Bernstein–Greene–Kruskal modes explained

Bernstein–Greene–Kruskal modes (a.k.a. BGK modes) are nonlinear electrostatic waves that propagate in a collisionless plasma. They are nonlinear solutions to the Vlasov-Poisson system of equations 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 unmagnetized case in 1957.[2]

BGK modes have been studied extensively in numerical simulations for two- and three-dimensional cases,[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] [11] and double layers[12] in space plasmas, as well as in scattering experiments in the laboratory.[13]

Small-amplitude limit: Van Kampen modes ?

It is generally claimed that in the linear limit BGK modes (e.g. in the small amplitude approximation) reduce to what is known as Van Kampen modes,[14] named after Nico van Kampen who derived the solutions in 1955.[15]

This is wrong, however, since such a transition from a nonlinear to a linear mode does not take place even in the infinitesimal amplitude limit. A harmonic hole equilibrium of the Vlasov-Poisson system, which is correctly described as a complete solution, i.e. inclusively its phase velocity, by the Schamel method,[16] shows that nonlinearity persists even in the small amplitude limit. The area of trapped particles in the phase space never vanishes in this limit and there is no moment in which the distribution of trapped particles is transformed (or collapses) into a delta-function.[17] [18] [19] [20] Another indication that this claim is unfounded is that nonlinear single modes prove to be unconditionally marginal stable in current-carrying plasmas regardless of the drift velocity between electrons and ions. Landau's theory, as a linear wave theory, is obviously not applicable in case of coherent waves such as BGK modes valid even in the harmonic single wave limit.[21] The advantage of the Schamel method over the BGK method, including the unlimited class of so-called undisclosed modes not covered by the BGK method, is discussed in [22] and.[23]

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.[24] The solutions for the QBGK modes were put forth by Lange et al. in 1996,[25] with potential applications to quantum plasmas.[26] [27] Classical and quantum BGK modes as well as their appearance in charged particle beams in storage rings and circular accelerators have been summarized in.[28]

