Hostname: page-component-77c89778f8-vpsfw Total loading time: 0 Render date: 2024-07-21T02:33:39.707Z Has data issue: false hasContentIssue false
  • English
  • Français

Electrostatic oscillations in cold inhomogeneous plasma I. Differential equation approach

Published online by Cambridge University Press:  13 March 2009

Z. Sedláček
Z. Sedláček
Affiliation:
Institute of Plasma Physics, Czechoslovak Academy of Sciences, Prague
Get access Rights & Permissions [Opens in a new window]

Extract

Small amplitude electrostatic oscillations in a cold plasma with continuously varying density have been investigated. The problem is the same as that treated by Barston (1964) but instead of his normal-mode analysis we employ the Laplace transform approach to solve the corresponding initial-value problem. We construct the Green function of the differential equation of the problem to show that there are branch-point singularities on the real axis of the complex frequency-plane, which correspond to the singularities of the Barston eigenmodes and which, asymptotically, give rise to non-collective oscillations with position-dependent frequency and damping proportional to negative powers of time. In addition we find an infinity of new singularities (simple poles) of the analytic continuation of the Green function into the lower half of the complex frequency-plane whose position is independent of the spatial co-ordinate so that they represent collective, exponentially damped modes of plasma oscillations. Thus, although there may be no discrete spectrum, in a more general sense a dispersion relation does exist but must be interpreted in the same way as in the case of Landau damping of hot plasma oscillations.

Type
Articles
Information
Journal of Plasma Physics , Volume 5 , Issue 2 , March 1971 , pp. 239 - 263
Copyright
Copyright © Cambridge University Press 1971

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Barston, E. M. 1964 Ann. Phys. (N.Y.) 29, 282.CrossRefGoogle Scholar
Briggs, R. J. & Paik, S. F. 1967 Int. J. Electron. 23, 163.Google Scholar
Case, K. M. 1959 Ann. Phys. (N.Y.) 7, 349.Google Scholar
Case, K. M. 1960 Phys. Fluids 3, 143.CrossRefGoogle Scholar
Case, K. M. 1960 Phys. Fluids 3, 149.CrossRefGoogle Scholar
Sikii, L. A. 1960 Dokiady Akad. Nauk S.S.S.R. 135, 1068.Google Scholar
Dolph, C. L. 1961 Bull. Am. Math. Soc. 67, 1.CrossRefGoogle Scholar
Dolph, C. L. 1963 Ann. Acad. Sci. Fennicae, Series Al, no. 336/9.Google Scholar
Engevik, L. 1966 Department of Applied Mathematics, University of Bergen, Reports nov. 11, 12.Google Scholar
Engevik, L. 1967 Department of Applied Mathematics, University of Bergen, Report no. 13.Google Scholar
Erdélyt, A. & Bateman, H. 1953 Higher Transcendental Functions, vol. 2. New York, Toronto and London: McGraw.HilI Book Co.Google Scholar
Friendman, B. 1956 Principles and Techniques of Applied Mathematics. New York: John Wiley and Sons.Google Scholar
Friendrichs, K. O. 1948 Communs Pure Appl. Math. 1, 361.CrossRefGoogle Scholar
Friendriches, K. O. 1966 Spectral Perturbation Phenomena. In Perturbation Theory and its Applications in Quantum Theory. New York: John Wiley and Sons.Google Scholar
Gakhov, F. D. 1958 Kraevye zadatshi. Moscow: Fizmatgiz.Google Scholar
Hayes, J. 1961 Phys. Fluids 4, 1387.Google Scholar
Kampen, N. G. Van 1955 Physica 21, 949.Google Scholar
Kostin, V. M. & Timofeev, A. V. 1967 Zh. Exp. Teoret. Fiz. 53, 1378.Google Scholar
Landau, L. D. 1946 Zh. Exp. Teoret. Fiz. 10, 25.Google Scholar
Lighthill, M. J. 1964 An Introduction to Fourier Analysis and Generalized Functions. Cambridge University Press.Google Scholar
Rebut, P. H. 1966 Ondes dane lee plasmas inhomogènes en géométrie cylindrique. These. Paris: Facuité des Sciences do 1'Université de Paris.Google Scholar
Rosencrans, S. I. & Sattinger, D. H. 1966 J. Math. Phys. 45, 289.CrossRefGoogle Scholar
Stepanov, K. M. 1965 Zh. Tekh. Fiz. 35, 1002.Google Scholar
Titchmarsh, E. C. 1951 Proc. Roy. Soc. Lond. A 207, 321.Google Scholar