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Non-linear dispersion of cold plasma waves

Published online by Cambridge University Press:  13 March 2009

Christopher K. W. Tam
Christopher K. W. Tam
Affiliation:
Laboratory for Plasma Physics and Space Sciences, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology
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Abstract

The propagation of a packet of weakly non-linear dispersive cold plasma waves is studied by means of a two-time scale expansion. The effects of amplitude dispersion and the coupling to the mean plasma motion are taken into account. The governing equations are put into a conservation form. It is found that this system of equations is elliptic or hyperbolic depending on the wave-number of the dispersive waves. In the elliptic case modulations in the wave train grow exponentially in time and a periodic wave train will be unstable in this sense. In the hyperbolic case, slow variations in the wave train propagate and the characteristic velocities give a non-linear generalization of the linear group velocity. It is shown that except for waves which have their wave vectors nearly at right angle to the unperturbed magnetic field only fast waves with wave-number less than 0.585 (Ωi Ωe)½/Vα are stable; where Va is the Alfvén velocity and Ωι Ωε, are the ion and electron cyclotron frequencies respectively. A process of steepening of these waves into shocks with dissipation due to wave turbulence at the head of the wave packet is suggested.

Type
Research Article
Information
Journal of Plasma Physics , Volume 4 , Issue 1 , February 1970 , pp. 109 - 125
Copyright
Copyright © Cambridge University Press 1970

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References

REFERENCES

Adlam, J. H. & Allen, J. E. 1958 Phil. Mag. 3, 448.CrossRefGoogle Scholar
Benjamin, T. B. & Lighthill, M. J. 1954 Proc. Roy. Soc. Lond. A 224, 448.Google Scholar
Cordey, J. G. & Saffman, P. G. 1967 J. Plasma Phys. 1, 129.CrossRefGoogle Scholar
Davis, L., Lüst, R. & Schüiter, A. 1958 Z. Naturforsch 13 a, 916.CrossRefGoogle Scholar
Ferraro, V. C. A. 1956 Proc. Roy. Soc. A 233, 310.Google Scholar
Jeffreys, H. & Jeffreys, B. S. 1956 Methods of Mathematical Physics (3rd ed.). Cambridge University Press.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1966 Electrodynamics of Continuous Media. London: Pergamon Press.Google Scholar
Lighthill, M. J. 1965 J. Inst. Math. Appl. 1, 269.CrossRefGoogle Scholar
Luke, J. C. 1966 Proc. Roy. Soc. Lond. A 292, 403.Google Scholar
Montgomery, D. 1959 Phys. Fluids 2, 585.CrossRefGoogle Scholar
Saffman, P. G. 1961 a J. Fluid Mech. 11, 552.CrossRefGoogle Scholar
Saffman, P.G. 1961 b J. Fluid Mech. 11, 16.CrossRefGoogle Scholar
Sagdeev, R. 1966 In Review of Plasma Physics, vol. 4 ed. by Leontovich, M. A.. New York: Consultants Bureau.Google Scholar
Tam, C. K. W. 1969 Phys. Fluids, 12, 1028.CrossRefGoogle Scholar
Whitham, G. B. 1965 a Proc. Roy. Soc. Lond. A 283, 238.Google Scholar
Whitham, G. B. 1965 b J. Fluid Mech. 22, 273.CrossRefGoogle Scholar
Whitham, G. B. 1967 a Proc. Roy. Soc. Lond. A 299, 6.Google Scholar
Whitham, G. B. 1967 b J. Fluid Mech. 27, 399.CrossRefGoogle Scholar
Yeh, T. 1968 Phys. Fluids 11, 2406.CrossRefGoogle Scholar