Notes and References

  1. Ng. C. S.. Bhattacharjee. A.. 2005. Bernstein-Greene-Kruskal Modes in a Three-Dimensional Plasma. Physical Review Letters. 95. 24. 245004. 10.1103/physrevlett.95.245004. 16384391. 0031-9007. 2005PhRvL..95x5004N.
  2. Bernstein. Ira B.. Greene. John M.. Kruskal. Martin D.. 1957. Exact Nonlinear Plasma Oscillations. Physical Review. 108. 3. 546–550. 10.1103/PhysRev.108.546. 1957PhRv..108..546B. 2027/mdp.39015095115203. free.
  3. Demeio. Lucio. Holloway. James Paul. 1991. Numerical simulations of BGK modes. Journal of Plasma Physics. en. 46. 1. 63–84. 10.1017/S0022377800015956. 1469-7807. 1991JPlPh..46...63D. 123050224 .
  4. Manfredi. Giovanni. Bertrand. Pierre. 2000. Stability of Bernstein–Greene–Kruskal modes. Physics of Plasmas. en. 7. 6. 2425–2431. 10.1063/1.874081. 1070-664X. 2000PhPl....7.2425M.
  5. Berk. H. L.. Breizman. B. N.. Candy. J.. Pekker. M.. Petviashvili. N. V.. 1999. Spontaneous hole–clump pair creation. Physics of Plasmas. en. 6. 8. 3102–3113. 10.1063/1.873550. 1070-664X. 1999PhPl....6.3102B.
  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. en. 101. A2. 2685–2697. 10.1029/95ja03145. 0148-0227. 1996JGR...101.2685O.
  7. Dieckmann. M. E.. Eliasson. B.. Shukla. P. K.. 2004. Streaming instabilities driven by mildly relativistic proton beams in plasmas. Physics of Plasmas. en. 11. 4. 1394–1401. 10.1063/1.1649996. 1070-664X. 2004PhPl...11.1394D.
  8. Schamel . H. . 1979 . Theory of Electron Holes . Physica Scripta . 20 . 3–4 . 336–342 . 10.1088/0031-8949/20/3-4/006.
  9. Turikov. V. A.. 1984. Electron Phase Space Holes as Localized BGK Solutions. Physica Scripta. en. 30. 1. 73–77. 10.1088/0031-8949/30/1/015. 1402-4896. 1984PhyS...30...73T. 250769529 .
  10. 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. 10.1103/PhysRevLett.101.255003. 19113719. 2008PhRvL.101y5003F.
  11. 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. en. 24. 6. 062311. 10.1063/1.4989717. 1070-664X. 2017PhPl...24f2311V.
  12. Quon. B. H.. Wong. A. Y.. 1976. Formation of Potential Double Layers in Plasmas. Physical Review Letters. 37. 21. 1393–1396. 10.1103/physrevlett.37.1393. 0031-9007. 1976PhRvL..37.1393Q.
  13. 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. 10.1103/PhysRevLett.87.155001. 11580704. 2001PhRvL..87o5001M.
  14. Book: Chen, Francis F.. Introduction to plasma physics and controlled fusion. 1984. Plenum Press. 0306413329. 2nd. New York. 261–262. 9852700.
  15. Van Kampen. N. G.. 1955. On the theory of stationary waves in plasmas. Physica. en. 21. 6–10. 949–963. 10.1016/S0031-8914(55)93068-8. 0031-8914. 1955Phy....21..949V.
  16. Schamel . H. . 1972 . Stationary solitary, snoidal and sinusoidal ion acoustic waves . Plasma Physics . 14 . 10 . 905–924 . 10.1088/0032-1028/14/10/002 . 0032-1028.
  17. Schamel . H. . 2012 . Cnoidal electron hole propagation: Trapping, the forgotten nonlinearity in plasma and fluid dynamics . Physics of Plasmas . 19 . 2 . 10.1063/1.3682047 . 1070-664X.
  18. Schamel . H. . 2015 . Particle trapping: A key requisite of structure formation and stability of Vlasov–Poisson plasmas . Physics of Plasmas . 22 . 4 . 10.1063/1.4916774 . 1070-664X.
  19. Schamel . H. . Mandal . D. . Sharma . D. . 2020 . Evidence of a new class of cnoidal electron holes exhibiting intrinsic substructures, its impact on linear (and nonlinear) Vlasov theories and role in anomalous transport . Physica Scripta . 95 . 5 . 055601 . 10.1088/1402-4896/ab725d . 0031-8949. free .
  20. Schamel . H. . Mandal . D. . Sharma . D. . 2020 . Diversity of solitary electron holes operating with non-perturbative trapping . Physics of Plasmas . 27 . 6 . 10.1063/5.0007941 . 1070-664X.
  21. Schamel . H. . 2018 . Unconditionally marginal stability of harmonic electron hole equilibria in current-driven plasmas . Physics of Plasmas . 25 . 6 . 10.1063/1.5037315 . 1070-664X.
  22. Schamel . H. . 2023 . Pattern formation in Vlasov–Poisson plasmas beyond Landau caused by the continuous spectra of electron and ion hole equilibria . Reviews of Modern Plasma Physics . en . 7 . 1 . 11 . 10.1007/s41614-022-00109-w . 2367-3192. 2110.01433 .
  23. Schamel . H. . Chakrabarti . N. . 2023 . Response to Comment on "On the Evolution Equations of Nonlinearly Permissible, Coherent Hole Structures Propagating Persistently in Collisionless Plasmas" [Ann. Phys. (Berlin) 2023, 2300102] . Annalen der Physik. 2023 . 2300441 . 3 . 10.1002/andp.202300441. free .
  24. Demeio. L.. 2007. Quantum Corrections to Classical BGK Modes in Phase Space. Transport Theory and Statistical Physics. en. 36. 1–3. 137–158. 10.1080/00411450701456857. 0041-1450. 2007TTSP...36..137D. 122915619 .
  25. Lange. Horst. Toomire. Bruce. Zweifel. P. F.. 1996. Quantum BGK modes for the Wigner-poisson system. Transport Theory and Statistical Physics. en. 25. 6. 713–722. 10.1080/00411459608222920. 0041-1450. 1996TTSP...25..713L.
  26. Haas. F.. Manfredi. G.. Feix. M.. 2000. Multistream model for quantum plasmas. Physical Review E. 62. 2. 2763–2772. 10.1103/PhysRevE.62.2763. 11088757. cond-mat/0203405. 2000PhRvE..62.2763H. 42012068 .
  27. Luque. A.. Schamel. H.. Fedele. R.. 2004. Quantum corrected electron holes. Phys. Lett. A . 324. 2–3 . 185–192. 10.1016/j.physleta.2004.02.049 . physics/0311126.
  28. Luque. A.. Schamel. H.. 2005. Electrostatic trapping as a key to the dynamics of plasmas, fluids and other collective systems. Physics Reports . 415. 5–6 . 261–359. 10.1016/j.physrep.2005.05.002 .

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Bernstein–Greene–Kruskal modes".

Except where otherwise indicated, Everything.Explained.Today is © Copyright 2009-2024, A B Cryer, All Rights Reserved. Cookie policy.