossRefGoogle Scholar
[26]Núñez, M., Álvarez, J. and Rojo, J., Blowup of certain analytic solutions of the Hall magnetohy-drodynamic equations, Phys. Plasmas 15, 062104 (2008).CrossRefGoogle Scholar
[27]Parker, E.N., Sweet’s mechanism for merging magnetic fields in conducting fluids, J. Geophys. Res. 62, 509–520 (1957).Google Scholar
[28]Rossi, B. and Olbert, S., Introduction to the Physics of Space, McGraw-Hill, New York (1970).Google Scholar
[29]Shay, M.A., Drake, J.F., Rogers, B.N. and Denton, R.E., Alfvénic collisionless magnetic reconnection and the Hall term, J. Geophys. Res. 106, 3759–3772 (2001).Google Scholar
[30]Shivamoggi, B.K., Evolution of current sheets near a hyperbolic magnetic neutral point, Phys. Fluids 29, 769–772 (1986).Google Scholar
[31]Shivamoggi, B.K., Hall magnetohydrodynamics near an X-type magnetic neutral line, Europhys. Lett. 85, 25001 (2009).CrossRefGoogle Scholar
[32]Shivamoggi, B.K., Steady and unsteady Hall magnetohydrodynamics near an X-type magnetic neutral line, Phys. Plasmas 18, 052304 (2011).Google Scholar
[33]Simakov, A.N. and Chacón, L., Quantitative analytical model for magnetic reconnection in Hall magnetohydrodynamics, Phys. Plasmas 16, 055701 (2009).CrossRefGoogle Scholar
[34]Smets, R., Aunai, N., Belmont, G., Boniface, C. and Fuchs, J., On the relationship between quadrupolar magnetic field and collisionless reconnection, Phys. Plasmas 21, 062111 (2014).Google Scholar
[35]Sonnerup, B.U.Ö., Magnetic field reconnection, in Solar System Plasma Physics, vol. 3, Lanzerotti, L.J., Kennel, C.F. and Parker, E.N. (eds.), pp. 45–108, North-Holland, Amsterdam (1979).Google Scholar
[36]Spitzer, L., Physics of Fully Ionized Gases, 2 ed., Interscience, New York (1962).Google Scholar
[37]Sulem, P.L., Frisch, U., Pouquet, A. and Meneguzzi, M., On the exponential flattening of current sheets near neutral X-points in two-dimensional ideal MHD flow, J. Plasma Phys. 33, 191–198 (1985).Google Scholar
[38]Sweet, P.A., The neutral point theory of solar flares, in Electromagnetic Phenomena in Cosmical Physics, IAU Symp., vol. 6, edited by Lehnert, B., pp. 123–134, Cambridge University Press, London (1958).Google Scholar
[39]Uberoi, M.S., Some exact solutions of magnetohydrodynamics, Phys. Fluids 6, 1379–1381 (1963).Google Scholar
[40]Uberoi, M.S., Reply to comments by S. Chapman and P. C. Kendall, Phys. Fluids 9, 2307 (1966).CrossRefGoogle Scholar
[41]Uzdensky, D.A., On the physical interpretation of Malyshkin’s (2008) model of resistive Hall magnetohydrodynamic reconnection, Phys. Plasmas 16, 040702 (2009).Google Scholar
[42]Velikovich, A.L., Rayleigh-Taylor instability of a plasma-vacuum boundary in the limit of a large Larmor radius, Phys. Fluids B 3, 492–494 (1991).CrossRefGoogle Scholar
[43]Wang, X., Bhattacharjee, A. and Ma, Z.W., Collisionless reconnection: Effects of Hall current and electron pressure gradient, J. Geophys. Res. 105, 27633–27648 (2000).Google Scholar
[44]Yamada, M., Kulsrud, R., and Ji, H., Magnetic reconnection, Rev. Mod. Phys. 82, 603 (2010).Google Scholar
[45]Zocco, L., Chacón, A. and Simakov, A.N., Current sheet bifurcation and collapse in electron mag-netohydrodynamics, Phys. Plasmas 16, 110703 (2009).CrossRefGoogle Scholar
[46]Zweibel, E.G. and Yamada, M., Magnetic reconnection in astrophysical and laboratory plasmas, Annu. Rev. Astron. Astrophys. 47, 291–332 (2009).CrossRefGoogle Scholar