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Formation of electron holes in the long-time evolution of the bump-on-tail instability

Published online by Cambridge University Press:  12 December 2017

Magdi Shoucri
Magdi Shoucri*
Affiliation:
Institut de recherche d'Hydro-Québec (IREQ), Varennes, Québec, J3X1S1, Canada
*
Address correspondence and reprint requests to: M. Shoucri, Institut de recherche d'Hydro-Québec (IREQ), Varennes, Québec, J3X1S1, Canada. E-mail: magshoucri@gmail.com
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Abstract

An Eulerian Vlasov code is applied for the numerical solution of the one-dimensional Vlasov–Poisson system of equations for electrons, and with ions forming an immobile background. We study the non-linear evolution of the bump-on-tail instability in the case when the system length L is greater than the wavelength λ of the unstable mode, with a beam density of 10% of the total density, nb = 0.1. We follow the growth and the saturation of an initially unstable wave perturbation, and the formation of a traveling Bernstein–Greene–Kruskal (BGK) mode, which evolves out of the instability. This first stage is followed by sidebands growing from round-off errors which develop and disrupt the BGK equilibrium. In the excited spectrum, mode coupling is mediated by the oscillating resonant particles and results in the electric energy of the system flowing to the longest wavelengths (inverse cascade), and reaching in the asymptotic state a steady state with constant amplitude oscillation modulated by the persistent oscillation of the trapped particles. Coherent phase-space electron holes are formed, which are localized phase-space regions of reduced density on trapped electron orbits, where the electron density is lower than the surrounding plasma electron density. The distribution function evolves to a shape with stationary inflection points of zero slope, at the phase velocities of the excited waves. The longest wavelengths show oscillations at frequencies below the plasma frequency, with phase velocities higher than that of the injected beam, which can accelerate electrons to energies in excess of the initial beam energy. The present work makes a connection between the formation of electron holes, the existence of inflection points of zero slopes in the electron distribution function at the phase velocities of the dominant waves, and at frequencies below the plasma frequency. A fine resolution grid is used in the Eulerian Vlasov code in the phase space and time to allow an accurate calculation of the time history of the system and of the dynamic and oscillation of the trapped particles in the low-density regions of the phase space, and of those particles at the separatrix regions of the vortex structures which evolve periodically between trapping and untrapping states and which can only be accurately studied using a fine-resolution phase-space grid.

Keywords

Bump-on-tail instabilityElectron holesEulerian Vlasov codeInverse cascade
Type
Research Article
Information
Laser and Particle Beams , Volume 35 , Issue 4 , December 2017 , pp. 706 - 721
Copyright
Copyright © Cambridge University Press 2017 

