Skip to main content Accessibility help
Hostname: page-component-5959bf8d4d-qtfcj Total loading time: 0.216 Render date: 2022-12-08T05:43:04.903Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

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.CrossRefGoogle Scholar
Basu, B. 2002 Moment equation description of weibel instability. Phys. Plasmas 9 (12), 51315134.CrossRefGoogle 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.CrossRefGoogle Scholar
Brizard, A. J. 2000 New variational principle for the Vlasov–Maxwell equations. Phys. Rev. Lett. 84 (25), 5768.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar
Cheng, C. Z. & Johnson, J. R. 1999 A kinetic-fluid model. J. Geophys. Res. 104 (A1), 413427.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar
Gary, S. P. & Karimabadi, H. 2006 Linear theory of electron temperature anisotropy instabilities: whistler, mirror, and weibel. J. Geophys. Res. 111, A11224.CrossRefGoogle Scholar
Ghizzo, A., Sarrat, M. & Del Sarto, D. 2017 Vlasov models for kinetic weibel-type instabilities. J. Plasma Phys. 83, 705830101.CrossRefGoogle Scholar
Grad, H. 1949 On the kinetic theory of rarefied gases. Commun. Appl. Maths 2 (4), 331407.CrossRefGoogle 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.CrossRefGoogle Scholar
Hesse, M., Kuznetsova, M. & Birn, J. 2004 The role of electron heat flux in guide-field magnetic reconnection. Phys. Plasmas 11 (12), 53875397.CrossRefGoogle Scholar
Hesse, M. & Winske, D. 1993 Hybrid simulations of collisionless ion tearing. Geophys. Res. Lett. 20 (12), 12071210.CrossRefGoogle Scholar
Hesse, M. & Winske, D. 1994 Hybrid simulations of collisionless reconnection in current sheets. J. Geophys. Res. 99 (A6), 1117711192.CrossRefGoogle Scholar
Holm, D. D. & Tronci, C. 2012 Euler-poincare formulation of hybrid plasma models. Commun. Math. Sci. 10, 191222; (EPFL-ARTICLE-174831).CrossRefGoogle 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.CrossRefGoogle Scholar
Kuznetsova, M. M., Hesse, M. & Winske, D. 2000 Toward a transport model of collisionless magnetic reconnection. J. Geophys. Res. 105 (A4), 76017616.CrossRefGoogle Scholar
Littlejohn, R. G. 1983 Variational principles of guiding centre motion. J. Plasma Phys. 29 (01), 111125.CrossRefGoogle 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.CrossRefGoogle Scholar
Morrison, P. J. 1998 Hamiltonian description of the ideal fluid. Rev. Mod. Phys. 70 (2), 467.CrossRefGoogle 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.CrossRefGoogle Scholar
Schlickeiser, R. & Shukla, P. K. 2003 Cosmological magnetic field generation by the weibel instability. Astrophys. J. Lett. 599 (2), L57.CrossRefGoogle Scholar
Swisdak, M. 2016 Quantifying gyrotropy in magnetic reconnection. Geophys. Res. Lett. 43 (1), 4349.CrossRefGoogle Scholar
Thyagaraja, A. & McClements, K. G. 2009 Plasma physics in noninertial frames. Phys. Plasmas 16 (9), 092506.CrossRefGoogle Scholar
Tronci, C. 2013 A Lagrangian kinetic model for collisionless magnetic reconnection. Plasma Phys. Control. Fusion 55 (3), 035001.CrossRefGoogle Scholar
Tronci, C. & Camporeale, E. 2015 Neutral Vlasov kinetic theory of magnetized plasmas. Phys. Plasmas 22 (2), 020704.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar
Yin, L. & Winske, D. 2003 Plasma pressure tensor effects on reconnection: hybrid and hall-magnetohydrodynamics simulations. Phys. Plasmas 10 (5), 15951604.CrossRefGoogle 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.CrossRefGoogle Scholar
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the or variations. ‘’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Electron inertia and quasi-neutrality in the Weibel instability
Available formats

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Electron inertia and quasi-neutrality in the Weibel instability
Available formats

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Electron inertia and quasi-neutrality in the Weibel instability
Available formats

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *