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Second-order oscillations of a Vlasov–Poisson plasma in Fourier-transformed velocity space

Published online by Cambridge University Press:  13 March 2009

Zdeněk Sedláček  and
Luigi Nocera
Zdeněk Sedláček
Affiliation:
Institute of Plasma Physics, Czechoslovak Academy of Sciences, Pod vodárenskou věží 4, CS-182 11 Prague 8, Czech Republic
Luigi Nocera
Affiliation:
Institute of Plasma Physics, Czechoslovak Academy of Sciences, Pod vodárenskou věží 4, CS-182 11 Prague 8, Czech Republic
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Abstract

We study the Vlasov–Poisson system of equations in Fourier-transformed velocity space. First we reformulate some results of the linear theory: in the new representation the van Kampen–Case eigenmodes are found to be ordinary functions with convenient continuity properties. We give a transparent derivation of the free-streaming temporal echo in terms of the kinematics of wave packets in Fourier-transformed velocity space. We further extend this analysis to include Coulomb interactions, which allows us to establish a connection between the echo theory, the second-order oscillations of Best and the phenomenon of linear side bands. The calculation of the time evolution of the global second-order electric field is performed in detail in the case of a Maxwellian equilibrium distribution function. We conclude that the phenomenon of linear side bands may be properly explained in terms of the intrinsic features of the equilibrium distribution function.

Type
Research Article
Information
Journal of Plasma Physics , Volume 48 , Issue 3 , December 1992 , pp. 367 - 389
Copyright
Copyright © Cambridge University Press 1992

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References

Armstrong, T. P., Harding, R. C., Knorr, G. & Montgomery, D. 1970 Methods in Computational Physics, vol. 9. Plasma Physics (ed. Adler, B., Fernbach, S. & Rotenberg, M.). Academic.Google Scholar
Bernstein, I. B., Trehan, S. K. & Weenink, M. P. H. 1964 Nucl. Fusion 4, 61.Google Scholar
Best, R. W. B. 1966 Proceedings of the 7th International Conference on Phenomena in Ionized Gases, Beograd, vol. II, p. 278.Google Scholar
Best, R. W. B. 1973 Physica 64, 387.Google Scholar
Best, R. W. B. 1974 b FOM Instituut voor Plasmafysica, Utrecht, Internal Report 74/056.Google Scholar
Best, R. W. B. 1974 b Physica 74, 183.CrossRefGoogle Scholar
Bremermann, H. 1965 Distributions, Complex Variables and Fourier Transforms. Addison-Wesley.Google Scholar
Case, K. M. 1959 Ann. Phys. (NY) 7, 349.CrossRefGoogle Scholar
Crawford, J. D. 1991 Modern Mathematical Methods in Transport Theory (Operator Theory: Advances and Applications, vol. 51 (ed. Greenberg, W. & Polewczak, J.), p. 97. Birkhäuser.Google Scholar
Crawford, J. D. & Hislop, P. D. 1989 Ann. Phys. (NY) 189, 26.CrossRefGoogle Scholar
Davidson, R. C. 1972 Methods in Nonlinear Plasma Physics. Academic.Google Scholar
Ecker, G. & Frömling, G. 1974 Z. Naturf. 29a, 1863.CrossRefGoogle Scholar
Friedman, B. 1956 Principles and Techniques of Applied Mathematics. Wiley.Google Scholar
Gautschi, W. 1972 Handbook of Mathematical Functions (ed. Abramowitz, M. & Stegun, I. A.). Dover.Google Scholar
Gnavi, G. & Gratton, F. 1984 IEEE Trans. Plasma Sci. 12, 223.CrossRefGoogle Scholar
Gould, R. W., O'Neil, T. M. & Malmberg, J. H. 1967 Phys. Rev. Lett. 19, 219.Google Scholar
Gros, M., Bertrand, P. & Baumann, G. 1978 J. Plasma Phys. 20, 465–478.CrossRefGoogle Scholar
Hochstrasser, U. W. 1972 Handbook of Mathematical Functions. (ed. Abramowitz, M. & Stegun, I. A.). Dover.Google Scholar
Knorr, G. 1963 Z. Naturf. 13, 1304.Google Scholar
Lambert, A. D. J., Best, R. W. B. & Sluijter, F. W. 1981 Beitr. Plasmaphys. 22 (H2), 101.CrossRefGoogle Scholar
Livshits, M. S. 1973 Operators, Oscillations, Waves. Open Systems. American Mathematical Society.Google Scholar
Malmberg, J. H., Wharton, C. B., Gould, R. W. & O'Neil, T. M. 1968 Phys. Fluids 11, 1147.Google Scholar
Morales, G. J. & Malmberg, J. H. 1974 Phys. Fluids 17, 609.CrossRefGoogle Scholar
Morales, G. J. & O'Neil, T. M. 1972 Phys. Rev. Lett. 28, 417.CrossRefGoogle Scholar
O'Neil, T. M. & Gould, R. W. 1968 Phys. Fluids 11, 134.CrossRefGoogle Scholar
Pavlenko, V. N. & Sitenko, A. G. 1988 Ekhovye Yavlenia v Plazme i Plazmopodobnykh Sredakh. Nauka.Google Scholar
Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T. 1986 Numerical Recipes. Cambridge University Press.Google Scholar
Roos, W. B. 1969 Analytic functions and Distributions in Physics and Engineering. Wiley.Google Scholar
SedlÁČek, Z. & Nocera, L. 1991 Institute of Plasma Physics, Czechoslovak Academy of Sciences, Prague, Report IPPCZ-312.Google Scholar
Smirnov, V. I. 1975 Course de mathematiques supérieures, t. 4, partie 1. MIR.Google Scholar
Trocheris, M. 1980 a J. Math. Phys. 21, 932.Google Scholar
Trocheris, M. 1980 b J. Math. Phys. 21, 941.CrossRefGoogle Scholar
Van Kampen, N. G. & Felderhof, B. U. 1967 Theoretical Methods in Plasma Physics. North Holland.Google Scholar
Wharton, C. B., Malmberg, J. H. & O'Neil, T. M. 1968 Phys. Rev. Lett. 11, 1761.Google Scholar
Ye, H. & Kaufman, A. N. 1988 Phys. Rev. Lett. 60, 1642.Google Scholar