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References

REFERENCES

Arber, T.D., Sircombe, N.J. & Vann, R.G.L. (2011). Eulerian conservative advection schemes for Vlasov solvers. In Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, (Shoucri, M., ed.), p. 65. New York: Nova Science Publishers.Google Scholar
Berk, H.L. & Roberts, K.V. (1967). Nonlinear study of Vlasov's equation for a special class of distribution functions. Phys. Fluids 10, 1595–1597.Google Scholar
Bernstein, M., Greene, J.M. & Kruskal, M.D. (1957). Exact nonlinear plasma oscillations. Phys. Rev. 108, 546–550.Google Scholar
Bertrand, P., Ghizzo, A., Feix, M., Fijalkow, E., Mineau, P., Suh, N.D. & Shoucri, M. (1988). Computer simulations of phase-space hole dynamics. In Nonlinear Phenomena in Vlasov Plasmas, (Doveil, F., ed.), pp. 109–125. Orsay, France: Editions de Physique.Google Scholar
Besse, N., Elskens, Y., Escande, D.F. & Bertrand, P. (2011). Validity of quasilinear theory: refutations and new numerical confirmation. Plasma Phys. Control. Fusion 53, 025012.Google Scholar
Birdsall, C.K. & Langdon, A.B. (1991). Plasma Physics via Computer Simulation. Bristol: IOP Publishing.CrossRefGoogle Scholar
Briand, C. (2015). Langmuir waves across the heliosphere. J. Plasma Phys. 81, 1–27.Google Scholar
Brunetti, M., Califano, F. & Pegoraro, F. (2000). Asymptotic evolution of nonlinear Landau damping. Phys. Rev. E 62, 4109–4114.Google Scholar
Buchanan, M. & Dorning, J. (1995). Nonlinear electrostatic waves in collisionless plasmas. Phys. Rev. E 52, 3015–3033.Google Scholar
Califano, F., Galeotti, A. & Mangeney, A. (2006). The Vlasov-Poisson model and the validity of a numerical approach. Phys. Plasmas 13, 082102.Google Scholar
Califano, F. & Lantano, M. (1999). Vlasov-Poisson simulations of strong wave-plasma interaction in condition of relevance for radio frequency plasma heating. Phys. Rev. Lett. 83, 96–99.CrossRefGoogle Scholar
Califano, F., Pegoraro, F. & Bulanov, S.V. (2000). Impact of kinetic processes on the macroscopic nonlinear evolution of the electromagnetic-beam-plasma instability. Phys. Rev. Lett. 84, 3602–3605.CrossRefGoogle ScholarPubMed
Cheng, C.Z. & Knorr, G. (1976). The integration of the Vlasov equation in configuration space. J. Comput. Phys. 22, 330–351.CrossRefGoogle Scholar
Crouseilles, N., Respaud, T. & Sonnendrücker, E. (2009). A forward semi-Lagrangian method for the numerical solution of the Vlasov equation. Comput. Phys. Commun. 180, 1730–1745.CrossRefGoogle Scholar
Dawson, J.M. & Shanny, R. (1968). Some investigations of nonlinear behaviour in one-dimensional plasmas. Phys. Fluids 11, 1506–1523.CrossRefGoogle Scholar
Demeio, L. & Holloway, J.P. (1991). Numerical simulations of BGK modes. J. Plasma Phys. 46, 63.CrossRefGoogle Scholar
Denavit, J. & Kruer, W.L. (1971). Comparison of numerical solutions of the Vlasov equation with particle simulations of collisionles plasmas. Phys. Fluids 14, 1782–1791.Google Scholar
Doveil, F., Firpo, M.-C., Elskens, Y., Guyomarc'h, D., Poleni, M. & Bertrand, P. (2001). Trapping oscillations, discrete particle effects and kinetic theory of collisionless plasma. Phys. Lett. A 284, 279–285.Google Scholar
Eliasson, B. & Shukla, P.K. (2006). Formation and dynamics of coherent structures involving phase-space vortices in plasmas. Phys. Rep. 422, 225–290.Google Scholar
Elskens, Y. & Escande, D. (2003). Microscopic Dynamics of Plasmas and Chaos. England: IOP.CrossRefGoogle Scholar
Escande, D.F., Besse, N., Doveil, F. & Elskens, Y. (2012). Picard iteration technique to self-consistent wave-particle interaction in plasmas. In Proc. 39th EPS Conf. and 16th Int. Congress Plasma Phys., Eur. Phys. Soc., (van Milligen, B. and Fasoli, A., Eds), paper P.4.161. Stockholm, Sweden: Royal Institute of Technology.Google Scholar
Gagné, R & Shoucri, M. (1977). A splitting scheme for the numerical solution of the Vlasov equation. J. Comput. Phys. 24, 445–449.Google Scholar
Ghizzo, A., Izrar, B., Bertrand, P., Fijalkow, E., Feix, M.R. & Shoucri, M.M. (1988 a). Stability of Bernstein-Green-Kruskal equilibria. Numerical experiments over a long time. Phys. Fluids 31, 72.Google Scholar
Ghizzo, A., Shoucri, M.M., Bertrand, P., Feix, M. & Fijalkow, E. (1988 b). Nonlinear evolution of the beam-plasma instabilities. Phys. Lett. A 129, 453–458.Google Scholar
Holloway, J.P. & Dorning, J.J. (1991). Undamped plasma waves. Phys. Rev. A 44, 3856.Google Scholar
Hutchinson, I.H. (2017). Electron holes in phase space: what they are and why they matter. Phys. Plasma, 24, 055601/1-13.CrossRefGoogle Scholar
Joyce, G., Knorr, G. & Burns, T. (1971). Nonlinear behavior of the one-dimensional weak beam plasma system. Phys. Fluids 14, 797–801.CrossRefGoogle Scholar
