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On the analytical description of the nonlinear stage of the Weibel instability in collisionless anisotropic plasma

Published online by Cambridge University Press:  08 November 2023

A.A. Nechaev Open the ORCID record for A.A. Nechaev [Opens in a new window] ,
A.A. Kuznetsov Open the ORCID record for A.A. Kuznetsov [Opens in a new window]  and
Vl.V. Kocharovsky Open the ORCID record for Vl.V. Kocharovsky [Opens in a new window]
A.A. Nechaev*
Affiliation:
Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod, 603950, Russia
A.A. Kuznetsov
Affiliation:
Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod, 603950, Russia
Vl.V. Kocharovsky
Affiliation:
Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod, 603950, Russia
*
†Email address for correspondence: a.nechaev@ipfran.ru
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Abstract

On the basis of the energy invariants for the Weibel instability in a collisionless non-relativistic plasma, an analytical relation is obtained between the instantaneous values of the space-averaged energy density of the magnetic field, its dominant wavenumber and the average plasma anisotropy parameter. The relation is valid for arbitrary particle velocity distribution functions, including ones that vary with time at the nonlinear stage of instability. It is obtained under the assumption that the plasma is homogeneous along a certain axis, determined, for example, by an external magnetic field, and taking into account only the modes with wavevectors orthogonal to this axis. We give an estimate of the maximum magnetic field energy achievable during the Weibel instability and show that its ratio to the initial longitudinal energy of particles, even for a large initial plasma anisotropy parameter, cannot exceed the value approximately equal to 0.2. The obtained analytical relation is verified using two-dimensional particle-in-cell simulations.

Keywords

plasma instabilitiesplasma nonlinear phenomenaspace plasma physics
Type
Letter
Information
Journal of Plasma Physics , Volume 89 , Issue 6 , December 2023 , 175890601
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press

