Bernstein–Greene–Kruskal modes

Bernstein–Greene–Kruskal modes (a.k.a. BGK modes) are nonlinear electrostatic waves that propagate in a unmagnetized, collisionless plasma. They are nonlinear solutions to the Vlasov–Poisson equation in plasma physics,[1] and are named after physicists Ira B. Bernstein, John M. Greene, and Martin D. Kruskal, who solved and published the exact solution for the one-dimensional case in 1957.[2]

BGK modes have been studied extensively in numerical simulations for two- and three-dimensional cases,[1][3][4][5] and are believed to be produced by the two-stream instability.[6][7] They have been observed as electron phase space holes (electrostatic solitary structures)[8][9][10] and double layers[11] in space plasmas, as well as in scattering experiments in the laboratory.[12]

Small-amplitude limit: Van Kampen modes

In the linear limit of BGK modes (e.g. in the small amplitude approximation), the solutions reduce to what is known as Van Kampen modes,[13] named after Nico van Kampen who derived the solutions in 1955.[14] The phase mixing of Van Kampen modes gives rise to Landau damping.[15]

Quantum BGK (QBGK) modes

BGK modes have been generalized to quantum mechanics, in which the solutions (called quantum BGK modes) solve the quantum equivalent of the Vlasov–Poisson system known as the Wigner–Poisson system, with periodic boundary conditions.[16] The solutions for the QBGK modes were put forth by Lange et al. in 1996,[17] with potential applications to quantum plasmas.[18]

References

  1. 1 2 Ng, C. S.; Bhattacharjee, A. (2005). "Bernstein-Greene-Kruskal Modes in a Three-Dimensional Plasma". Physical Review Letters. 95 (24): 245004. doi:10.1103/physrevlett.95.245004. ISSN 0031-9007. PMID 16384391.
  2. ↑ Bernstein, Ira B.; Greene, John M.; Kruskal, Martin D. (1957). "Exact Nonlinear Plasma Oscillations". Physical Review. 108 (3): 546–550. doi:10.1103/PhysRev.108.546.
  3. ↑ Demeio, Lucio; Holloway, James Paul (1991). "Numerical simulations of BGK modes". Journal of Plasma Physics. 46 (1): 63–84. doi:10.1017/S0022377800015956. ISSN 1469-7807.
  4. ↑ Manfredi, Giovanni; Bertrand, Pierre (2000). "Stability of Bernstein–Greene–Kruskal modes". Physics of Plasmas. 7 (6): 2425–2431. doi:10.1063/1.874081. ISSN 1070-664X.
  5. ↑ Berk, H. L.; Breizman, B. N.; Candy, J.; Pekker, M.; Petviashvili, N. V. (1999). "Spontaneous hole–clump pair creation". Physics of Plasmas. 6 (8): 3102–3113. doi:10.1063/1.873550. ISSN 1070-664X.
  6. ↑ Omura, Y.; Matsumoto, H.; Miyake, T.; Kojima, H. (1996). "Electron beam instabilities as generation mechanism of electrostatic solitary waves in the magnetotail". Journal of Geophysical Research: Space Physics. 101 (A2): 2685–2697. doi:10.1029/95ja03145. ISSN 0148-0227.
  7. ↑ Dieckmann, M. E.; Eliasson, B.; Shukla, P. K. (2004). "Streaming instabilities driven by mildly relativistic proton beams in plasmas". Physics of Plasmas. 11 (4): 1394–1401. doi:10.1063/1.1649996. ISSN 1070-664X.
  8. ↑ Turikov, V. A. (1984). "Electron Phase Space Holes as Localized BGK Solutions". Physica Scripta. 30 (1): 73–77. doi:10.1088/0031-8949/30/1/015. ISSN 1402-4896.
  9. ↑ Fox, W.; Porkolab, M.; Egedal, J.; Katz, N.; Le, A. (2008). "Laboratory Observation of Electron Phase-Space Holes during Magnetic Reconnection". Physical Review Letters. 101 (25): 255003. doi:10.1103/PhysRevLett.101.255003. PMID 19113719.
  10. ↑ Vasko, I. Y.; Kuzichev, I. V.; Agapitov, O. V.; Mozer, F. S.; Artemyev, A. V.; Roth, I. (2017). "Evolution of electron phase space holes in inhomogeneous plasmas". Physics of Plasmas. 24 (6): 062311. doi:10.1063/1.4989717. ISSN 1070-664X.
  11. ↑ Quon, B. H.; Wong, A. Y. (1976). "Formation of Potential Double Layers in Plasmas". Physical Review Letters. 37 (21): 1393–1396. doi:10.1103/physrevlett.37.1393. ISSN 0031-9007.
  12. ↑ Montgomery, D. S.; Focia, R. J.; Rose, H. A.; Russell, D. A.; Cobble, J. A.; Fernández, J. C.; Johnson, R. P. (2001). "Observation of Stimulated Electron-Acoustic-Wave Scattering". Physical Review Letters. 87 (15): 155001. doi:10.1103/PhysRevLett.87.155001. PMID 11580704.
  13. ↑ Chen, Francis F. (1984). Introduction to plasma physics and controlled fusion (2nd ed.). New York: Plenum Press. pp. 261–262. ISBN 0306413329. OCLC 9852700.
  14. ↑ Van Kampen, N. G. (1955). "On the theory of stationary waves in plasmas". Physica. 21 (6–10): 949–963. doi:10.1016/S0031-8914(55)93068-8. ISSN 0031-8914.
  15. ↑ Ng, C. S.; Bhattacharjee, A.; Skiff, F. (2006). "Weakly collisional Landau damping and three-dimensional Bernstein-Greene-Kruskal modes: New results on old problems". Physics of Plasmas. 13 (5): 055903. doi:10.1063/1.2186187. ISSN 1070-664X.
  16. ↑ Demeio, L. (2007). "Quantum Corrections to Classical BGK Modes in Phase Space". Transport Theory and Statistical Physics. 36 (1–3): 137–158. doi:10.1080/00411450701456857. ISSN 0041-1450.
  17. ↑ Lange, Horst; Toomire, Bruce; Zweifel, P. F. (1996). "Quantum BGK modes for the Wigner-poisson system". Transport Theory and Statistical Physics. 25 (6): 713–722. doi:10.1080/00411459608222920. ISSN 0041-1450.
  18. ↑ Haas, F.; Manfredi, G.; Feix, M. (2000). "Multistream model for quantum plasmas". Physical Review E. 62 (2): 2763–2772. arXiv:cond-mat/0203405. doi:10.1103/PhysRevE.62.2763.
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