Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T14:18:38.431Z Has data issue: false hasContentIssue false
  • English
  • Français

On the nonlinear development of the Langmuit modulational instability

Published online by Cambridge University Press:  13 March 2009

R. Bingham  and
C. N. Lashmore
R. Bingham
Affiliation:
Culham Laboratory, Abingdon, Oxfordshire Euratom/ UKAEA Fusion Association
C. N. Lashmore
Affiliation:
Culham Laboratory, Abingdon, Oxfordshire Euratom/ UKAEA Fusion Association
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the nonlinear development of a long-wavelength finite-amplitude Langmuir wave. The wavenumber k0 of the initial Langmuir wave is chosen such that the three-wave decay is forbidden. We then describe the coupling of the initial Langmuir wave to Stokes and anti-Stokes Langmuir perturbations (with wavenumbers k0 ∓ ks) due to the presence of a low-frequency density perturbation of wavenumber ks. We then show that for a wide range of experimental conditions, the Stokes and anti-Stokes Langmuir waves are generated with wavenumbers well separated from k0. In order to describe the nonlinear evolution of these perturbations and the pump wave we make the static approximation for the ions and describe the high-frequency waves by three distinct wave envelopes. These coupled nonlinear differential equations are then solved exactly for a number of special cases. For the temporal evolution, we obtain periodic solutions and, when damping is included, we find a slow exponential decay of the amplitudes with a corresponding increase in the nonlinear period of oscillation. The stationary spatially varying solutions are shown to include four basic types of behaviour: periodic, solitary wave, phase jump and shock-like profiles. These latter solutions are of interest since they are obtained for zero dissipation and for a coherent wave interaction.

Type
Research Article
Information
Journal of Plasma Physics , Volume 21 , Issue 1 , February 1979 , pp. 51 - 69
Copyright
Copyright © Cambridge University Press 1979

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

Armstrong, J. A., Bloembergen, N., Ducuing, J. & Persham, P. S. 1962 Phys. Rev. 127, 1918.CrossRefGoogle Scholar
Baumgartel, K. & Sauer, K. 1977 J. Plasma Phys. 17, 185.CrossRefGoogle Scholar
Berkhoer, A. L. & Zakharov, V. E. 1970 Soviet Phys. JETP, 31, 486.Google Scholar
Bingham, R. & Lashmore-Davies, C. N. 1976 Nuci. Fusion, 16, 67.CrossRefGoogle Scholar
Bingham, R. & Lashmore-Davies, C. N. 1977 Culham Preprint CLM-P479.Google Scholar
Buchelnikova, N. S. & Matochkin, E. P. 1977 Proceedings of the Thirteenth International Conference on Phenomena in Ionized Gases, Berlin.Google Scholar
Byrd, P. F. & Friedman, M. D. 1971 Handbook of Elliptic Integrals for Engineers and Scientists. Springer-Verlag.CrossRefGoogle Scholar
Inoue, Y. 1975 J. Phys. Soc. Japan, 39, 1092.CrossRefGoogle Scholar
Khakimov, F. KH. & Tsytovich, V. N. 1976 Soviet Phys. JETP, 43, 929.Google Scholar
Lashmore-Davies, C. N. 1975 Nucl. Fusion, 15, 213.CrossRefGoogle Scholar
Morales, G. J. & Lee, Y. C. 1976 Phys. Fluids, 19, 690.CrossRefGoogle Scholar
Nicholson, D. R. & Goldman, M. V. 1976 Phys. Fluids, 19, 1621.CrossRefGoogle Scholar
Nishikawa, K., Lee, Y. C. & Liu, C. S. 1975 Comm. Plasma Phys. 2, 63.Google Scholar
Rudakov, L. I. 1973 Soviet Phys. Doklady, 17, 1166.Google Scholar
Vedenov, A. A. & Rudakov, L. I. 1965 Soviet Phys. Doklady, 9, 1073.Google Scholar
Watanabe, M. & Nishikawa, K. 1976 J. Phys. Soc. Japan, 41, 1029.CrossRefGoogle Scholar
Wong, A. Y. & Quon, B. H. 1975 Phys. Rev. Lett. 34, 1499.CrossRefGoogle Scholar
Zakharov, V. E. 1972 Soy. Phys. JETP, 35, 908.Google Scholar