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References

Arber, T.D., Bennett, K., Brady, C.S., Lawrence-Douglas, A., Ramsay, M.G., Sircombe, N.J., Gillies, P., Evans, R.G., Schmitz, H., Bell, A.R. & Ridgers, C.P. 2015 Contemporary particle-in-cell approach to laser-plasma modelling. Plasma Phys. Control. Fusion 57 (11), 113001.CrossRefGoogle Scholar
Borodachev, L.V., Garasev, M.A., Kolomiets, D.O., Kocharovsky, V.V., Martyanov, V.Yu. & Nechaev, A.A. 2017 Dynamics of a self-consistent magnetic field and diffusive scattering of ions in a plasma with strong thermal anisotropy. Radiophys. Quantum El. 59 (12), 991–999.CrossRefGoogle Scholar
Davidson, R.C. 1989 Kinetic waves and instabilities in a uniform plasma. In Basic Plasma Physics: Selected Chapters from the Handbook of Plasma Physics (ed. A.A. Galeev & R.N. Sudan), vol. 1 and 2, p. 229. North-Holland Publishing Company.Google Scholar
Davidson, R.C. & Hammer, D.A. 1971 Energy constants associated with the nonlinear theory of electromagnetic instabilities. Phys. Fluids 14 (7), 1452–1455.CrossRefGoogle Scholar
Dieckmann, M.E., Lerche, I., Shukla, P.K. & Drury, L.O.C. 2007 Aspects of self-similar current distributions resulting from the plasma filamentation instability. New J. Phys. 9, 10.CrossRefGoogle Scholar
Fleishman, G.D., Nita, G.M., Chen, B., Yu, S. & Gary, D.E. 2022 Solar flare accelerates nearly all electrons in a large coronal volume. Nature 606 (7915), 674–677.CrossRefGoogle Scholar
Frederiksen, J.T., Hededal, C.B., Haugbølle, T. & Nordlund, Å. 2004 Magnetic field generation in collisionless shocks: pattern growth and transport. Astrophys. J. 608 (1), L13–L16.CrossRefGoogle Scholar
Göde, S., Rödel, C., Zeil, K., Mishra, R., Gauthier, M., Brack, F.-E., Kluge, T., MacDonald, M., Metzkes, J., Obst, L., et al. 2017 Relativistic electron streaming instabilities modulate proton beams accelerated in laser-plasma interactions. Phys. Rev. Lett. 118 (19), 194801.CrossRefGoogle ScholarPubMed
Gruzinov, A. 2001 Gamma-ray burst phenomenology, shock dynamo, and the first magnetic fields. Astrophys. J. 563 (1), L15–L18.CrossRefGoogle Scholar
Huntington, C.M., Manuel, M.J. -E., Ross, J.S., Wilks, S.C., Fiuza, F., Rinderknecht, H.G., Park, H. -S., Gregori, G., Higginson, D.P., Park, J., et al. 2017 Magnetic field production via the weibel instability in interpenetrating plasma flows. Phys. Plasmas 24 (4), 041410.CrossRefGoogle Scholar
Kocharovsky, V.V., Kocharovsky, Vl.V., Martyanov, V.Yu. & Tarasov, S.V. 2016 Analytical theory of self-consistent current structures in a collisionless plasma. Phys. Usp. 59 (12), 1165–1210.CrossRefGoogle Scholar
Kuznetsov, A.A., Nechaev, A.A., Garasev, M.A. & Kocharovsky, Vl.V. 2023 Quasilinear modeling of the evolution of weibel turbulence in anisotropic collisionless plasma. J. Exp. Theor. Phys. 137 (6) (in press).Google Scholar
Lemons, D.S., Winske, D. & Gary, S.P. 1979 Nonlinear theory of the Weibel instability. J. Plasma Phys. 21 (2), 287–300.CrossRefGoogle Scholar
Lyubarsky, Y. & Eichler, D. 2006 Are gamma-ray burst shocks mediated by the weibel instability? Astrophys. J. 647 (2), 1250–1254.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. 618 (2), L75–L78.CrossRefGoogle Scholar
Montes, C. & Peyraud, J. 1972 Thermodynamics of the relaxation of a temperature anisotropy in a collisionless plasma. J. Plasma Phys. 7 (1), 67–79.CrossRefGoogle Scholar
Nechaev, A.A., Garasev, M.A., Kocharovsky, V.V. & Kocharovsky, Vl.V. 2020 Weibel mechanism of magnetic-field generation in the process of expansion of a collisionless-plasma bunch with hot electrons. Radiophys. Quantum El. 62 (12), 830–848.CrossRefGoogle Scholar
Ossakow, S.L., Ott, E. & Haber, I. 1972 Nonlinear evolution of whistler instabilities. Phys. Fluids 15 (12), 2314–2326.CrossRefGoogle Scholar
Pokhotelov, O.A. & Amariutei, O.A. 2011 Quasi-linear dynamics of Weibel instability. Ann. Geophys. 29 (11), 1997–2001.CrossRefGoogle Scholar
Romanov, D.V., Bychenkov, V.Y., Rozmus, W., Capjack, C.E. & Fedosejevs, R. 2004 Self-organization of a plasma due to 3D evolution of the Weibel instability. Phys. Rev. Lett. 93 (21), 215004.CrossRefGoogle ScholarPubMed
Ruyer, C., Bolaños, S., Albertazzi, B., Chen, S.N., Antici, P., Böker, J., Dervieux, V., Lancia, L., Nakatsutsumi, M., Romagnani, L., et al. 2020 Growth of concomitant laser-driven collisionless and resistive electron filamentation instabilities over large spatiotemporal scales. Nat. Phys. 16 (9), 983–988.CrossRefGoogle Scholar
Ruyer, C., Gremillet, L., Debayle, A. & Bonnaud, G. 2015 Nonlinear dynamics of the ion Weibel-filamentation instability: an analytical model for the evolution of the plasma and spectral properties. Phys. Plasmas 22 (3), 032102.CrossRefGoogle Scholar
Silva, T., Afeyan, B. & Silva, L.O. 2021 Weibel instability beyond bi-Maxwellian anisotropy. Phys. Rev. E 104 (3), 035201.CrossRefGoogle ScholarPubMed
Silva, T., Schoeffler, K., Vieira, J., Hoshino, M., Fonseca, R.A. & Silva, L.O. 2020 Anisotropic heating and magnetic field generation due to raman scattering in laser-plasma interactions. Phys. Rev. Res. 2, 023080.CrossRefGoogle Scholar
Spitkovsky, A. 2008 Particle acceleration in relativistic collisionless shocks: fermi process at last? Astrophys. J. 682 (1), L5–L8.CrossRefGoogle Scholar
Stockem, A., Dieckmann, M.E. & Schlickeiser, R. 2010 PIC simulations of the temperature anisotropy-driven Weibel instability: analysing the perpendicular mode. Plasma Phys. Control. Fusion 52 (8), 085009.CrossRefGoogle Scholar
Takabe, H. 2023 Theory of magnetic turbulence and shock formation induced by a collisionless plasma instability. Phys. Plasmas 30 (3), 030901.CrossRefGoogle Scholar
Vagin, K.Y. & Uryupin, S.A. 2014 On the growth rate of aperiodic instability in plasma with an anisotropic bi-maxwellian electron velocity distribution. Plasma Phys. Rep. 40 (5), 393–403.CrossRefGoogle Scholar
Weibel, E.S. 1959 Spontaneously growing transverse waves in a plasma due to an anisotropic velocity distribution. Phys. Rev. Lett. 2 (3), 83–84.CrossRefGoogle Scholar
Yang, T.-Y.B., Arons, J. & Langdon, A.B. 1994 Evolution of the Weibel instability in relativistically hot electron–positron plasmas. Phys. Plasmas 1 (9), 3059–3077.CrossRefGoogle Scholar
Zaitsev, V.V. & Stepanov, A.V. 2015 Particle acceleration and plasma heating in the chromosphere. Sol. Phys. 290, 3559–3572.CrossRefGoogle Scholar
Zaitsev, V.V. & Stepanov, A.V. 2017 Acceleration and storage of energetic electrons in magnetic loops in the course of electric current oscillations. Sol. Phys. 292, 141.CrossRefGoogle Scholar
Zhang, C., Hua, J., Wu, Y., Fang, Y., Ma, Y., Zhang, T., Liu, S., Peng, B., He, Y., Huang, C.-K., et al. 2020 Measurements of the growth and saturation of electron Weibel instability in optical-field ionized plasmas. Phys. Rev. Lett. 125 (25), 255001.CrossRefGoogle ScholarPubMed
Zhou, M., Wu, D.H., Loureiro, N.F. & Uzdensky, D.A. 2021 Statistical description of coalescing magnetic islands via magnetic reconnection. J. Plasma Phys. 87 (6), 905870620.CrossRefGoogle Scholar
Zhou, S., Bai, Y., Tian, Y., Sun, H., Cao, L. & Liu, J. 2018 Self-organized kilotesla magnetic-tube array in an expanding spherical plasma irradiated by kHz femtosecond laser pulses. Phys. Rev. Lett. 121 (25), 255002.CrossRefGoogle Scholar