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Higher order approximations in the theory of longitudinal plasma oscillations

Published online by Cambridge University Press:  13 March 2009

F. Einaudi
Affiliation:
Cornell-Sydney University Astronomy Center, Cornell University, Ithaca, New York, U.S.A.
W. I. Axford
Affiliation:
Cornell-Sydney University Astronomy Center, Cornell University, Ithaca, New York, U.S.A.

Abstract

The non-linear behaviour of one-dimensional electrostatic oscillations in a homogeneous, unbounded, collisionless and fully ionized plasma is considered for the case in which a single wave of small, but finite amplitude is excited initially. The Vlasov–Poisson equations are solved using the method of strained co-ordinates in which the independent variable t, the electric field and the distribution function are expanded in the form of asymptotic series, the terms of which are founded by an iterative procedure. An ordering parameter e is introduced, which is proportional to the initial amplitude of the electric field given by linear theory. Differential equations are derived which can be solved sequentially to obtain uniformly valid solutions to all orders in ε. Solutions are given to second order and applied to the case in which the background distribution function is Maxwellian. It is found that the changes in the real and imaginary part of the frequency are small in comparison to the values obtained in the linear theory; that the free-streaming terms decay exponentially in time with a damping rate proportional to ε2, in contrast with the linear theory where they are Un- damped; and that the analysis allows us to calculate the changes in the background distribution function for large time, resulting from particle-wave interactions.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1967

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References

REFERENCE

Al'tshul', L. M. & Karpman, V. I. 1966 JETP 22, 361.Google Scholar
Einaudi, F. 1967 Ph.D. Thesis, Cornell University. Also Cornell-Sydney University Astronomy Center, Rept no 86, 1967.Google Scholar
Faddeeva, V. N. & Terent'ev, N. M. 1954 Tables of the Probability Integral of Complex Argument (ed. Fock, V. A.). Moscow.Google Scholar
Frieman, E. 1963 J. Math. Phys. 4, 410.Google Scholar
Frieman, E. & Rutherford, P. 1964 Ann. Phys. 28, 134.Google Scholar
Jackson, J. D. 1960 J. Nucl. Energy, Part C 1 171.Google Scholar
Kryloff, N. & Bogoliuboff, N. 1947 Introduction to Nonlinear Mechanics. Princeton University Press.Google Scholar
Kuo, Y. H. 1953 J. Math. Phys. 32, 83.Google Scholar
Landau, L. D. 1946 J. Physic, U.S.S.R. 10, 25.Google Scholar
Lighthill, M. J. 1949 Phil. Mag. Ser. 7, 40, 1179.Google Scholar
Montgomery, D. 1961 Phys. Rev. 123, 1077.Google Scholar
Montgomery, D. 1963 Phys. Fluids 6, 1099.CrossRefGoogle Scholar
O'Nell, T. 1965 Phys. Fluids 8, 2255.Google Scholar
Poincaré, H. 1892 Les methodes nouvelles de la méchanique céleste 1, ch. 3, Paris.Google Scholar
Sturrook, P. A. 1957 Proc. Roy. Soc. Lond. A242, 277.Google Scholar
Tsien, H. S. 1956 Advances in Applied Mechanics, vol. 4, p. 281.CrossRefGoogle Scholar
Van Dyke, M. 1964 Perturbation Methods in Fluid Mechanics. Academic Press.Google Scholar
Van Kampen, N. C. 1955 Physica 21, 949.Google Scholar