Knorr, G. (1977). Two-dimensional turbulence of electrostatic Vlasov plasmas. Plasma Phys. 19, 529–538.Google Scholar
Kruer, W.L., Dawson, J.M. & Sudan, R.N. (1969). Trapped-particle instability. Phys. Rev. Lett. 23, 838.CrossRefGoogle Scholar
Manfredi, G. (1997). Long-time behavior of non-linear Landau damping. Phys. Rev. Lett. 79, 2815–2818.CrossRefGoogle Scholar
Manfredi, M., Shoucri, M., Shkarofsky, I., Ghizzo, A., Bertrand, P., Fijalkow, E., Feix, M., Karttunen, S., Pattikangas, T. & Salomaa, R. (1996). Collisioless diffusion of particles and current across a magnetic field in a beam-plasma interaction. Fusion Tech. 29, 244–260.Google Scholar
Nakamura, T & Yabe, T. (1999). Cubic interpolated propagation scheme for solving the hyper-dimensional Vlasov-Poisson equation in phase-space. Comput. Phys. Commun. 120, 122–154.Google Scholar
Okuda, H., Horton, R., Ono, M. & Wong, K.L. (1985). Effects of beam plasma instability on current drive via injection of an electron beam into a torus. Phys. Fluids 28, 3365–3379.Google Scholar
Pohn, E., Shoucri, M. & Kamelander, G. (2005). Eulerian Vlasov codes. Commun. Comput. Phys. 166, 81–93.Google Scholar
Schamel, H. (2000). Hole equilibria in Vlasov-Poisson systems: a challenge to wave theories of ideal plasmas. Phys. Plasmas 7, 4831–4844.Google Scholar
Shadwick, B.A. & Morrison, P.J. (1994). On neutral plasma oscillations. Phys. Lett. A 184, 277–282.Google Scholar
Shoucri, M. (1978). Computer simulation of the sideband instability. Phys. Fluids 21, 1359–1365.CrossRefGoogle Scholar
Shoucri, M. (1979). Nonlinear evolution of the bump-on-tail instability. Phys. Fluids 22, 2038–2039.Google Scholar
Shoucri, M. (1980). Destruction of trapping oscillations by sideband instability. Phys. Fluids 23, 2030–2033.Google Scholar
Shoucri, M. (2006). The sidebands instability. J. Plasma 72, 861–864.Google Scholar
Shoucri, M. (2008). Numerical Solution of Hyperbolic Differential Equations. New-York: Nova Science Publishers Inc.Google Scholar
Shoucri, M. (2009). The application of the method of characteristics for the numerical solution of hyperbolic differential equations. In Numerical Simulation Research Progress. (Colombo, S.P. and Rizzo, C.L. Eds.), pp. 1–98. New-York: Nova Science Publishers.Google Scholar
Shoucri, M. (2010). The bump-on-tail instability. In Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas. (Shoucri, M., ed.), pp. 317–358. New York: Nova Science Publishers.Google Scholar
Shoucri, M. (2011). Numerical simulation of the bump-on-tail instability. In Numerical Simulations, Applications, Examples, and Theory. (Angermann, L., ed.), pp. 3–38. Croatia: InTech Publishers.Google Scholar
Shoucri, M. & Storey, L.R.O. (1986). Motion of an electron bunch through a plasma. Phys. Fluids 29, 262.CrossRefGoogle Scholar
Siminos, E., Bénisti, D. & Grémillet, L. (2011). Stability of nonlinear Vlasov-Poisson equilibria through spectral deformation and Fourier-Hermite expansion. Phys. Rev. E 83, 056402/1-13.CrossRefGoogle ScholarPubMed
Sircombe, N.J., Arber, T.D. & Dendy, R.O. (2006). Aspects of electron acoustic wave physics in laser backscatter from plasmas. Plasma Phys. Control. Fusion 48, 1141–1153.Google Scholar
Strozzi, D.J., Williams, E.A., Langdon, A.B. & Bers, A. (2007). Kinetic enhancement of Raman backscatter, and electron acoustic Thomson scatter. Phys. Plasmas 14, 013104/1-13.Google Scholar
Strozzi, D.J., Williams, E.A., Langdon, A.B., Bers, A. & Brunner, S. (2010). Eulerian-Lagrangian kinetic simulations of laser-plasma interactions. In Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas. (Shoucri, M. ed.), p. 89, New-York: Nova Science Publishers.Google Scholar
Umeda, T. (2008). A conservative and non-oscillatory scheme for Vlasov code simulations. Earth Planets Space 60, 773–779.Google Scholar
Umeda, T., Omura, Y., Yoon, P.H., Gaelzer, R. & Matsumoto, H. (2003). Harmonic Langmuir waves. III. Vlasov simulation. Phys. Plasmas 10, 382–391.Google Scholar
Valentini, F., Carbone, V., Veltri, P. & Mangeney, A. (2005). Wave-particle interaction and nonlinear Landau damping in collisionless electron plasmas. Transit. Theory Stat. Phys. 34, 89–101.Google Scholar
Valentini, F., O'Neill, T.M. & Dubin, D.H. (2006). Excitation of nonlinear electron acoustic waves. Phys. Plasmas 13, 052303/1-7.Google Scholar
Valentini, F., Perrone, D., Califano, F, Pegorare, P., Veltri, P., Morrison, P.J. & O'Neil, T.M. (2012). Undamped electrostatic plasma waves. Phys. Plasmas 19, 092103/1-10.Google Scholar
Yin, L., Daughton, W., Albright, B.J., Bezerrides, B., DuBois, D.F., Kindel, J.M. & Vu, H.X. (2006). Nonlinear development of stimulated Raman scattering from electrostatic modes excited by self-consistent non-Maxwellian velocity distribution. Phys. Rev. E 73, 025401/1-4.Google Scholar