Hostname: page-component-77c89778f8-vpsfw Total loading time: 0 Render date: 2024-07-16T11:51:51.009Z Has data issue: false hasContentIssue false

Electron inertia and quasi-neutrality in the Weibel instability

Published online by Cambridge University Press:  05 June 2017

Enrico Camporeale*
Center for Mathematics and Computer Science (CWI), 1098 XG Amsterdam, The Netherlands
Cesare Tronci
Department of Mathematics, University of Surrey, Guildford GU2 7XH, United Kingdom
Email address for correspondence:


While electron kinetic effects are well known to be of fundamental importance in several situations, the electron mean-flow inertia is often neglected when length scales below the electron skin depth become irrelevant. This has led to the formulation of different reduced models, where electron inertia terms are discarded while retaining some or all kinetic effects. Upon considering general full-orbit particle trajectories, this paper compares the dispersion relations emerging from such models in the case of the Weibel instability. As a result, the question of how length scales below the electron skin depth can be neglected in a kinetic treatment emerges as an unsolved problem, since all current theories suffer from drawbacks of different nature. Alternatively, we discuss fully kinetic theories that remove all these drawbacks by restricting to frequencies well below the plasma frequency of both ions and electrons. By giving up on the length scale restrictions appearing in previous works, these models are obtained by assuming quasi-neutrality in the full Vlasov–Maxwell system.

Research Article
© Cambridge University Press 2017 

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.)


Aunai, N., Hesse, M. & Kuznetsova, M. 2013 Electron nongyrotropy in the context of collisionless magnetic reconnection. Phys. Plasmas 20 (9), 092903.Google Scholar
Basu, B. 2002 Moment equation description of weibel instability. Phys. Plasmas 9 (12), 51315134.Google Scholar
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. et al. 2001 Geospace environmental modeling (gem) magnetic reconnection challenge. J. Geophys. Res. 106 (A3), 37153719.Google Scholar
Brizard, A. J. 2000 New variational principle for the Vlasov–Maxwell equations. Phys. Rev. Lett. 84 (25), 5768.Google Scholar
Burby, J. W.2015 Chasing Hamiltonian structure in gyrokinetic theory. PhD thesis, Princeton University.Google Scholar
Cai, H.-J. & Lee, L. C. 1997 The generalized ohms law in collisionless magnetic reconnection. Phys. Plasmas 4 (3), 509520.Google Scholar
Camporeale, E. & Burgess, D. 2016 Comparison of linear modes in kinetic plasma models. J. Plasma Phys. 83, 535830201.Google Scholar
Camporeale, E. & Lapenta, G. 2005 Model of bifurcated current sheets in the earth’s magnetotail: equilibrium and stability. J. Geophys. Res. 110, A07206.Google Scholar
Cazzola, E., Innocenti, M. E., Goldman, M. V., Newman, D. L., Markidis, S. & Lapenta, G. 2016 On the electron agyrotropy during rapid asymmetric magnetic island coalescence in presence of a guide field. Geophys. Res. Lett. 43 (15), 78407849.Google Scholar
Cendra, H., Holm, D. D., Hoyle, M. J. W. & Marsden, J. E. 1998 The Vlasov–Maxwell equations in Euler–Poincaré form. J. Math. Phys. 39 (6), 31383157.Google Scholar
Cheng, C. Z. & Johnson, J. R. 1999 A kinetic-fluid model. J. Geophys. Res. 104 (A1), 413427.Google Scholar
Degond, P., Deluzet, F. & Doyen, D. 2017 Asymptotic-preserving particle-in-cell methods for the Vlasov–Maxwell system in the quasi-neutral limit. J. Comput. Phys. 330, 467492.Google Scholar
Fonseca, R. A., Silva, L. O., Tonge, J. W., Mori, W. B. & Dawson, J. M. 2003 Three-dimensional weibel instability in astrophysical scenarios. Phys. Plasmas 10 (5), 19791984.Google Scholar
Gary, S. P. & Karimabadi, H. 2006 Linear theory of electron temperature anisotropy instabilities: whistler, mirror, and weibel. J. Geophys. Res. 111, A11224.Google Scholar
Ghizzo, A., Sarrat, M. & Del Sarto, D. 2017 Vlasov models for kinetic weibel-type instabilities. J. Plasma Phys. 83, 705830101.Google Scholar
Grad, H. 1949 On the kinetic theory of rarefied gases. Commun. Appl. Maths 2 (4), 331407.Google Scholar
Haynes, C. T., Burgess, D. & Camporeale, E. 2014 Reconnection and electron temperature anisotropy in sub-proton scale plasma turbulence. Astrophys. J. 783 (1), 38.Google Scholar
Hesse, M., Kuznetsova, M. & Birn, J. 2004 The role of electron heat flux in guide-field magnetic reconnection. Phys. Plasmas 11 (12), 53875397.Google Scholar
Hesse, M. & Winske, D. 1993 Hybrid simulations of collisionless ion tearing. Geophys. Res. Lett. 20 (12), 12071210.Google Scholar
Hesse, M. & Winske, D. 1994 Hybrid simulations of collisionless reconnection in current sheets. J. Geophys. Res. 99 (A6), 1117711192.Google Scholar
Holm, D. D. & Tronci, C. 2012 Euler-poincare formulation of hybrid plasma models. Commun. Math. Sci. 10, 191222; (EPFL-ARTICLE-174831).Google Scholar
Krall, N. A. & Trivelpiece, A. W. 1973 Principles of Plasma Physics. McGraw Hill.Google Scholar
Kuznetsova, M. M., Hesse, M. & Winske, D. 1998 Kinetic quasi-viscous and bulk flow inertia effects in collisionless magnetotail reconnection. J. Geophys. Res. 103 (A1), 199213.Google Scholar
Kuznetsova, M. M., Hesse, M. & Winske, D. 2000 Toward a transport model of collisionless magnetic reconnection. J. Geophys. Res. 105 (A4), 76017616.Google Scholar
Littlejohn, R. G. 1983 Variational principles of guiding centre motion. J. Plasma Phys. 29 (01), 111125.Google Scholar
Low, F. E. 1958 A Lagrangian formulation of the Boltzmann–Vlasov equation for plasmas. Proc. R. Soc. Lond. A 248, 282287; The Royal Society.Google Scholar
Morrison, P. J. 1998 Hamiltonian description of the ideal fluid. Rev. Mod. Phys. 70 (2), 467.Google Scholar
Newcomb, W. A. 1962 Lagrangian and Hamiltonian methods in magnetohydrodynamics. Nucl. Fusion 451463.Google Scholar
Sarrat, M., Del Sarto, D. & Ghizzo, A. 2016 Fluid description of weibel-type instabilities via full pressure tensor dynamics. Europhys. Lett. 115 (4), 45001.Google Scholar
Schlickeiser, R. & Shukla, P. K. 2003 Cosmological magnetic field generation by the weibel instability. Astrophys. J. Lett. 599 (2), L57.Google Scholar
Swisdak, M. 2016 Quantifying gyrotropy in magnetic reconnection. Geophys. Res. Lett. 43 (1), 4349.Google Scholar
Thyagaraja, A. & McClements, K. G. 2009 Plasma physics in noninertial frames. Phys. Plasmas 16 (9), 092506.Google Scholar
Tronci, C. 2013 A Lagrangian kinetic model for collisionless magnetic reconnection. Plasma Phys. Control. Fusion 55 (3), 035001.Google Scholar
Tronci, C. & Camporeale, E. 2015 Neutral Vlasov kinetic theory of magnetized plasmas. Phys. Plasmas 22 (2), 020704.Google Scholar
Wang, L., Hakim, A. H., Bhattacharjee, A. & Germaschewski, K. 2015 Comparison of multi-fluid moment models with particle-in-cell simulations of collisionless magnetic reconnection. Phys. Plasmas 22 (1), 012108.Google Scholar
Wang, X., Bhattacharjee, A. & Ma, Z. W. 2000 Collisionless reconnection: effects of hall current and electron pressure gradient. J. Geophys. Res. 105 (A12), 2763327648.Google Scholar
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
Winske, D. & Hesse, M. 1994 Hybrid modeling of magnetic reconnection in space plasmas. Physica D 77 (1–3), 268275.Google Scholar
Yin, L. & Winske, D. 2003 Plasma pressure tensor effects on reconnection: hybrid and hall-magnetohydrodynamics simulations. Phys. Plasmas 10 (5), 15951604.Google Scholar
Yin, L., Winske, D., Gary, S. P. & Birn, J. 2001 Hybrid and hall-mhd simulations of collisionless reconnection: dynamics of the electron pressure tensor. J. Geophys. Res. 106 (A6), 1076110775.Google Scholar